HIGH ENERGY DENSITY LABORATORY ASTROPHYSICSastro · 2015-11-24 · Astrophys Space Sci (2007) 307:1...

HIGH ENERGY DENSITY LABORATORY ASTROPHYSICS

Edited by:

SERGEY V. LEBEDEV

Imperial College London, United Kingdom

Reprinted from Astrophysics and Space Science

Volume 307, Nos. 1–3, 2007

Library of Congress Cataloging-in-Publication Data is available

ISBN 978-1-4020-6054-0 (hardbook)

ISBN 978-1-4020-6055-7 (eBook)

Published by Springer,

P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Credit cover figure, left: Credit: NASA, ESA, John Krist (STScI <http://www.stsci.edu/>/JPL), Karl Stapelfeldt (JPL),

Jeff Hester (Arizona State Univ.), Chris Burrows (ESA/STScI)

Credit cover figures, right: Imperial College London, Andrea Ciardi (Paris Observatory/Imperial College)

and Sergey Lebedev (Imperial College)

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TABLE OF CONTENTS

Preface 1

Astrophysical jets, high-Mach-number flows, magnetized radiative jets, magnetic reconnection

P. Hartigan / An Introduction to Observations Relevant to Astrophysical Jets and Nebulae 3–6

Eric G. Blackman / Distinguishing Propagation vs. Launch Physics of Astrophysical Jets and the Role of

Experiments 7–10

Yoshiaki Kato / Magnetic-Tower Jet Solution for Launching Astrophysical Jets 11–15

A. Ciardi, S.V. Lebedev, A. Frank, E.G. Blackman, D.J. Ampleford, C.A. Jennings, J.P. Chittenden, T. Lery,

S.N. Bland, S.C. Bott, G.N. Hall, J. Rapley, F.A. Suzuki Vidal and A. Marocchino / 3D MHD Simula-

tions of Laboratory Plasma Jets 17–22

Akira Mizuta, Tatsuya Yamasaki, Shigehiro Nagataki, Shin Mineshige / Outflow Propagation in Collapsars:

Collimated Jets and Expanding Outflows 23–27

David J. Ampleford, Andrea Ciardi, Sergey V. Lebedev, Simon N. Bland, Simon C. Bott, Jeremy P. Chittenden,

Gareth N. Hall, Adam Frank and Eric Blackman / Jet Deflection by a Quasi-Steady-State Side Wind in the

Laboratory 29–34

A. Frank / Hypersonic Swizzle Sticks: Protostellar Turbulence, Outflows and Fossil Outflow Cavities 35–39

U. Shumlak, B.A. Nelson and B. Balick / Plasma Jet Studies via the Flow Z-Pinch 41–45

S. Sublett, J.P. Knauer, I.V. Igumenshchev, A. Frank and D.D. Meyerhofer / Double-Pulse Laser-Driven Jets on

OMEGA 47–50

D.J. Ampleford, S.V. Lebedev, A. Ciardi, S.N. Bland, S.C. Bott, G.N. Hall, N. Naz, C.A. Jennings, M. Sherlock,

J.P. Chittenden, A. Frank and E. Blackman / Laboratory Modeling of Standing Shocks and Radiatively

Cooled Jets with Angular Momentum 51–56

R.F. Coker, B.H. Wilde, J.M. Foster, B.E. Blue, P.A. Rosen, R.J.R. Williams, P. Hartigan, A. Frank and C.A. Back /

Numerical Simulations and Astrophysical Applications of Laboratory Jets at Omega 57–62

Ikkoh Funaki, Hidenori Kojima, Hiroshi Yamakawa, Yoshinori Nakayama and Yukio Shimizu / Laboratory

Experiment of Plasma Flow Around Magnetic Sail 63–68

Markus Bottcher / Astrophysical Jets of Blazars and Microquasars 69–75

Ian J. Parrish and James M. Stone / Simulation of the Magnetothermal Instability 77–82

Lucas F. Wanex and Erik Tendeland / Sheared Flow as a Stabilizing Mechanism in Astrophysical Jets 83–86

Ph. Nicolaı, V.T. Tikhonchuk, A. Kasperczuk, T. Pisarczyk, S. Borodziuk, K. Rohlena and J. Ullschmied / How to

Produce a Plasma Jet Using a Single and Low Energy Laser Beam 87–91

Radu Presura, Stephan Neff and Lucas Wanex / Experimental Design for the Laboratory Simulation of Magnetized

Astrophysical Jets 93–98

V.I. Sotnikov, R. Presura, V.V. Ivanov, T.E. Cowan, J.N. Leboeuf and B.V. Oliver / Excitation of Electromagnetic

Flute Modes in the Process of Interaction of Plasma Flow with Inhomogeneous Magnetic Field 99–101

B. Loupias, E. Falize, M. Koenig, S. Bouquet, N. Ozaki, A. Benuzzi-Mounaix, C. Michaut, M. Rabec le Goahec, W.

Nazarov, C. Courtois, Y. Aglitskiy, A. YA. Faenov and T. Pikuz / Plasma Jet Experiments Using LULI 2000

Laser Facility 103–107

D. Martinez, C. Plechaty and R. Presura / Magnetic Fields for the Laboratory Simulation of Astrophysical Objects 109–114

Supernova remnants, shock processing, radiative shocks

C.C. Kuranz, R.P. Drake, T.L. Donajkowski, K.K. Dannenberg, M. Grosskopf, D.J. Kremer, C. Krauland, D.C.

Marion, H.F. Robey, B.A. Remington, J.F. Hansen, B.E. Blue, J. Knauer, T. Plewa and N. Hearn / Assessing

Mix Layer Amplitude in 3D Decelerating Interface Experiments 115–119

A.B. Reighard and R.P. Drake / The Formation of a Cooling Layer in a Partially Optically Thick Shock 121–125

A.D. Edens, R.G. Adams, P.K. Rambo, I.C. Smith, J.L. Porter and T. Ditmire / Measurement of the Growth of

Perturbations on Blast Waves in a Mixed Gas 127–130

Roland A. Smith, James Lazarus, Matthias Hohenberger, Alastair S. Moore, Joseph S. Robinson, Edward T. Gumbrell

and Mike Dunne / Colliding Blast Waves Driven by the Interaction of a Short-Pulse Laser with a Gas of

Atomic Clusters 131–137

Alastair S. Moore, James Lazarus, Matthias Hohenberger, Joseph S. Robinson, Edward T. Gumbrell, Mike Dunne

and Roland A. Smith / Investigating the Astrophysical Applicability of Radiative and Non-Radiative Blast

wave Structure in Cluster Media 139–145

J.F. Hansen, H.F. Robey, R.I. Klein and A.R. Miles / Mass-Stripping Analysis of an Interstellar Cloud by a Supernova

Shock 147–152

Vikram V. Dwarkadas / Hydrodynamics of Supernova Evolution in the Winds of Massive Stars 153–158

C. Michaut, T. Vinci, L. Boireau, M. Koenig, S. Bouquet, A. Benuzzi-Mounaix, N. Osaki, G. Herpe, E. Falize, B.

Loupias and S. Atzeni / Theoretical and Experimental Studies of Radiative Shocks 159–164

Matthew G. Baring and Errol J. Summerlin / Electrostatic Potentials in Supernova Remnant Shocks 165–168

X. Ribeyre, L. Hallo, V.T. Tikhonchuk, S. Bouquet and J. Sanz / Non-Stationary Rayleigh-Taylor Instabilities in

Pulsar Wind Interaction with a Supernova Shell 169–172

Compact object accretion disks, x-ray photoionized plasmas

D.D. Ryutov, J.O. Kane, A. Mizuta, M.W. Pound and B.A. Remington / Phenomenological Theory of the Photoe-

vaporation Front Instability 173–177

R.J.R. Williams / Photoionized Flows from Magnetized Globules 179–182

Akira Mizuta, Jave O. Kane, Marc W. Pound, Bruce A. Remington, Dmitri D. Ryutov and Hideaki Takabe / Nonlinear

Dynamics of Ionization Fronts in HII Regions 183–186

Marc W. Pound, Jave O. Kane, Dmitri D. Ryutov, Bruce A. Remington and Akira Mizuta / Pillars of Heaven 187–190

T. Sano / The Evolution of Channel Flows in MHD Turbulence Driven by Magnetorotational Instability 191–195

David Alexander / Laboratory Exploration of Solar Energetic Phenomena 197–202

S. Nagataki / Explosion Mechanism of Core-Collapse Supernovae and Collapsars 203–206

Stellar evolution, stellar envelopes, opacities, radiation transport

John I. Castor / Astrophysical Radiation Dynamics: The Prospects for Scaling 207–211

P.A. Rosen, J.M. Foster, M.J. Taylor, P.A. Keiter, C.C. Smith, J.R. Finke, M. Gunderson and T.S. Perry / Experiments

to Study Radiation Transport in Clumpy Media 213–217

Supernovae, gamma-ray bursts, exploding systems, strong shocks, turbulent mixing

J.F. Hansen, M.J. Edwards, D.H. Froula, A.D. Edens, G. Gregori and T. Ditmire / Laboratory Observation of

Secondary Shock Formation Ahead of a Strongly Radiative Blast Wave 219–225

Nathan C. Hearn, Tomasz Plewa, R. Paul Drake and Carolyn Kuranz / FLASH Code Simulations of Rayleigh-Taylor

and Richtmyer-Meshkov Instabilities in Laser-Driven Experiments 227–231

Markus Bottcher and Charles D. Dermer / Models of Very-High-Energy Gamma-Ray Emission from the Jets of

Microquasars: Orbital Modulation 233–236

S. Gupta and M. Bottcher / Time-Dependent Synchrotron and Compton Spectra from Microquasar Jets 237–240

Sergei S. Orlov and Snezhana I. Abarzhi / New Experimental Platform for Studies of Turbulence and Turbulent

Mixing in Accelerating and Rotating Fluids at High Reynolds Numbers 241–244

Mikhail V. Medvedev / Weibel Turbulence in Laboratory Experiments and GRB/SN Shocks 245–250

M. Herrmann and S. I. Abarzhi / Diagnostics of the Non-Linear Richtmyer-Meshkov Instability 251–255

Planetary Interiors, high-pressure EOS, dense plasma atomic physics

M. Koenig, A. Ravasio, A. Benuzzi-Mounaix, B. Loupias, N. Ozaki, M. Borghesi, C. Cecchetti, D. Batani, R.

Dezulian, S. Lepape, P. Patel, H.S. Park, D. Hicks, A. Mckinnon, T. Boehly, A. Schiavi, E. Henry, M. Notley,

R. Clark and S. Bandyopadhyay / Density Measurements of Shock Compressed Matter Using Short Pulse

Laser Diagnostics 257–261

G. Chabrier, D. Saumon and C. Winisdoerffer / Hydrogen and Helium at High Density and Astrophysical

Implications 263–267

Raymond F. Smith, K. Thomas Lorenz, Darwin Ho, Bruce A. Remington, Alex Hamza, John Rogers, Stephen

Pollaine, Seokwoo Jeon, Yun-Suk Nam and J. Kilkenny / Graded-Density Reservoirs for Accessing High

Stress Low Temperature Material States 269–272

S. Mazevet, M. Challacombe, P. M. Kowalski and D. Saumon / He Conductivity in Cool White Dwarf

Atmospheres 273–277

Jonathan J. Fortney / The Structure of Jupiter, Saturn, and Exoplanets: Key Questions for High-Pressure

Experiments 279–283

J. Hawreliak, J. Colvin, J. Eggert, D.H. Kalantar, H.E. Lorenzana, S. Pollaine, K. Rosolankova, B.A. Reming-

ton, J. Stolken and J.S. Wark / Modeling Planetary Interiors in Laser Based Experiments Using Shockless

Compression 285–289

Ultrastrong fields, particle acceleration, collisionless shocks

D.D. Ryutov and B.A. Remington / Scaling Laws for Collisionless Laser–Plasma Interactions of Relevance to

Laboratory Astrophysics 291–296

Matthew G. Baring / Topical Issues for Particle Acceleration Mechanisms in Astrophysical Shocks 297–303

Koichi Noguchi and Edison Liang / Three-Dimensional Particle Acceleration in Electromagnetic Cylinder and

Torus 305–308

Edison Liang / Simulating Poynting Flux Acceleration in the Laboratory with Colliding Laser Pulses 309–313

Koichi Noguchi and Edison Liang / Three-Dimensional Particle Acceleration in Electromagnetic Dominated Outflows

with Background Plasma and Clump 315–318

K.-I. Nishikawa, C.B. Hededal, P.E. Hardee, G.J. Fishman, C. Kouveliotou and Y. Mizuno / 3-D RPIC Simulations

of Relativistic Jets: Particle Acceleration, Magnetic Field Generation, and Emission 319–323

Justin D. Finke and Markus Bottcher / Spectral Features of Photon Bubble Models of Ultraluminous X-ray

Sources 325–327

Felicie Albert, Kim TaPhuoc, Rahul Shah, Frederic Burgy, Jean Philippe Rousseau and Antoine Rousse / Polychro-

matic X-ray Beam from the Acceleration of Energetic Electrons in Ultrafast Laser-Produced Plasmas 329–333

T. Baeva, S. Gordienko and A. Pukhov / Scalable Dynamics of High Energy Relativistic Electrons: Theory, Numerical

Simulations and Experimental Results 335–340

Sebastien Le Pape, Daniel Hey, Pravesh Patel, Andrew Mackinnon, Richard Klein, Bruce Remington, Scott Wilks,

Dmitri Ryutov, Steve Moon and Marc Foord / Proton Radiography of Megagauss Electromagnetic Fields

Generated by the Irradiation of a Solid Target by an Ultraintense Laser Pulse 341–345

Astrophys Space Sci (2007) 307:1

DOI 10.1007/s10509-006-9276-8

Preface

C© Springer Science + Business Media B.V. 2007

The 6th International Conference on High Energy Density

Laboratory Astrophysics was held on March 11–14, 2006

at Rice University in Houston, Texas. This is a continu-

ation of the very successful previous conferences, held in

1996 in Pleasanton, California, in 1998 at the University of

Arizona, in 2000 at Rice University, in 2002 at the University

of Michigan, and in 2004 at the University of Arizona

(organized by the University of Rochester).

During the past decade, research teams around the world

have developed astrophysics-relevant research utilizing high

energy-density facilities such as intense lasers and z-pinches.

Research is underway in many areas, such as compressible

hydrodynamic mixing, strong shock phenomena, radiation

flow, radiative shocks and jets, complex opacities, equa-

tions of state, superstrong magnetic fields, and relativistic

plasmas. Ongoing research is producing exciting results us-

ing the Omega laser at the University of Rochester, the Z

machine at Sandia National Laboratories, and other facilities

worldwide. Future astrophysics-related experiments are now

being planned for the 2 MJ National Ignition Facility (NIF)

laser at Lawrence Livermore National Laboratory, the 2 MJ

Laser Megajoule (LMJ) in Bordeaux, France; petawatt-class

lasers now under construction in several countries, and future

Z pinches.

The conference brought together scientists interested in

this emerging research area with topics including:

Stellar evolution, stellar envelopes, opacities, radiation trans-

port

Planetary Interiors, high-pressure EOS, dense plasma atomic

physics

Supernovae, gamma-ray bursts, exploding systems, strong

shocks, turbulent mixing

Supernova remnants, shock processing, radiative shocks

Astrophysical jets, high-Mach-number flows, magnetized ra-

diative jets, magnetic reconnection

Compact object accretion disks, x-ray photoionized plasmas

Ultrastrong fields, particle acceleration, collisionless shocks

These proceedings cover many of the invited and con-

tributed talks presented at the conference. Of over 100 papers

that were presented at the conference, 62 are included in this

publication.

The conference was organized by:

Edison Liang, Rice University, Houston, TX

Paul Drake, University of Michigan, Ann Arbor, MI

George Kyrala, LANL, Los Alamos, NM

Sergey Lebedev, Imperial College London, UK

Bruce Remington, LLNL, Livermore, CA

Hideaki Takabe, Osaka University, Japan

The organizers would like to thank Umbe Cantu for the

conference administration, as well as the sponsor and endors-

ing organisations:

Rice University, Houston, TX

Los Alamos National Laboratory, Physics Division

Lawrence Livermore National Laboratory, High Energy

Density Program

APS Division of Plasma Physics

APS Topical Group for Plasma Astrophysics

DOE–NNSA

Finally, the editor would like to thank all the authors and

the referees for their contribution, time and effort.

Sergey Lebedev

Guest Editor

London, 2006

Springer

Astrophys Space Sci (2007) 307:3–6

DOI 10.1007/s10509-006-9228-3

O R I G I N A L A R T I C L E

An Introduction to Observations Relevant to Astrophysical Jetsand Nebulae

P. Hartigan

Received: 30 May 2006 / Accepted: 25 July 2006C© Springer Science + Business Media B.V. 2006

Abstract This article reviews the basic physics and jargon

associated with astronomical observations of nebulae, with

an emphasis on processes relevant to shock waves in astro-

physical jets.

Keywords Astronomical observations · Shock waves ·Nebulae

1 Motivation

The HEDLA meetings bring together laboratory experimen-

talists, numerical modeling experts and observational astro-

physicists to study how plasmas and fluids behave in a wide

variety of conditions in nature. While this synthesis presents

unique opportunities for collaborative research, communica-

tion between the different disciplines can be problematic, as

each field has jargon and conventions that are not immedi-

ately transparent to scientists in other fields. As the oldest

science, it is perhaps not surprising that astronomical con-

ventions can be particularly arcane, though with a bit of back-

ground they quickly become second nature.

This contribution provides a brief overview of the physics,

conventions, and nomenclature used when describing and in-

terpreting astronomical images of nebulae, with the objective

to make it easier for a non-specialist to understand an obser-

vational astronomical talk on this subject. Most astronomi-

cal observations are either images or spectra (though some

are both!), and in what follows I treat each of these in turn.

For more information about physical processes that influence

spectra I refer the reader to the classic texts ‘Radiative Pro-

cesses in Astrophysics’ by Rybicki and Lightman (1979) for

P. HartiganDepartment of Physics and Astronomy, Rice University, P.O. Box1892, Houston, Texas 77251-1892, USA

continuum processes, and to ‘Astrophysics of Gaseous Neb-

ulae and Active Galactic Nuclei’ by Osterbrock (1989) for

emission line spectra.

2 Images

Astronomical images can be spectacularly beautiful, but what

they tell us physically about the objects is determined to a

large degree by the answers to two questions: (i) What is the

scale/resolution?, and (ii) What do the colors represent?

2.1 Distance scales and units

Astronomers typically use cgs-Gaussian units, but for conve-

nience we like to represent planetary and solar-system scale

distances in Astronomical Units (1 AU = 1.495 × 1013 cm),

stellar distances in parsecs (1 pc = 3.09 × 1018 cm = 3.26

light years), and masses and luminosities in solar units (1 MO

= 1.99 × 1033 gm, 1 LO = 3.83 × 1033 erg s−1). The AU

is the distance from the Earth to the Sun, and a parsec is

a typical distance between stars in the solar neighborhood.

For reference, the average distance from the Sun to Pluto is

about 40 AU, to the nearest star is 1.3 pc, the nearest region

of massive star formation (Orion) is 460 pc, the center of our

galaxy 8.5 kpc, and the nearest large external galaxy (M31)

0.77 Mpc.

Distances are determined from a variety of methods, but

the most direct one, applicable for the closest objects, is to

observe the angular shift, known as the parallax, of the object

relative to distant background stars as the Earth moves around

the Sun (Fig. 1). A parsec is defined by the distance an object

would have to be in order to have a parallax of 1 arcsecond.

Hence, the number of AU in a pc is 206265, the same as the

number of arcseconds in a radian.

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4 Astrophys Space Sci (2007) 307:3–6

Fig. 1 Definition of a parsec

The theoretical spatial resolution limit of an image is

roughly λ/D, where λ is the wavelength of the light and D the

diameter of the telescope. However, atmospheric turbulence

limits all ground-based optical images to ∼1 arcsecond; ob-

servations in the near-infrared fare somewhat better and rou-

tinely obtain 0.5 arcseconds at a good site. With the use of

adaptive optics to correct for atmospheric distortion one can

approach the theoretical limit of λ/D, but such images have

very small fields of view at present. A good rule of thumb

is that a ground-based image is 1 arcsecond, and a space-

based or radio image is λ/D (e.g. Hubble Space Telescope is

∼0.07 arcseconds).

By combining the above considerations it is straightfor-

ward to quickly infer distance scales for any astronomical

image, provided you know the distance to the source. For ex-

ample, if you see a ground-based image of the Orion Nebula,

then the spatial resolution is ∼1′′, which at 460 pc corre-

sponds to 460 AU, or about 6 times the diameter of our solar

system. An HST image of a large region of ionized gas at

2 kpc has a resolution of 2000 × 0.07 = 140 AU, and so on.

2.2 Colors, magnitudes, photometry, broadband

and narrowband images

Modern multicolor astronomical images are created by load-

ing individual images into red, blue, and green channels, each

image consisting of continuum and emission lines transmit-

ted by the filter. Often the longest wavelength is put into

the red channel, and the shortest into the blue, but one can

also put images of the lowest ionization states in red and the

highest ionization states in blue, or simply choose whichever

channels make the result pleasing aesthetically. Hence, color

composites can provide a great deal of information about the

physics of the region, or none at all, depending on the com-

posite. It is also possible to create a ‘false color’ image where

the intensities of a single image are assigned a specific color.

Such images give no more information than greyscale images

or contour plots do, and in this case the colors are completely

arbitrary. X-ray and radio continuum images often appear in

false color.

The Earth’s atmosphere transmits from about 0.35 µm to

about 1 µm, and this range is typically broken up into five

bandpasses roughly 0.1 µm in width labeled U, B, V, R, and

I for ultraviolet, blue, visual, red, and infrared, respectively.

The Earth’s atmosphere is also transparent in several ‘win-

dows’ throughout the near- and mid-infrared, including the

three near-IR bandpasses J, H, and K at 1.25, 1.65, and 2.2

µm, respectively, and several bandpasses at mid-IR wave-

lengths that range out to 20µm. However, ground-based tele-

scopes emit thermal radiation at mid-IR wavelengths, which

is why a small space telescope like Spitzer, cooled to cryo-

genic temperatures, is much more sensitive than are larger

telescopes on the ground at these wavelengths.

To quantify brightnesses in these broad bandpass filters,

astronomers define a magnitude at each wavelength as mag

= 2.5 log10(F2/F1), where mag is the magnitude difference

between objects with fluxes F1 and F2. The scale is defined

so brighter is smaller, with the Sun having an apparent mag-

nitude of about −26.5 at V, while the stars in the Big Dipper

are about +2. The zero point for each wavelength is approxi-

mately the brightness of the star Vega in the northern summer

sky. Color indices are defined as the difference between two

magnitudes, e.g. B−V, where by convention the bluer filter

is first so that redder objects have more positive colors. For

objects like stars that radiate nearly like blackbodies, bluer

colors mean higher surface temperatures (Fig. 2). In stellar

evolution studies one often plots an ‘H-R diagram’, of ei-

ther magnitude vs. color, or log(L) vs. log(T), where L is the

luminosity (erg/s) and T the surface temperature of the star.

If the object is a nebula that emits primarily line radia-

tion and not continuum, then the apparent brightness simply

depends on how many emission lines fall within the bandpass

Fig. 2 Examples of three optical spectra of stars that have different surface temperatures. Blackbody fits to the spectra appear as dotted curves

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Astrophys Space Sci (2007) 307:3–6 5

of the filter. Typically one uses a narrowband filters to iso-

late specific emission lines of interest. Narrowband images

also minimize the relative brightness of stars, which, being

continuum sources, emit at all wavelengths including those

transmitted by narrow band filters.

3 Spectra

3.1 Emission line spectra, densities,

temperatures, velocities

The most direct way to determine physical conditions within

a nebula is to obtain a spectrum at each point. Many spec-

trographs disperse the light perpendicular to a long slit, and

generate a spectrum at each position along the slit. Mapping

the slit across the entire nebula produces a ‘data cube’ for each

emission line, where the counts at a fixed wavelength produce

an image at the corresponding radial velocity. Adding all the

velocities together gives an emission line image. Instruments

such as image slicers and Fabry-Perots give data cubes, as

do most molecular line maps obtained with radio telescopes.

Data cubes are both images and spectra, and, when combined

with proper motion measurements that show how the object

moves in the plane of the sky, represent the most thorough ob-

servational description of the object possible. Data cubes are

time-consuming to obtain and can be challenging to analyze.

The Doppler shift of emission lines in the spectra relative

to the velocity of the ambient medium (usually the parent

molecular cloud or a protostar in the case of stellar jets)

reveals the kinematics of the gas, and the observed emission

line width clarifies the gas dynamics such as temperature and

turbulence. Because nebulae are optically thin (transparent)

to most photons with energies a few eV, the emission lines

observed at one position are actually integrals over the entire

line of sight through the nebula.

Heating and cooling within nebulae are governed by non-

LTE processes. Nebular densities are very low, so nearly all

the atoms and ions are in their ground states. Typical nebu-

lar densities range 10–106 cm−3 (note that this number must

be multiplied by ∼1.67 × 10−24 to obtain g cm−3 for pure

H composition). Nebular gas cools primarily as free elec-

trons collide with atoms and ions and excite them to upper

states, which then decay back to the ground level. In this

way, kinetic energy of the electrons is converted to light,

which escapes, so the nebula cools. Heating is usually ac-

complished by photoionization or by shock waves. In an ‘H

II’ region, ultraviolet photons from a hot star ionize nebular

gas and deposit any energy in excess of the ionization thresh-

old into kinetic energy of the freed electron. Alternatively,

shock fronts suddenly decelerate gas and convert a fraction

of the bulk kinetic energy into heat.

The kinetic temperature is determined by a balance of

heating and cooling, and for an H II region is typically 104 K.

Temperatures immediately behind shock waves are propor-

tional to the square of the shock velocity, and the temperature

then declines in an extended ‘cooling zone’ as the gas radi-

ates emission lines. Astronomers refer to shocks that cool

by emitting photons as ‘radiative’, a definition which differs

from that used in high energy density physics, where radiative

shocks are those where radiation is an important component

to the total energy content of the gas. In astronomical nomen-

clature, a nonradiative shock is one where the timescales for

cooling are so long that the postshock gas cools by some

other means, such as expansion.

It is usually straightforward to measure the electron den-

sity and temperature in a nebula. Many of the most abundant

ionization states (e.g. O I, O II, O III, N I, N II, C I, S II, S III)

have electronic configurations which have 2, 3, or 4 electrons

in an outer p-shell. As shown in Fig. 3, p3 configurations al-

ways have a close doublet at a few eV above ground, and

transitions from these levels are particularly good for mea-

suring densities. Let us denote level 1 as the ground state,

and levels 2 and 3 as a closely spaced doublet excited state.

Because levels 2 and 3 have nearly identical excitation en-

ergies, their relative density n2/n3 is independent of tem-

perature. However, the flux ratio F21/F31 of doublet lines to

the ground state depends strongly on the density. In the low

density limit, every collisional excitation is followed by ra-

diative decay. In this limit, F21/F31 equals the ratio at which

the two levels are populated from the ground, which is typ-

ically the ratio of statistical weights g2/g3 (ν21/ν31 ∼1 for a

Fig. 3 Left – an energy level diagram for O I, an ion with 4 electrons inan outer p-shell. Line ratios between states highly separated in energy(e.g.λ5577/λ6300) constrain the temperature. Right – the same diagrambut for O II, which has 3 electrons in the p-shell. The emission lineratio of λ3727/λ3729 determines the electron density between the lowdensity limit (∼50 cm−3) and the high density limit (∼2 × 104 cm−3).All the lines depicted in this figure are forbidden transitions (electricquadrupole or magnetic dipole), with wavelengths in Angstroms

Springer


closely-spaced doublet). When the density exceeds some crit-

ical value where collisional deexcitation rates exceed those

for spontaneous decay, n2/n3 approaches the LTE value of

g2/g3, and the flux ratio F21/F31 = (n2A21ν21)/(n3A31ν31) ∼(g2A21)/(g3A31), which differs by a factor of A21/A31 from

the value in the low density limit. Hence, measuring the line

ratio from a closely-spaced doublet gives the electron den-

sity if the density lies between the low density limit and high

density limit for the transition (see Osterbrock 1989, p134,

for examples of diagnostic curves).

In general, an observed flux ratio between two transitions

from an ion or atom defines a curve in Ne–Te space. The

temperature dependence is more pronounced when the levels

are more separated in energy. Line ratios between different

ions determine the ionization fraction, and between different

elements measure the relative abundances. A good historical

example of such an analysis applied to shocks in jets is the

classic paper by Brugel et al. (1981).

In shock waves, the gas gradually becomes more neutral

as it recombines. A hot star that illuminates a dark cloud of

gas and dust sets up a similar stratified ionization structure,

where higher ionization states occur on the side of the nebula

which faces the star, and more neutral species dominate at

greater distances into the cloud where ultraviolet light from

the star becomes attenuated. In photoionized regions there is

a balance between the ionization rate, which is proportional

to aνFνNXi , and the recombination rate αNXi+1Ne. Here, aν

is the photoionization cross section, α is the recombination

rate coefficient, Fν is the ionizing flux, Ne, NXi , and NXi+1

are, respectively, the density of electrons, atoms in ionization

state i, and atoms in ionization state i + 1. Equating these

rates, the ionization parameter NXi+1/NXi is proportional to

aνFν /Ne, which is why lower density nebulae have higher

ionization fractions for a given ionizing flux.

3.2 Continuous spectra

Nebulae also emit continuum radiation from a variety of pro-

cesses. Although less diagnostic than emission lines, contin-

uum radiation also provides information concerning the state

of the plasma responsible for the emission. Continuum radi-

ation is characterized by the spectral index α, where Fν ∼να . For example, α = 2 on the Rayleigh-Jeans portion of a

blackbody spectrum. A spectral index is convenient because

as long as one observes the source at two different frequen-

cies, it is possible to fit a power law through those two points

and obtain a spectral index. Of course, the source is probably

not emitting according to a power law, but one can always

define α = d ln Fν /d ln ν.

Spectral indices for continuum processes change depend-

ing on whether or not the optical depth τ of the source is

thin (≪1 or thick ≫1) For example, optically thick ther-

mal free-free radiation from a plasma is identical to that

from a blackbody, while optically thin free-free has a nearly

flat spectral index (α ∼ −0.1 at optical/IR wavelengths). For

synchrotron radiation, which occurs as electrons spiral rela-

tivistically around magnetic field lines, optically thick emis-

sion at long wavelengths has a spectral index of 5/2, but α

= –(p−1)/2 at short wavelengths, where p represents the en-

ergy spectrum of the electrons, n(E)dE ∼ E−p (cf. Rybicki

and Lightman, 1979).

References

1. Brugel, E., Bohm, K.-H., Mannery, E.: ApJS 47, 117 (1981)2. Osterbrock, D.: Astrophysics of Gaseous Nebulae and Active Galac-

tic Nuclei. University Science Books, Sausalito (1989)3. Rybicki, G., Lightman, A.: Radiative Processes in Astrophysics. Wi-

ley, New York (1979)

Springer


DOI 10.1007/s10509-006-9205-x


Distinguishing Propagation vs. Launch Physics of AstrophysicalJets and the Role of Experiments

Eric G. Blackman

Received: 15 April 2006 / Accepted: 26 June 2006C© Springer Science + Business Media B.V. 2006

Abstract The absence of other viable momentum sources

for collimated flows leads to the likelihood that magnetic

fields play a fundamental role in jet launch and/or collima-

tion in astrophysical jets. To best understand the physics of

jets, it is useful to distinguish between the launch region

where the jet is accelerated and the larger scales where the jet

propagates as a collimated structure. Observations presently

resolve jet propagation, but not the launch region. Simu-

lations typically probe the launch and propagation regions

separately, but not both together. Here, I identify some of the

physics of jet launch vs. propagation and what laboratory jet

experiments to date have probed. Reproducing an astrophysi-

cal jet in the lab is unrealistic, so maximizing the benefit of the

experiments requires clarifying the astrophysical connection.

Keywords Astrophysical jets . Experiments . Magnetic

fields

1. Distinguishing jet launch vs. jet propagation

physics

Jets in astrophysics emanate from accretion disk engines. The

available jet mechanical luminosity is inversely proportional

to the radius from the central engine, so the jet power is

drawn from the inner most regions of the disk. Material must

be accelerated to outflow speeds comparable to the escape

speeds at the launch point. Radiation pressure is typically

incapable of providing the directed momentum and magnetic

launch models are favored (see Livio, 2004; Pudritz, 2004;

Lynden-Bell, 2003).

E. G. BlackmanDepartment of Physics and Astronomy, University of Rochester,Rochester, NY, 14627

Magnetic models take different forms. In steady-state

“fling” models (e.g. Blandford and Payne, 1982), mass flux

is sustained by centrifugal and toroidal magnetic pressure

forces along the poloidal field. Explosive “spring” models

(e.g. Wheeler et al., 2002; Matt et al., 2004, 2006; Moiseenko

et al., 2006) also thrive on a gradient of magnetic field pres-

sure, but are time dependent and do not require an initially

imposed mass flux. Such “springs” may operate in gamma-

ray bursts (GRB) and maybe supernovae. In both spring and

fling models, the launch region is Poynting flux dominated

but on scales ∼<50Rin, (where Rin is the scale of the inner

engine) the jet becomes flow dominated.

Springs and flings can be further distinguished from

magnetic tower Poynting flux dominated outflow models

(Lynden-Bell, 2003; Uzdensky and MacFadyen, 2006); for

the latter, Poynting flux domination remains even in the prop-

agation region (R ∼> 50Rin). Related models have been ap-

plied to GRB and active galactic nuclei (AGN) assuming the

baryon loading is low. In the relativistic jets of AGN, mi-

croquasars, and GRB it is not certain how far in the prop-

agation region the outflow remains PF dominated, In the

non-relativistic jets of young stellar object (YSO), jets are

baryon rich and likely flow dominated outside the launch

region.

Presently, observations do not resolve the launch region

at R ∼< 50Rin for any source, although best indirect evidence

for MHD launch perhaps comes from rotation of YSO jets

∼<100 AU scales (Coffey et al., 2004; Woitas et al., 2005).

That B-fields are important to jet launching (R ∼< 50Rin) is

more widely agreed upon than the role of B-fields in the

asymptotic propagation region (despite the dearth of resolved

observations of the former.) For example, if, by ∼50Rin, a

magnetically collimated supersonic launch accelerates ma-

terial to its asymptotic directed supersonic speed, then the

tangent of the opening angle is just the inverse Mach number

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and the dynamical role of magnetic fields at larger radii may

be inconsequential.

In all standard magnetic jet models, the magnetic field is

dominant in a corona above the rotator, and the magnetic

field has a large scale, at least compared to the scale of the

turbulence in the rotator into which it is anchored. In recent

magnetic tower models (Lynden-Bell, 2003; Uzdensky and

McFadyen, 2006), the tower has both signs of vertical mag-

netic flux since it is composed of loops anchored with both

footpoints in the engine. Traditional MHD models which

start with a large scale dipole field, produce a jet composed

of one sign of magnetic flux and the return flux meanders at

large distances, being dynamically unimportant. Instabilities

in both geometries can disconnect blobs and produce knotty

jets.

In short, the physics of the launch region (not yet resolved

by astrophysical telescopes) involves such issues as: (1) Ori-

gin of magnetic fields, field buoyancy to coronae, magnetic

helicity injection and relaxation into larger coronal struc-

tures, (2) physics of centrifugal+magnetic acceleration of

material from small to super-Alfvenic speeds, or Poynting

flux driven bursts of acceleration, (3) criteria for steady or

bursty jets, and (4) assessment of the extent of Poynting flux

domination.

The physics of the propagation region (resolved by as-

trophysical telescopes) involves such issues as: (1) Propa-

gation, instability formation, and sustenance of collimation

in as a function of internal vs. external density and strength

of magnetic fields, (2) bow shocks, cocoon physics, particle

acceleration, (3) effect of cooling on morphology, and (4)

interaction with ambient media, stars, or cross-winds.

2. Insights on launch from the sun and

a two-stage paradigm for jet fields

Coronal holes and the solar wind provide an analogy to the

more extreme jet launching from accretion disks. The launch

region of the solar wind IS resolved. The coronal magnetic

carpet (e.g. Schrijver and Zwaan, 2000) is composed of large

scale “open” field lines as well as smaller scale “closed”

loops. Both reverse every 11 years, so we know that the field

is not a residue of flux freezing and must be produced by a

dynamo.

There are three types of dynamos in astrophysics (e.g.

Blackman and Ji, 2006): (1) Velocity driven small scale dy-

namos, for which magnetic energy amplification occurs with-

out sustained large scale flux on spatial or temporal scales

larger than the largest scale of the turbulence, (2) veloc-

ity driven large scale dynamos which can amplify field on

scales larger than the largest turbulent scale, and (3) magnet-

ically dominated large scale dynamos, also known as mag-

netic relaxation, whereby an already strong field, adjusts its

geometry and such that any twists migrate to large scales.

Both type (2) and type (3) involve magnetic helicity and an

associated mean turbulent electromotive force aligned with

the local mean magnetic field.

Type 1 and type 2 operate in the interior of a rotator,

but some version of type 2, followed by a type 3 dynamo,

provides the observable coronal field of the sun: First, a ve-

locity driven helical dynamo amplifies fields of large enough

scale that they buoyantly rise to the corona without shredding

from turbulent diffusion. Once in the magnetically dominated

corona, continued footpoint motions twist the field and inject

magnetic helicity. In response, the loops incur instabilities

which open up them or make them rise. Fields that power

jets from disks may arise similarly.

The sun and disk are helicity injecting boundaries to their

magnetically dominated corona, (analogous to spheromak

helicity injection (Bellan, 2000)). The type 2 dynamo occurs

beneath the launch region and type 3 occurs in the launch

region. Neither occurs in the propagation region.

3. Insights on propagation and launch

from jet experiments

Astrophysical jet experiments are in their first incarnation,

and presently involve non-relativistic jet motion. We cannot

expect any experiment to reproduce any astrophysical source,

but rather, address specific physics pieces. To gain insight on

astrophysical problems, a careful assessment of how a given

experiment specifically relates to the Section 1 distinction

between jet formation and propagation is required.

3.1. Insights from coaxial gun helicity injection

experiments: Launch

Hsu and Bellan (2002) employ a coaxial plasma gun com-

posed of two coaxial electrodes linked by an axisymmetric

vacuum magnetic field. This is analogous to an accretion disk

with a dense set of poloidal magnetic loops, axisymmetrically

distributed with zero initial toroidal field. At eight azimuthal

locations, plasma is injected onto to the field lines while an

electric potential is driven across the anchoring electrodes.

An E × B toroidal rotation of the plasma results which then

twists the poloidal field, amplifying a toroidal component.

Equivalently, magnetic helicity is injected along the field.

Once the twist is injected and the toroidal field ampli-

fied, the loops rise and merge on the axis. (This is related

to a type 3 dynamo, defined above.) A twisted unipolar core

tower forms, rises, and remains collimated by hoop stress.

The force free parameter αin j ≡ J · B/B2 = I/ψ (where I

is the current from the imposed voltage across the elec-

trodes and ψ is the initial poloidal magnetic flux) measures

the amount of twist injected. The measurements roughly

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agree with theoretical expectations of the Kruskal-Shafranov

kink instability criterion. For αin j ≤ 4π/L , where L is the

magnetic column length, the collimated structure is sta-

ble. When αin j ∼> 4π/L , the magnetic tower forms exhibits

a kink instability but the structure stays connected. For

αin j ≫ 4π/L , the magnetic tower forms, a kink instability

occurs, and a disconnected magnetic blob forms.

The experiments show that a kink instability need not im-

mediately destroy jet collimation, even when disconnected

blobs are produced. Real jets might be a series of ejected

magnetically blobs, rather than a continuous flow. In astro-

physics, pressure confinement may play an important role of

collimating any magnetic tower.

The experiments probe jet formation in a plasma with

β ∼ 0.02 − 0.1, T ∼ 5 − 20eV , fields of ∼1kG, and num-

ber density n = 1014/cm3. The Alfven Mach number <1 so

this is a launch region experiment not a propagation region

experiment. The value of αin j in real astrophysical system is

determined by shear, resistivity, and coronal density.

3.2. Insights from pulsed wire array experiments:

Launch and propagation

For the pulsed power machine MAGPIE, Lebedev et al.

(2005) set up radial array of tungsten wires arranged like

spokes on a wheel (with a modest concavity) and applied

a radial current. The current ablates the wires and produces

a magnetic field around each. The mean magnetic field has

a net toroidal component in the plane above the array and a

net toroidal field of opposite sign below the array. The J × B

force from the toroidal magnetic pressure accelerates mate-

rial vertically. Toward the axis of the array, plasma is denser

and an initial hydrodynamic precursor jet forms.

As the wires ablate and the magnetic force accelerates the

plasma, a magnetically dominated cavity forms, Hoop stress

collimates the denser plasma along the axis of the tower.

The axis becomes a β ∼ 1 plasma surrounded by β < 1

toroidal magnetic field dominated cavity. Outside of the cav-

ity is a β > 1 ambient plasma supplied from the early stage

of wire ablation. The dimensionless Reynolds and magnetic

Reynolds numbers are larger than unity so the parameters are

crudely OK for MHD an astro-comparison. Cooling is im-

portant as the cooling length is short compared to dynamical

spatial scales. The experiment addresses principles of BOTH

launch AND propagation physics.

The very narrowly collimated β = 1 core jet has an in-

ternal Mach number of ∼4. The surrounding large toroidal

magnetic pressure driven cavity proceeds at Mach 10 with

respect to the weakly magnetized ambient medium while the

radial expansion is only Mach 3 so the cavity is collimated.

In the experiments, these Mach numbers are already reached

even when the tower height is only of order ∼Rin, where Rin

the array diameter (∼4 mm.). That the vertical expansion is

supersonic with respect to the ambient medium implies that

the jet head has evolved from its formation region into its

propagation region.

The structures produced in the experiments are analogous

to pressure confined magnetic tower models (Lynden-Bell

2003; Uzdensky and MacFadyen 2006), however there is

very little polodial field, and the net toroidal field is produced

from poloidal loops oriented perpendicular to the radial di-

rection. A magnetically dominated tower encircling a β = 1

highly collimated core may also apply to astrophysical jets.

At later stages of the evolution, the magnetic tower be-

comes kink unstable and a magnetic blob is ejected. But,

as in Hsu and Bellan (2002), here too the instability does

not destroy the collimation of the tower. In this case, the

ambient thermal pressure slows radial expansion. Blob for-

mation again highlights the importance of time dependent

dynamics, and that disconnected blobs may be the true na-

ture of magnetized jets. Were more material available from

the wires, the blob formation process in the experiment could

repeat.

Though not the main focus of Lebedev et al. (2005), it is

important to emphasize the precursor jet which precedes the

magnetic cavity and results from the initially ablated plasma

from the inner region of the wire array. This jet is hydrody-

namic and collimated by radiative cooling. In fact, the ana-

logue of this precursor jet is a close cousin to the the main

focus of earlier conical wire array experiments of Lebedev

et al. (2002) and Lebedev (2004). In these experiments, the

conical array was more nearly cylindrical (concave at angles

of 30 deg. with respect to the array axis rather than 80 deg.).

Once the current is driven, this lower inclination implies an

increased density on the axis of the jet compared to Lebedev

(2005), thereby increasing cooling enough to break the flux

freezing.

Lebedev et al. (2002, 2004) are thus supersonic hydrody-

namic jet experiments. Given the discussion of Section 1, ex-

periments for which the magnetic field is not important inside

of jet are relevant at most to the propagation region, not the

launch region. The particular hydrodynamic jet experiments

do show that that collimated supersonic launch may obvi-

ate the need for asymptotic magnetic collimation of a given

jet when cooling is important. The collimation is enhanced

when the wire material has a larger ion charge, enhancing ra-

diative losses. This is consistent with a model of asymptotic

protostellar jet collimation discussed in Tenorio-Tagle et al.

(1988).

The Lebedev et al. (2002, 2004) experiments show Mach

number ≥15 jets. Jet deflection and shock propagation are

studied in Lebedev et al. (2004) experiments, where an ad-

ditional cross-wind is introduced into across the propagat-

ing jet flow. Generally, the hydrodynamic cooling-collimated

jets seem to be relatively stable to non-axisymmetric

perturbations.

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3.3. Insights from laser ablation experiments:

Propagation

Another class of hydrodynamic jet experiments have been

performed in laser inertial confinement facilities (Blue et al.,

2005; Foster et al., 2005). These probe aspects of the jet

propagation regime only. The experiments involve laser illu-

mination of a thin metal disk such as titanium or aluminum.

The thin target is placed flush against a washer about 6 times

thicker, often of the same material. The lasers ablate the

thin target disk and the ablated plasma is driven through the

washer hole, exiting the hole in the form of a supersonic jet.

The jet then propagates into a foam. A variety of features

can be studied from the jet propagation into the foam using

X-ray radiography and X-ray back-lighting.

Blue et al. (2005) report on experiments performed at NIF,

They studied aspects of nozzle angle on jet structure by com-

paring axially symmetric (2-D) vs. titled (3-D) nozzles. The

3-D case leads to an earlier transition to turbulence than in

the 2-D case. Code testing of 2-D vs 3-D effects and the

efficacy of the 3-D radiative HD code HYDRA (Marinak

et al., 96) was confirmed, although the Reynolds numbers of

the experiment are R = 107 while only R = 102–103 in the

simulations.

Similar experiments performed by Foster et al. (2005),

on OMEGA obtain Mach numbers as high as 5. The im-

ages are somewhat clearer than in Blue et al. (2005). Tur-

bulent flows, dense plasma jets, bow shock structures are

seen. Modeling was done using 2-D hyrdo simulations with

RAGE (Baltrusaitis et al., 1996). These experiments probe a

jet and foam density ratio of ρ j/ρa ∼ 1. This is intermediate

between YSO jets which have ρ j > ρa vs. AGN jets which

may have ρ j < ρa . The latter however, are relativistic, and

the experiments involve only non-relativistic flows.

References

Baltrusaitis, R.M., Gittings, M.L., Weaver, R.P., Benjamin, R.F.,Budzinski, J.M.: Phys. Fluids 8, 2471 (1996)

Bellan, P.M.: Spheromaks, Imperial College Press, London (2000)Blackman, E.G., Ji, H.: in press, MNRAS, astro-ph/0604221 (2006)Blandford, R.D., Payne, D.G.: MNRAS 199, 883 (1982)Blue B.E., et al.: Physical Review Letters 94, 095005 (2005)Coffey, D., Bacciotti, F., Woitas, J., Ray, T.P., Eisloffel, J.: ApJ 604, 758

(2004)Foster J.M., et al.: ApJL 634, L77 (2005)Hsu, S.C., Bellan, P.M.: MNRAS 334, 257 (2002)Lebedev et al.: ApJ 563, 113 (2002)Lebedev et al.: ApJ 616, 988 (2004)Lebedev et al.: MNRAS (2005)Livio, M.: Baltic Astronomy 13, 273 (2004)Lynden-Bell, D.: MNRAS 341, 1360 (2003)Matt, S., Frank, A., Blackman, E.G.: ASP Conf. Ser. 313: Asymmetrical

Planetary Nebulae III: Winds, Structure and the Thunderbird, 449(2004)

Matt, S., Frank, A., Blackman, E.G.: In press ApJ Lett. (2006)Moiseenko, S.G., Bisnovatyi-Kogan, G.S., Ardeljan, N.V.: Submitted

to MNRAS, astro-ph/0603789 (2006)Pudritz, R.E.: Astrophys. Space Science 292, 471 (2004)Schrijver, C.J., Zwaan, C.: Solar and Stellar Magnetic Activity, Cam-

bridge Univ. Press, Cambridge (2000)Tenorio-Tagle, G., Canto, J., Rozyczka, M.: A & A 202, 256 (1988)Uzdensky, D.A., MacFadyen, A.I.: Submitted to ApJ, astro-ph/0602419

(2006)Wheeler, J.C., Meier, D.L., Wilson, J.R.: ApJ 568, 807 (2002)Woitas, J., Bacciotti, F., Ray, T.P., Marconi, A., Coffey, D., Eisloffel, J.:

A&A, 432, 149 (2005)

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DOI 10.1007/s10509-006-9220-y

O R I G I NA L A RT I C L E

Magnetic-Tower Jet Solution for Launching Astrophysical Jets

The formation of the first jets in the universe

Yoshiaki Kato

Received: 23 April 2006 / Accepted: 17 July 2006C© Springer Science + Business Media B.V. 2006

Abstract In spite of the large number of global three-

dimensional (3-D) magnetohydrodynamic (MHD) simula-

tions of accretion disks and astrophysical jets, which have

been developed since 2000, the launching mechanisms of

jets is somewhat controversial. Previous studies of jets have

concentrated on the effect of the large-scale magnetic fields

permeating accretion disks. However, the existence of such

global magnetic fields is not evident in various astrophys-

ical objects, and their origin is not well understood. Thus,

we study the effect of small-scale magnetic fields confined

within the accretion disk. We review our recent findings on

the formation of jets in dynamo-active accretion disks by us-

ing 3-D MHD simulations. In our simulations, we found the

emergence of accumulated azimuthal magnetic fields from

the inner region of the disk (the so-called magnetic tower) and

also the formation of a jet accelerated by the magnetic pres-

sure of the tower. Our results indicate that the magnetic tower

jet is one of the most promising mechanisms for launching

jets from the magnetized accretion disk in various astrophys-

ical objects. We will discuss the formation of cosmic jets in

the context of the magnetic tower model.

Keywords Accretion . Accretion disks . Black hole

physics . ISM: jets and outflows . MHD . Relativity

PACS: First, Second, More

Y. KatoUniversity of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki305-8577, Japane-mail: [email protected]

1. Introduction

Jets in active galactic nuclei (AGNs) are one of the most

largest single objects in the universe, and also they are

ubiquitous in different systems such as X-ray binaries

(XRBs) and young stellar objects (YSOs). The comprehen-

sive understanding of astrophysical jets is that they are driven

by the gravitational energy of material accreting towards cen-

tral objects, such as stars and compact objects. In fact, many

observations suggest that astrophysical jets are associated

with disks/tori, that may feed inner accretion disks. There-

fore, the launching mechanism of jets strongly depends on

the underlying accretion flows. In this review, we mainly fo-

cus on the development of a theory of accretion disks and jets

after the early 1990s. In the followings, we first remark on

magnetohydrodynamical (MHD) studies of accretion flows

and outflows. We then move on to studies of jets from ac-

cretion disks using global MHD simulations. In Section 2,

we introduce our recent discovery of the formation of mag-

netic tower jets in dynamo-active accretion disks. Finally, in

Section 3, we summarize the study of magnetic tower jets.

Although an alpha-viscosity prescription provides an con-

venient tool for representing a steady structure of the accre-

tion flows, the magnitude of viscosity is not a free parameter

and may not be a constant in space and time. Since magnetic

fields provide a source of disk viscosity, as a consequence

of magneto-rotational instability (MRI; Balbus and Hawley,

1991), we expect that magnetic fields play crucial roles in the

dynamics of accretion flows (see Stone and Pringle, 2001).

That is, the magnetohydrodynamical (MHD) approach is

indispensable. The first global 3-D MHD simulations of

non-radiative accretion flows were performed by Matsumoto

(1999). He calculated the evolution of magnetic fields and

structural changes of a torus which is initially threaded by

toroidal magnetic fields. Hawley (2000), on the other hand,

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calculated the evolution of a magnetized torus which confines

poloidal magnetic fields. After 2000, many global 3-D MHD

disk simulations starting with locally confined fields were

published (e.g., Machida et al., 2000; Hawley and Krolik,

2001; Hawley, 2001; Hawley et al., 2001). All of these global

simulations revealed that MRI maintains turbulent flows in

the magnetized accretion disk and provides spontaneous gen-

eration of subthermal magnetic fields, however, the connec-

tion between the disks and the outflows was not resolved by

these simulations (see Balbus, 2003 for a review).

The acceleration mechanism of the MHD jet has been

studied extensively by many groups. Blandford and Payne

(1982), for the first time, suggested a disk wind driven by in-

teraction between disks and magnetic fields permeating the

disk as the origin of the jets (see also Pudritz and Norman,

1983, 1986; Lovelace et al., 1987). They assumed that the

poloidal magnetic field is much stronger than the toroidal

magnetic field in the surface layer of the disk or in the disk

corona, where plasma-β is low, and jets are accelerated by

a magneto-centrifugal force along the magnetic field line.

In this case, the plasma corotates with the magnetic field

lines until the Alfven point, beyond which toroidal fields

start to dominate and hence collimation begins via the mag-

netic pinch effect. The pioneering simulations of MHD jets

from accretion disks were performed by Uchida and Shibata

(1985; see also Shibata and Uchida, 1986). They calculated

the evolution of a disk threaded with vertical fields extending

to infinity and found the propagation of a torsional Alfven

wave along the magnetic field lines, where the jet was acceler-

ated by a twisted magnetic field (see also Shibata and Uchida,

1985; Meier et al., 2001: they named this process the “sweep-

ing magnetic twist mechanism”). Accordingly, they proposed

another kind of magnetically driven jet, in which the toroidal

magnetic field is dominant everywhere (see also Shibata et

al., 1990; Fukue, 1990; Fukue et al., 1991; Contopoulos,

1995; Kudoh and Shibata, 1995, 1997), where the jets are

accelerated by the magnetic pressure. If this is the case, the

Alfven point is embedded in the disk or there is no Alfven

point, and the Blandford-Payne mechanism cannot be ap-

plied to such jets. Later, many 2-D MHD simulations of jets

driven by large-scale magnetic fields permeating disks were

performed (e.g., Matsumoto et al., 1996; Kudoh et al., 1998;

Casse and Keppens, 2002, 2004). On the contrary to toroidal

field dominated jets, in these simulations, the Alfven point is

far from the disk surface indicating that the jet is primarily

accelerated by the magneto-centrifugal force (see Kudoh et

al., 1998). It has alos been argued that such toroidal field

dominated jets are very unstable to kink instabilities in real

three-dimensional space and cannot exist in actual situations

(e.g., Spruit et al., 1997). In order to study the structure and

the stability of outflows driven by large-scale magnetic fields

beyond the Alfven point, some groups carried out MHD sim-

ulations of outflows from disks treated as boundary condi-

tions (e.g., in 2-D: Todo et al., 1992; Ustyugova et al., 1995;

Ouyed and Pudritz, 1997a,b, 1999; in 3-D: Ouyed et al.,

2003; Ouyed, 2003). In relation to MHD disk simulations,

these simulations were more concerned with the jet structure

driven by vertical magnetic fields, where the disk only plays

a passive role. Since angular momentum can be efficiently

extracted from the surface of the accretion disks by the ver-

tical fields, a surface avalanche produces anomalous mass

accretion in those simulations. Thus, we need to be careful

as to whether or not the launching mechanism of a jet de-

pends on magnetic fields, which are provided externally or

generated internally. This is the first stage in the research of

astrophysical jets.

Previous studies of jets concentrated on the effects of

large-scale magnetic fields permeating accretion disks. One

may ask what the origin of such a large scale field is? Unfor-

tunately, the origin of such a magnetic field is poorly under-

stood (see Kronberg, 1994 and references therein). In addi-

tion, large-scale jet models predict that the direction of the

jets are expected to be aligned with that of the large-scale

magnetic field lines. Recent observations, however, show

that the direction of large-scale magnetic fields are not cor-

related with the direction of the jets in young stellar ob-

jects (Menard and Duchene, 2004). Rather, we expect that

the magnetic fields generated by the disk itself are the most

promising sources of magnetic fields that drive outflows. In

order to study the outflows from the magnetized disk, some

groups carried out 2-D MHD simulations of outflows from

dynamo-active disks treated as boundary conditions (e.g.,

Turner et al., 1999; von Rekowski et al., 2003). On the other

hand, Kudoh et al. (2002) carried out 2-D axisymmetric MHD

simulations of a thick torus involving poloidal magnetic fields

and found a rising magnetic loop, which behaves like a jet,

from the torus. This is the second stage in the research of

astrophysical jets.

Recently, outflows have also appeared in 3-D MHD

simulations of accretion disks. Hawley and Balbus (2002;

hereafter HB02) calculated the evolution of a torus with

initial poloidal fields and found three well-defined dynami-

cal components: a hot, thick, rotationally supported, high-β

Keplerian disk; a surrounding hot, unconfined, low-β coro-

nal envelope; and a magnetically confined unbound high-β

jet along the centrifugal funnel wall (see also Igumenshchev

et al., 2003) These studies are a key to developing the next

stage of magnetic jet models; what we call a magnetic tower

jet. Now, we have entered the third stage in the research of

jets in magnetized accretion flows.

2. Magnetic tower jets

Lynden-Bell and Boily (1994: hereafter LB94) studied the

evolution of force-free magnetic loops anchored to the star

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and the disk. They obtained self-similar solutions for the evo-

lution of magnetic loops. They found that the loop is unstable

against twist injection from rotating disks and that the loop

expands along a direction of 60 degrees from the rotation

axis of the disk (see also Uzdensky et al., 2002a,b; Uzden-

sky, 2002). Lovelace et al. (1995) pointed out that the dipole

magnetic field of the star deforms itself into an open magnetic

field due to the differential rotation between the star and the

disk. Hayashi et al. (1996: hereafter HSM96) carried out, for

the first time, MHD simulations of the magnetic interaction

between a protostar and its surrounding accretion disk. They

discovered an outflow driven by expanding magnetic loops

and a magnetic flare as a result of magnetic reconnection in

the loop. Later, Goodson et al. (1997) carried out similar sim-

ulations and found the density collimation along the rotation

axis of the disk, which looks like a jet. (see also Goodson

et al., 1999; Goodson and Winglee, 1999; Fendt and Elstner,

1999; Keppens and Goedbloed, 2000; Matt et al., 2002). Al-

though they found the expanding magnetic loops, such mag-

netic loops are not collimated. Subsequently, magnetostatic

configuration of collimated magnetic loops (a so-called as

magnetic tower) anchored between the star and the disk were

studied by Lynden-Bell (1996, 2003: hereafter L96, L03, re-

spectively). He showed a solution of a magnetic tower sur-

rounding by external plasma with finite pressure (see also

Li et al., 2001). However, the formation and the evolution

of such a magnetic tower have not been resolved until 2004.

In the followings, we review the published simulations of

magnetic tower jets.

2.1. Formation of a magnetic tower in the

magnetosphere of a neutron star

Kato et al. (2004a: hereafter KHM04) extended HSM96 and

studied the magnetic interaction between a neutron star and

a disk by using 2-D axisymmetric MHD simulations. Initial

models of their study are illustrated in Fig. 1a. They assume

a rotating torus surrounding a weakly magnetized neutron

star with a dipole magnetic field. Outside the torus, they

assume an isothermal, hot, low-density hydrostatic corona.

They found an expansion of the magnetic loops as a result

of the twist injection from the disk, due to the differential

rotation of the disk and the star. The magnetic loop ceases to

splay out when the magnetic pressure balances with the am-

bient gas pressure. Afterwards, the expanding magnetic loop

forms a cylindrical tower of helical magnetic fields whose

height increases with time (Fig. 2a). A key discrimination

from previous simulations is the ambient corona. In previ-

ous MHD simulations of disk-star magnetic interactions, the

magnetic tower structure was not so prominent, because the

ambient gas pressure was too low to confine the magnetic

tower inside the computational box. It is interesting to note

that expanding magnetic loops can also be collimated by

large-scale vertical magnetic fields, if they are associated

with accretion disks (see Matt et al., 2003). Lastly, KHM04

discovered, for the first time, the formation and evolution

of a magnetic tower, which is consistent with that proposed

in L96. Independently, Romanova et al. (2004) also found

the formation of a magnetic tower in a magnetosphere of a

protostar in the propeller regime (see also Romanova et al.,

2005).

2.2. Formation of a magnetic tower in a black hole

accretion flow

Kato et al. (2004b; hereafter KMS04) studied the struc-

ture of non-radiative MHD flows starting with a rotating

torus with initially poloidal localized fields around a non-

spinning black hole by using the pseudo-Newtonian potential

(Paczynski and Wiita, 1980). Initial models of their study are

Fig. 1 Initial models of oursimulations: (a) A rotating torus(light-blue region) is surroundedby a weakly magnetized neutronstar (metallic-gray region). Solidlines indicate a dipole magneticfield threading the torus. (b) Arotating torus (light-blue region)is surrounded by a non-rotatingblack hole. Solid lines indicatesubthermal poloidal magneticfields confined within the torus

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Fig. 2 Snapshots of oursimulations: (a) Formation of amagnetic tower jet in themagnetosphere of a neutron star.(b) Formation of a magnetictower jet in the magnetizedaccretion disk around a blackhole. In both figures, the blueregion, light-blue region, andsolid lines indicate theisovolume of the density of theaccretion disk, the isovolume ofthe Poynting flux, and magneticfield lines, respectively. To seeanimations of our simulations,the interested reader is directedto the link at http://www.ccs.tsukuba.ac.jp/

people/ykato/researches/

illustrated in Fig. 1b. They found the emergence of a magnetic

tower from the magnetized accretion flows, when the bulk of

the torus material reaches the innermost region close to the

central black hole (Fig. 2b). The fields are mostly toroidal

in the rim regions of the jets, whereas poloidal (vertical)

fields dominate in the inner core of the jet. The collimation

width of the magnetic tower depends on the coronal pres-

sure; the more enhanced the coronal pressure is, the more

collimated the jet is. Non-negligible coronal pressure tends

to suppress the emergence of MHD jets. In contrast to mag-

netic towers in the magnetoshere of neutron stars, which are

generated by winding-up a dipole magnetic field, the mag-

netic tower in black hole accretion flows is generated by

inflating toroidal magnetic fields accumulated inside the ac-

cretion disk. Our 3-D magnetic tower solution in black hole

accretion flows is basically the same as LB96 proposed. A

magnetic tower jet in KMS04 is consistent with the toroidal

field dominated jet, since the magnetic tower is made of a

toroidal field generated by dynamo action within the disk.

KMS04 showed, for the first time, that such toroidal field

dominated jets survive at least for a few orbital periods of

the initial torus. The most striking feature of a magnetic

tower jet in KMS04 is the natural emergence of magnetic

fields from the disk, that can accelerate the jets, and hence

a magnetic tower jet is a promising model for launching as-

trophysical jets from accretion disks in various astrophysical

objects.

Independently, De Villiers et al. (2003) carried out 3-D

general relativistic MHD simulations of the magnetized ac-

cretion flows plunging into the spinning black hole and found

the formation of magnetically dominated evacuated region

near the poles where outflows exist (they called it a funnel:

see also McKinney and Gammie, 2004; Hirose et al., 2004).

In contrast to HB02, the funnel is magnetically dominated,

indicating that the funnel is the main product of the emer-

gence of a magnetic tower from the disk (see also De Villiers

et al., 2005; Hawley and Krolik, 2006).

3. Conclusion

In these proceedings, we have briefly reviewed the MHD

study of accretion flows and jets and have discussed recent

progress in the study of magnetic tower jets. We should re-

mark on the definition of a magnetic tower jet, because the

formation process of magnetic towers is different in the mag-

netosphere of a star as compared to that in a dynamo-active

accretion flow. A magnetic tower is generated by a twisted

magnetic loop, supported by an external force, anchored

between differential rotation mediums (see L03), however,

many MHD simulations of magnetized accretion flows indi-

cate that a magnetic tower can also be produced via the emer-

gence of toroidal magnetic fields generated inside a dynamo-

active accretion disk. In other words, magnetic tower jets can

extend more than the scale of pre-existing magnetic fields

that drive the jet. Thus, jets that are accelerated by small-

scale magnetic fields may be appropriate for the definition

of magnetic tower jets. Magnetic tower jets could well be

the first jets formed in the early universe, because the large-

scale structure of strong magnetic fields are yet to develop

in the star forming regions and galaxies at high redshift. Fi-

nally, we expect that magnetic tower jets will give a standard

framework for the next stage in the research of launching

jets.

Acknowledgements The author would like to thank organizers forinviting me to a wonderful meeting. It was a great opportunity for meto present a talk for the laboratory astrophysics community. NumericalComputations were carried out on VPP5000 at the Astronomical DataAnalysis Center, ADAC, of the National Astronomical Observatory(ryk22a).

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DOI 10.1007/s10509-006-9215-8


3D MHD Simulations of Laboratory Plasma Jets

A. Ciardi · S.V. Lebedev · A. Frank · E.G. Blackman ·

D.J. Ampleford · C.A. Jennings · J.P. Chittenden ·

T. Lery · S.N. Bland · S.C. Bott · G.N. Hall · J. Rapley ·

F.A. Suzuki Vidal · A. Marocchino


Abstract Jets and outflows are thought to be an integral part

of accretion phenomena and are associated with a large vari-

ety of objects. In these systems, the interaction of magnetic

fields with an accretion disk and/or a magnetized central ob-

ject is thought to be responsible for the acceleration and colli-

mation of plasma into jets and wider angle flows. In this paper

we present three-dimensional MHD simulations of magnet-

ically driven, radiatively cooled laboratory jets that are pro-

duced on the MAGPIE experimental facility. The general

outflow structure comprises an expanding magnetic cavity

which is collimated by the pressure of an extended plasma

background medium, and a magnetically confined jet which

develops within the magnetic cavity. Although this structure

is intrinsically transient and instabilities in the jet and disrup-

tion of the magnetic cavity ultimately lead to its break-up, a

well collimated, “knotty” jet still emerges from the system;

A. Ciardi ()LUTH, Observatoire de Paris et UMR 8102 du CNRS, 92195Meudon, France

A. Ciardi . S.V. Lebedev . J.P. Chittenden . S.N. Bland . S.C. Bott .

G.N. Hall . J. Rapley . F.A.S. Vidal . A. MarocchinoThe Blackett Laboratory, Imperial College, London, SW7 2BW,UK

A. Frank . E.G. BlackmanDepartment of Physics and Astronomy, University of Rochester,Rochester, NY, USA

A. Frank . E.G. BlackmanLaboratory for Laser Energetics, University of Rochester,Rochester, NY, USA

T. LeryDublin Institute for Advanced Studies, Dublin, Ireland

D.J. Ampleford . C.A. JenningsSandia National Laboratory, Albuquerque, New Mexico, USA

such clumpy morphology is reminiscent of that observed in

many astrophysical jets. The possible introduction in the ex-

periments of angular momentum and axial magnetic field

will also be discussed.

Keywords MHD plasmas . Accretion . Accretion discs .

Laboratory astrophysics . Winds . Jets and outflows

Introduction

The formation and collimation of jets is a problem of great

interest in astrophysics. Jets are observed in a diversity of

often unrelated systems and range from the sub-parsec and

parsec scale in the case of young stellar object jets (Reipurth

and Bally, 2001) to the galactic scale jets, thought to be

powered by super-massive black holes present in the cen-

tre of active galactic nuclei (see Begelman et al., 1984). Jets

may also play a critical role in the formation of gamma-ray

bursts (see Piran, 2005 for a review) and by association su-

pernovae (Galama et al., 1998; Stanek et al., 2003; LeBlanc

and Wilson, 1970; Khokhlov et al., 1999; MacFadyen and

Woosley, 1999; Wheeler et al., 2002; Akiyama et al., 2003).

Such diversity, in otherwise similar outflow structures may

suggest the presence of a universal formation mechanism

and over the last twenty years, magnetic fields and the oc-

currence of rotation have been identified as the principal

agents for creating collimated outflows. A common feature

of the many variations of the magneto-rotational scenario

(Blandford and Payne, 1982; Pudritz and Norman, 1986; Pel-

letier and Pudritz, 1992; Wardle and Koenigl, 1993; Shu et

al., 1994; Ustyugova et al., 1999, 2000; Uchida and Shibata,

1985; Contopoulos and Lovelace, 1994; Ouyed et al., 1997;

Goodson et al., 1999; Goodson and Winglee, 1999; Kudoh

et al., 2002), is that a magnetic field can extract the rotational

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energy and launch the plasma from a gravitational potential

well to escape velocities. In many of the models, the winding

of an initially poloidal magnetic field results in a flow pattern

dominated by a toroidal magnetic field. In this context the

interaction of a dominant toroidal magnetic field with ther-

mal ambient plasma has been investigated in the laboratory

both experimentally and numerically (Lebedev et al., 2005b,

Ciardi et al., 2005). One of the aims of the present work is

to extend the numerical work to 3D MHD simulations and

to investigate the late stages of the evolution of “magnetic

tower” in the laboratory.

Laboratory experiments, performed on a variety of high

energy density facilities, are starting to address some impor-

tant astrophysical issue (Remington, 2005).

Conical wire arrays have been successfully used to pro-

duce radiatively cooled, hypersonic jets (Ciardi et al., 2002;

Lebedev et al., 2002) and to study their interaction with an

ambient medium (Lebedev et al., 2004; Ampleford et al.,

2005; Lebedev et al., 2005a). In these experiments a conical

cage of micron-sized metallic wires was driven on the MAG-

PIE pulsed-power facility, which delivers a current ∼1 MA

over ∼240 ns. The jet formation mechanism relies on the

combination of a high rate of radiative cooling together with

the redirection of flow across a conical shock. Magnetic fields

were not important in these jets and the formation process

was purely hydrodynamic. The current experiments are mod-

ified in order to introduce a dynamically significant magnetic

field in the system.

Numerical experiments

A radial wire array consists of two concentric electrodes con-

nected radially by 16 tungsten wires 13µm in diameter and

with the radius of the inner and outer electrodes 2 mm and

20 mm respectively. Figure 1 shows the initial set-up of a

three-dimensional simulation of a radial wire array. Mag-

netic field lines are also shown and the regions where the

toroidal global field dominates over the local field of the

wires can be clearly distinguished. Due to limitations in re-

solving the micron-sized wire cores in these large scale sim-

ulations, the wires are initiated as relatively cold dense gas

and not as solid metallic wires. Nevertheless, these artificial

initial conditions reproduce correctly the ablation rate of the

wires (Lebedev et al., 2001) and the rapid formation of a hot

coronal plasma surrounding the wires. The electrodes are

treated in the computations as highly conductive but ther-

mally insulated regions. The code solves on an Eulerian grid

the three-dimensional single fluid, two temperatures and re-

sistive MHD equations. The evolution of the electromagnetic

fields is followed through an explicit Runge-Kutta type time-

integration solver and corrected transport. The LTE ioniza-

tion is calculated using a Thomas-Fermi average atom model,

Fig. 1 A radial wire array consists of thin metallic wires connectingtwo concentric electrodes. Current flows along the wires and into thecentral electrode. The J×BGlobal force acting on the plasma ablatedfrom the wires is accelerated in the axial direction. The “global” mag-netic field, which dominates the system, is purely toroidal. The wires’“private” field is also plotted for some of the wires.

and we also include optically thin radiation as a loss term in

the electrons energy equation; the latter is coupled to the en-

ergy equation for the ions through an energy equilibration

term; more details on code are in (Chittenden et al., 2004).

A typical simulation of a radial wire array is shown in Fig-

ure 2. The ablation of the wires initially produces an ambient

plasma cloud which expands above the plane containing the

wires. This thermally dominated plasma provides the colli-

mating environment for the magnetic cavity. When sections

of the wire cores are fully ablated the proper magnetic tower

jet begins to form, consisting of a magnetic cavity with a

jet on its axis. Axial expansion of the cavity and instabili-

ties disrupt the system, leaving a clumpy and collimated jet

behind.

The formation of ambient plasma is due to the steady ab-

lation of the wires which produces hot plasma (∼10 eV) of

relatively low resistivity (η) with respect to the cold (∼1 eV)

wire cores. For a Spitzer like resistivity η ∼ T −3/2, where T

is the temperature of the plasma, a marked difference in the

resistivity develops in this two-component structure, with

currents preferentially flowing in the ablated plasma. The

global magnetic field (see Figure 1) accelerates the ablated

plasma in the axial direction, while the wire cores, which are

virtually force-free, act as a continuous but stationary source

of plasma. We note that resistive diffusion dominates over the

advection of the magnetic field up to a height of lR ∼2–3 mm

above the wires, this is approximately the length scale over

which the ablated plasma is magnetically accelerated to

characteristic velocities of vabl ∼130 km s−1. Close to the

wires the electron temperature is a few eV and the magnetic

Reynolds number ReM = vlR/DM ∼ 0.1; DM = η/µ0 is the

magnetic diffusivity and we used v ∼ 30 km s−1. At axial

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Fig. 2 Time evolution of a radial wire arrays composed of 16 tungsten wires 7.5µm in diameter. Mass density x-z slices from a 3D simulation areshown. The hatched areas indicate the electrodes. Regions with a density below 10−4 kg m−3 (white) are treated as vacuum.

positions above lR the magnetic Reynolds number increases

as a consequence of increasing plasma velocity, temperature

and diffusion length scales. Nevertheless, the magnetic field

rapidly decays above the wires to ∼10% its value calculated

in the vacuum region below the wires. The ambient plasma

has β > 1 and the thermal pressure will act to confine the

magnetic cavity that forms later. Near the wires the magnetic

field pressure dominates and the plasma β < 1. Over the

ablation time (tabl ∼ 250 ns) a region of height ∼30–40 mm

above the plane of the wires is filled with plasma, its density

varies as ∼1/r where r is radial distance from the array’s axis.

The axially peaked plasma distribution (ne ∼1018 cm−3 on

axis) occurs as the shock heated radially converging plasma

is cooled by radiation losses, resulting in a plasma “column”

that is hydrodynamically confined. In the axial direction, be-

cause of the time dependent ablation rate (∼I2) the density

decreases rapidly away from the plane of the wires. Because

of the discrete nature of the wires, the plasma distribution is

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Fig. 3 Four iso-densitycontours are shown at the sametime (235 ns). The densities are,from left to right, 5 × 10−4,5 × 10−3, 5 × 10−2,5 × 10−1 kg m−3, and can bethought of as being nested. Thebackground plasma is visible inthe leftmost panel, while in thetwo mid panels, the welldeveloped magnetic cavity canbe seen. The rightmost panelshows the jet that forms insidethe magnetic cavity.

Fig. 4 Magnetic field (yellow)and current density (red)distribution inside the magneticcavity at 225 ns (left) and 245 ns(right). To show the inside of themagnetic cavity the iso-densitycontours (same as in Figure 3)are sliced vertically.

highly modulated in the azimuthal direction (see Figure 3).

Nevertheless the evolution of the magnetic cavity is highly

symmetric and it is only at later times, as instabilities develop,

that asymmetric features become apparent.

Because the mass ablation rate decreases with the strength

of the global magnetic field as ∼1/r (Lebedev et al., 2001),

the highest ablation rate occurs in the proximity of the in-

ner electrode. It is there that during the current discharge

full ablation of millimetre-sized sections of the wire cores

takes place. Because of the disappearance of the force-free

wire cores and thus of the plasma source, the magnetic field

pressure associated with the global toroidal magnetic field is

now able to sweep the remaining plasma upwards and side-

ways. The magnetic field acts as a piston, snowploughing the

surrounding plasma and forming a magnetic “bubble” inside

the background plasma (Figure 3).

In Figure 4 the magnetic field (yellow lines) and the

current density (red lines) distribution inside the magnetic

cavity are shown for two distinct times. Similarly to the

experiments, astrophysical magnetic tower jets are domi-

nated by a toroidal magnetic field which is confined by the

pressure of an ambient plasma (Lynden-Bell, 1996, 2003;

Kato et al., 2004a, b). With the appearance of the magnetic

cavity, a current-carrying jet forms on axis and it is confined

by the magnetic field hoop stress. The characteristic den-

sity and temperature in the jet are ni ∼ 3 × 1019 cm−3 and

T ∼ 30 eV respectively. The characteristic velocity of the jet

is ∼150–200 km s−1, which is higher than the initial flow ve-

locity present before the appearance of the magnetic cavity

and indicating that the plasma is actually accelerated in the

jet formation process. Initially the plasma beta of the jet is

∼1 and the magnetic Reynolds number is ∼5–10. With the

exception of the jet, the magnetic cavity is mostly void of any

plasma. The principal current path is thus along the walls of

the magnetic cavity and through the jet (see Figure 4). As

noted above, in the jet itself, acceleration of material oc-

curs as plasma swept by the converging magnetic piston,

is compressed and redirected axially. In the simulation of

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tungsten arrays, radiation losses are such that the plasma

shell surrounding the magnetic cavity remains fairly thin.

Variation of the ambient plasma distribution and the driv-

ing magnetic field strength can significantly alter the growth

rate of the magnetic tower. The characteristic axial expansion

velocity of the magnetic tower is ∼200–400 km/s, while its

radial expansion occurs with a velocity of about 50 km/s.

The higher velocities are observed for arrays made with the

7.5µm wires, where the magnetic tower forms during the

rise of the current pulse and propagates in an environment

having a smaller axial extent. A dominant kink (m = 1) mode

instability develops immediately after the jet formation and

leads to its break-up. For typical jet parameter the growth

time (∼2.5 ns) of the instability is significantly smaller than

the evolution time (Lebedev et al., 2005b). Nevertheless the

combination of the axial expansion of the magnetic tower

and instabilities do not lead to the destruction of the jet; in-

stead a collimated, clumpy jet is launched out of the cavity.

During this transient phase the current and field distribution

change significantly (Figure 4): the current begins to con-

nect once again at the base of the magnetic cavity while the

magnetic field develops a significant axial component and

becomes highly tangled, thus promoting reconnection. Fi-

nally, the radiatively cooled, “knotty” jet emerging from the

cavity has typical velocities of 200–300 km s−1, Mach num-

bers of >10, plasma beta ∼1–10 and ReM ∼1–5. Because

of the high Mach number the jet will remain collimated over

long distances. In addition, the clumps that form the jet have

generally different axial velocity and will interact with each

through a series of internal shocks, reminiscent of the internal

shocks observed in proto-stellar jets (Hartigan et al., 2001).

The rapid development of instabilities in the jet may be

partly suppressed by the presence of a poloidal field in the jet

and we are currently developing a series of experiments to

investigate its effects. A typical radial array set-up involves

the presence of a solenoid-like electrode below the plane of

the wires which introduces a longitudinal magnetic field of

the order of ∼15% of the toroidal field. Field compression,

resistive diffusion in the plasma and electrode geometry can

all influence the actual topology of the field prior to the jet for-

mation; also the presence of an axial field introduces angular

momentum in the flow, further complicating the analysis. Al-

though the exact role of such effects has not yet been clarified,

preliminary numerical and experimental results indicate that

the inclusion of axial fields and angular momentum can have

a major effect on the overall evolution of magnetic towers and

on the jet collimation. These results open up the prospect of

significantly extending the range of jet studies that can be

performed in the laboratory.

Acknowledgements The present work was supported in part by theEuropean Community’s Marie Curie Actions – Human Resource andMobility within the JETSET network under contract MRTN-CT-2004

005592. The authors also wish to acknowledge the SFI/HEA IrishCentre for High-End Computing (ICHEC) and the London e-ScienceCentre (LESC) for the provision of computational facilities andsupport.

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DOI 10.1007/s10509-006-9256-z


Outflow Propagation in Collapsars: Collimated Jets andExpanding Outflows

Akira Mizuta · Tatsuya Yamasaki ·

Shigehiro Nagataki · Shin Mineshige

Received: 24 April 2006 / Accepted: 20 September 2006C© Springer Science + Business Media B.V. 2006

Abstract We investigate the outflow propagation in the col-

lapsar in the context of gamma-ray bursts (GRBs) with 2D

relativistic hydrodynamic simulations. We vary the specific

internal energy and bulk Lorentz factor of the injected out-

flow from non-relativistic regime to relativistic one, fixing

the power of the outflow to be 1051erg s−1. We observed

the collimated outflow, when the Lorentz factor of the in-

jected outflow is roughly greater than 2. To the contrary,

when the velocity of the injected outflow is slower, the ex-

panding outflow is observed. The transition from collimated

jet to expanding outflow continuously occurs by decreasing

the injected velocity. Different features of the dynamics of

the outflows would cause the difference between the GRBs

and similar phenomena, such as, X-ray flashes.

Keywords Hydrodynamics . Jet . GRBs . Supernovae .

Shock . Relativity

1 Introduction

The gamma-ray bursts (GRBs) are the most energetic phe-

nomena in the sky. A collimated and relativistic jet is

A. Mizuta ()Max-Planck-Institute fur Astrophysik, Karl-Schwarzschild-Str. 1,85741 Garching, Germanye-mail: [email protected]

T. Yamasaki · S. MineshigeYukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

S. NagatakiYukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan;KIPAC, Stanford University, P.O.Box 20450, MS 29, Stanford,CA, 94309, USA

necessary to explain the observational features of GRBs

(Piran, 2000). The central engine of the GRBs is not

fully understood yet. However, recent observations of the

long duration GBRs associated with SNe, for example,

GRB980425/SN1998bw (Galama et al., 1998; Iwamoto

et al., 1998) and GRB030329/SN2003dh (Hjorth et al., 2003;

Price et al., 2003; Stanek et al., 2003) link the GRBs and the

death of massive stars. Note, both SN1998bw and SN2003dh

are categorized to a sub-class of the SNe, such as, hypernovae

whose explosion energy is ∼1052 ergs which is one order

magnitude higher than that of normal supernova explosion.

The similar phenomena called as X-ray flashes (XRFs) are

also observed (Heise et al., 2001). XRFs have larger fluence

in the X-ray band than in the gamma-ray. Since the event

rate of XRFs is similar to GRBs, several hypotheses are pro-

posed to link these events. Nakamura (2000) proposed an

unified model that explains the different properties of GRBs

and XRFs by the different viewing angle of the collimated

outflow. Lamb et al. (2005) proposed a model that explains

the different properties by the different opening angle of the

outflow.

Theoretically the relation between the death of the mas-

sive stars and GRBs was predicted by Woosley (1993). That

is so called collapsar model. When an iron core of a rapidly

rotating massive star collapses, a proto neutron star or black

hole is formed in the center of the progenitor. Though the

gas along the rotational axis can freefall quickly, the gas

along the equatorial plane gradually falls into the center be-

cause of the large centrifugal force. As a result an accretion

disk is formed. MacFadyen and Woosley (1999) performed

hydrodynamic simulations of this model. They deposit ther-

mal energy in the polar region around the core, assuming

neutrino emission from the accretion disk, and neutrino and

anti-neutrino annihilation there. Then the gas expands and

forms an bipolar flow. Since the calculation was Newtonian

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one, the relativistic effects which are important for GRBs

are not included. Aloy et al. (2000) did relativistic hydro-

dynamic simulations of the same type of problems done by

MacFadyen and Woosley (1999). They showed collimated

and relativistic jet along the polar axis of the progenitor. The

jet finally breaks out from the progenitor, making a highly

Lorentz factor component up to Ŵ ∼ 40.

Another type of relativistic hydrodynamic simulations

have also done by Zhang et al. (2003, 2004), and Umeda

et al. (2005). They inject not only a thermal energy but also

a kinetic energy from the computational boundary, assum-

ing an outflow formation around the center of the progenitor.

They followed the outflow propagation in the progenitor and

interstellar medium. All their model were initially outflows

with a large thermal energy and showed successful eruption

from the progenitor, i.e., relativistic jets. But there still re-

main some issues on the propagation of the outflows in the

progenitor. Which type of the outflow can keep the collimated

structure and how do they keep the good collimation. In this

paper, we show the different types of outflows in the collapsar.

2 Model

We study the outflow propagation in the progenitor, assum-

ing an outflow formation after the core collapse. The ra-

dial mass profile of the progenitor developed by Hashimoto

(1995) is used. The progenitor had a mass of about 40 solar

masses in the main sequence and has 16 solar masses in the

pre-supernovae stage. The radius of the progenitor is 3.7 ×1010 cm. We use non-uniform grid points, assuming the ax-

isymmetric geometry (r − z). Logarithmically uniform 500

grid points are spaced for 2 × 108cm < z < 6.6 × 1010cm.

We also set uniform 120 zones for 0 < r < 1.2 × 109 cm

and logarithmically uniform 130 zones for 1.2 × 109 < r <

1.1 × 1010 cm. The inner boundary of the computational box

is located at the distance of 2 × 108 cm from the center of the

progenitor. In this study the origin of the coordinate corre-

sponds to the center of the progenitor. The boundary condi-

tions at the cylindrical axis (r = 0) and z = 2 × 108 cm, are

reflective one except 0 < r < 7 × 107 cm at z = 2 × 108 cm

where an outflow is injected. The boundary conditions at

other boundaries are outflow boundary condition. The mass

densities of the progenitor is ∼106g cm−3 (around the inner

boundary), ∼1g cm−3 (at the surface of the progenitor), and

10−6g cm−3 (constant outside of the progenitor).

The 2D special relativistic hydrodynamic equations are

solved, using our relativistic hydrodynamic code based on

Godunov-type scheme (Mizuta et al., 2004, 2006). An ideal

equation of state p = (γ − 1)ρǫ is also solved to close the

equations, where p is pressure, the constant γ (=4/3) is spe-

cific heat ratio, ρ is rest mass density, and ǫ is specific inter-

nal energy. As our current numerical code can handle only

constant specific heat ratio, we take precedence the state for

the relativistic temperature γ (=4/3) in this paper. Since the

timescale for the outflows to cross the progenitor is much

shorter than that of the freefall of the envelopes, we ignore

the gravitational potential by the formed black hole or proto

neutron star at the center of the progenitor. The initial gas

temperature of the envelope and outside of the surface is set

to be very low (ǫ/c2 = 10−9 and ǫ/c2 = 10−6).

We assume an outflow formation from the center of the

progenitor. It is also assumed that the outflow is parallel to

the cylindrical axis. We inject this outflow from the bound-

ary described above. Four parameters are necessary to define

the outflow condition. In this paper, we fixed two of them.

The first one is the power of the outflow which is fixed to

be 1051ergs s−1. The total energy by ten seconds injection

satisfies 1052 erg which is the energy of the hypernova ex-

plosion. The second one is the radius of the injected outflow

which is fixed to be 7 × 107 cm. We vary other two param-

eters, such as, the specific internal energy ǫ0 and the bulk

Lorentz factor Ŵ0, where subscripts ‘0’ stand for the values

of the injected outflows from the computational boundary.

The bulk Lorentz factor is varied from Ŵ0 = 1.05 to Ŵ0 = 5,

corresponding 3-velocity is from v0 = 0.3c to v0 = 0.98c,

where c is speed of light. The specific internal energy is

varied from ǫ0/c2 = 0.1 to ǫ0/c2 = 30. The outflow of the

model (Ŵ0, ǫ0) = (30, 5) is similar to the models used by

Zhang et al. (2003, 2004) and Umeda et al. (2005). This is the

most attractive model for GRBs, since the outflow contains

a large amount of thermal energy. Such an outflow could be

formed in the quickly rotating progenitor. The outflow of the

model (Ŵ0, ǫ0/c2) = (1.05, 0.1) is the most slowest and cold-

est one. The mass density of the injected outflow in model

(Ŵ0, ǫ0/c2) = (1.05, 0.1) is ∼104g cm−3 and the highest one

in all models. To the contrary, the mass density of the injected

outflow in model (Ŵ0, ǫ0/c2) = (5, 30) is ∼1g cm−3 and the

lowest one in all models. As the mass density of the progen-

itor (before an outflow comes) around the injection point is

106g cm−3, the outflows in the all models are so-called “light

jet” whose mass density is lower than that of the ambient gas.

Thus we can expect strong interaction between the outflow

and progenitor gas.

3 Results and discussions

Figure 1 shows the density (top) and Lorentz (bottom) con-

tours of two models [left panel : (Ŵ0, ǫ0/c2) = (5, 30) and

right panel : (Ŵ0, ǫ0/c2) = (1.05, 0.1)], when the outflow

breaks out from the progenitor surface. The outflow of the for-

mer model keeps good collimation in the progenitor, since the

high Lorentz factor is localized along the cylindrical axis. To

the contrary, the outflow of the latter model shows expanding

feature. In both cases, the bow shock which drives progenitor

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bow shock bow shock

reverse shockreverse shock

(Γ0,ε0/c2)=(5,30) (Γ0,ε0/c

2)=(1.05,0.1)

Fig. 1 The contours of rest mass density (top) and Lorentz factor(bottom) of two modes. Left panel shows collimated jet at t = 3.5s for case [(Ŵ0, ǫ0/c2) = (5, 30)]. Right panel shows expanding out-

flow at t = 10 s for case [(Ŵ0, ǫ0/c2) = (1.05, 0.1)] Figures are takenfrom Mizuta et al. (2006) and reproduced by permission of theAAS.

Fig. 2 The results of a series of calculations in which ǫ0/c2 is fixedto be 5. Models (Ŵ0, ǫ) = (5, 5), (4, 5), (3, 5), (2, 5), (1.4, 5), (1.25, 5)and (1.15, 5) are shown The contours of the rest mass density and

Lorentz factor in each models are presented as same as in Fig. 1. Figuresare taken from Mizuta et al. (2006) and reproduced by permission ofthe AAS.

Springer


gas to high pressure and temperature can be seen. Since the

bow shock is enough strong, the pressure driven by the bow

shock can keep the outflow to be collimated structure in case

of the collimated jet. The reverses shocks also appear in both

models. In case of the collimated jet this shock is close to the

bow shock, and located at the point where the bulk Lorentz

factor decreases to unity. To the contrary, in case of expand-

ing outflow, the distance between the bow shock and reverse

shock increases as time goes on.

A back flow which is an anti-parallel flow to the main jet

is observed in case of the collimated jet. This back flow be-

gins from the shock heated gas through the reverses shock.

Internal oblique shocks appear in the collimated jet which

helps the jets to keep the collimated structure during the

propagation in the progenitor (Norman, 1982; Falle, 1991;

Leahy, 1991). There are two possibilities to appear such in-

ternal structures. The first is the dynamical nonlinear effect of

Kelvin-Helmholtz instability which occurs at the boundary

of the jet and the back flow. The second is the shear flow insta-

bility which occurs in the jet itself (Urpin, 2002). We need to

do higher resolution calculations to identify the reason of the

internal structures. No back flow is observed in case of the ex-

panding outflow. In model (Ŵ0, ǫ0/c2) = (5, 30), the Lorentz

factor increases up to 34 during the propagation in the progen-

itor, and to more than 100 after the break. The narrow opening

angle for high Lorentz factor cases is good agreement with

theoretical estimate of the opening angle ∼1/Ŵ. The appear-

ance of such a high Lorentz factor component corresponds

to the feature of the GRBs. This acceleration is caused by the

energy conversion from the thermal energy to kinetic one.

Since the outflow of model (Ŵ0, ǫ0/c2) = (1.05, 0.1) does

not include so much thermal energy, no large acceleration is

occurs. The flow is non-relativistic one.

Figure 2 shows the results of a series of the calculations,

fixing ǫ0/c2 = 5 and various Ŵ0. The feature of the outflow

changes from the collimated jet to the expanding outflow by

decreasing the Lorentz factor of the injected outflow, i.e., Ŵ0.

The maximum Lorentz factor seen in each model also de-

creases from relativistic regime to non-relativistic regime by

decreasing the Ŵ0. A same continuous transition by changing

the Ŵ0 is observed in the series of the calculations in which

ǫ0/c2 is fixed to be 1 or 0.1 (Mizuta et al., 2006).

We have observed different types of the outflow propa-

gation in the progenitor. The outflows which can keep colli-

mated structure and becomes high Lorentz factor would be

observed as GRBs, since the properties correspond to those

of GRBs. Even if the outflows keeps collimated structure, the

Lorentz factor increases up to a few in some models. Such

outflows could be observed as XRFs. The outflows which do

not keep good collimation but are mildly relativistic flows

also would be the candidate of XRFs. The outflows which

have large opening angle and expanding features would be

observed as aspherical SNe (no accompanied GRBs).

Recently several types of laboratory experiments to pro-

duce jet like flows have been proposed and done by using

laser produced plasmas (Farley et al., 1999; Shigemori et al.,

2000; Mizuta et al., 2002; Foster et al.,2005) and Z-pinch

plasmas (Lebedev et al., 2002). Those are usually dense out-

flows and suitable to study the dynamics of protostar jets.

Wheres the all outflows presented in this paper are light jet

which shows a variety of properties of morphology and dy-

namics. We hope that we can produce such light jets in the

laboratory to study the different type of the morphology and

dynamics shown in this paper in the near future.

4 Conclusion

We investigate the outflow propagation in the collapsar in

the context of gamma-ray bursts (GRBs) with 2D relativis-

tic hydrodynamic simulations. We observed a variety of the

outflow properties by changing the specific internal energy

and bulk Lorentz factor of the injected outflow from non-

relativistic regime to relativistic one. The feature of the out-

flow changes from the collimated jets to expanding outflows

by decreasing the Ŵ0. The observed different features of the

dynamics possibly explain the different features of the simi-

lar phenomena such as, GRBs and XRFs. The production of

the light jet in the laboratory is expected to study the features

observed in this study.

Acknowledgment This work was carried out on NEC SX5, Cyberme-dia Center and Institute of Laser Engineering, Osaka University, andFujitsu VPP5000 of National Observatory of Japan. This work wassupported in part by the Grants-in-Aid of the Ministry of Education,Science, Culture, and Sport (14079205, A.M., S.M.) and (14102004,14079202, and 16740134, S.N.), This work was supported by the Grant-in-Aid for the 21st Century COE “Center for Diversity and Universalityin Physics” from the Ministry of Education, Culture, Sports, Scienceand Technology (MEXT) of Japan.

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DOI 10.1007/s10509-006-9238-1


Jet Deflection by a Quasi-Steady-State Side Windin the Laboratory

David J. Ampleford · Andrea Ciardi · Sergey V. Lebedev · Simon N. Bland ·

Simon C. Bott · Jeremy P. Chittenden · Gareth N. Hall · Adam Frank · Eric Blackman

Received: 16 May 2006 / Accepted: 18 August 2006C© Springer Science + Business Media B.V. 2006

Abstract We present experimental data on the steady state

deflection of a highly supersonic jet by a side-wind in the lab-

oratory. The use of a long interaction region enables internal

shocks to fully cross the jet, leading to the development of

significantly more structure in the jet than in previous work

with a similar setup (Lebedev et al., 2004). The ability to

control the length of the interaction region in the laboratory

allows the switch between a regime representing a clumpy

jet or wind and a regime similar to a slowly varying mass

loss rate. The results indicate that multiple internal oblique

shocks develop in the jet and the possible formation of a sec-

ond working surface as the jet attempts to tunnel through the

ambient medium.

Keywords Hydrodynamics . ISM . Herbig . Haro objects .

Methods . Laboratory . Stars . Winds . Outflows

1 Introduction

Astrophysical observations have shown that some jets pro-

duced by protostars are not straight, and instead exhibit a

steady curvature over a significant fraction of their length

D. J. Ampleford ()Sandia National Laboratories, Albuquerque, NM 87123-1106,USAe-mail: [email protected]

A. CiardiObservatoire de Paris, LUTH, Meudon, 92195, France

S. V. Lebedev · S. N. Bland · S. C. Bott · J. P. Chittenden ·G. N. HallBlackett Laboratory, Imperial College, London SW7 2BW, UK

A. Frank · E. BlackmanDepartment of Physics and Astronomy, Laboratory for LaserEnergetics, University of Rochester, Rochester, NY 14627, US

(many jet radii). Deflected jets normally occur as a pair of

counter-propagating jets from a common source. These de-

flected bipolar jets fall into two categories – those with S-

shaped (Reipurth et al., 1997) and those with C-shaped sym-

metries (Bally and Reipurth, 2001). The mechanisms behind

the deflection of the C-shaped jets has been the subject of var-

ious studies; these studies have indicated that the deflection

of the many of these jets cannot be explained by an ambi-

ent magnetic field (Hurka et al., 1999), photo-ablation of the

surface of the jet (Bally and Reipurth, 2001), or a pressure

gradient in the ISM (Canto and Raga, 1996). It has emerged

that the most likely explanation for the deflection of these

jets is the effect of a ram pressure due to a side-wind as dis-

cussed by Balsara and Norman (1992) and Canto and Raga

(1995). For protostellar jets such a wind may be produced

by differential motion of the source star and the surrounding

interstellar medium. This is substantiated by observations

which show that within a nebula many C-shaped jet struc-

tures are present, each with the jets deflected back towards

the central star forming region, hence the effective wind is

produced by the motion of the stars outward through the ISM

(Bally and Reipurth, 2001).

In previous experiments we have studied the deflection of

highly supersonic jets in the laboratory using conical wire ar-

ray z-pinches and a photo-ablated CH foil (Ampleford et al.,

2002; Lebedev et al., 2004; Frank et al., 2005). The previous

work indicated that these experiments are in the correct pa-

rameter regime to study the propagation of astrophysical jets

in a side-wind, similar to the mechanism for deflection of C-

shaped jets (the experiments aim to model the propagation of

one of the jets far from the source; the formation mechanism

and other jet are neglected). An important feature observed

in the previous experiments was the presence of shocks in

the jet during the deflection (as also shown by simulations

utilizing astrophysical codes (Frank et al., 2005; Lim and

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Fig. 1 (a) The experimentalsetup used by Lebedev et al.(2004), (b) illustration ofrequirements for a shock to crossthe jet and (c) the setup used inthis paper. The target is longcompared to the jet diameter,and angled to provide a uniformwind density on the jet

Raga, 1998)). In this paper we will use a modification of the

experimental setup used by Lebedev et al. (2004) to study

the deflection of a supersonic radiatively cooled jet by a side

wind that is steady state on the typical time scale of the jet;

shocks are allowed to fully evolve within the jet while the jet

is still subjected to a constant side wind.

2 Criteria for producing a steady state deflection and

experimental setup

In order to determine whether the interaction of a jet in a side

wind is steady state it is useful to consider an oblique shock

in the jet. If the jet is still influenced by the wind for the full

spatial scale required to allow a shock to fully cross the jet

then the interaction can be considered steady state. A shock

will cross the jet in a time

tcross = φ j

vs

, (1)

whereφ j is the jet diameter and vs is the transverse velocity of

the shock (see Fig. 1b for the setup and parameters discussed).

The maximum time that the jet is influenced by the side-

wind (of axial extent L) is Lv j

. Hence for a shock to be allowed

to cross the jet (and potentially be reflected or break-out) the

transit time of the shock should be less than the time that the

jet is influenced by the wind:

L

v j

>φ j

vs

(2)

φ j

cs

(3)

where it has been assumed that the transverse shock in the

jet is weakly driven, so the shock velocity vs can be approx-

imated as the sound speed cs . This can be reformulated to

incorporate the definition of the internal Mach number of the

jet (the axial Mach number) M = v j

cs:

L

φ j

M (4)

Satisfying Equation (4) guarantees that the interaction is

steady state (it should be noted that not satisfying Eq. (4)

does not necessarily indicate that the interaction is not steady

state). Depending on the clumpiness of the jet and wind, it

is possible that C-shaped protostellar jets could fall into the

steady-state and non-steady-state regimes. For the case dis-

cussed by Lebedev et al. (2004), assuming the jet remains

in a constant wind density for the full length of the foil

(L ∼ 5 mm), then the length of the interaction was ∼10 jet

diameters, however the jet Mach number was 20 (the ac-

tual Mach number depending on heating of the jet during the

interaction). This does not satisfy Equation (4), so shocks

were unlikely to be able to cross the jet, and the experimen-

tal data suggests that they did not (Lebedev et al., 2004). To

explore a steady state interaction a longer interaction region

is required.

The overall experimental setup used in this paper is

broadly similar to that used by Lebedev et al. (2004). Current

produced by the MAGPIE generator (1MA, 240ns described

by Mitchell et al. (1996)) is passed through a conical arrange-

ment of 16 fine tungsten wires (each 18µm in diameter). The

current and self-generated magnetic field of the array produce

a J × B force that acts on the low density coronal plasma

which surrounds each static wire producing a steady flow of

plasma (Lebedev et al., 2002a). This Lorentz J × B force has

components which are both radial and axial (Fig. 1a). The

formation of a conical shock on the array axis thermalizes the

kinetic energy associated with the radial component of the ve-

locity, leaving the axial component unaffected (Canto et al.,

1988). At the top of this conical shock a pressure gradient is

present which accelerates the flow; strong radiative cooling

enables the formation of a highly supersonic (Mach number

M 30), well collimated outflow (Lebedev et al., 2002b).

Data from two diagnostics will be discussed in this pa-

per. A 532 nm, 0.4 ns Nd-YAG laser is used for laser shad-

owgraphy, with a schlieren cut-off of 1 × 1020 cm−3. An

XUV imaging system which is sensitive to photon energies

hν > 30 eV and has an integration time of 3 ns (Bland et al.,

2004) is also fielded.

Following the previous discussion of the ability of shocks

to cross the jet in a characteristic time-scale, we note that the

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jet production process continues for many shock crossing

times (i.e. the jet itself can be considered steady-state if no

interaction occurs). Previously jets produced by this method

have been used to explore various aspects relevant to the un-

derstanding of protostellar jets, such as the effect of radiative

cooling and the effect of symmetry of convergent flows on

jet production (Lebedev et al., 2002b; Ciardi et al., 2002),

the effect of angular momentum on the jet (Ampleford et al.,

2006a), the effect of an ambient medium on jet propagation

(Ampleford et al., 2005) and the effect of a side-wind on the

jet (Lebedev et al., 2004). To impose such a side wind on the

jet a CH foil is photo-ablated by soft X-ray emission from the

wire array; the expansion of the foil causes the wind to im-

pact on the jet, as discussed in more detail by Lebedev et al.

(2004). In this paper we expand on our previous discussion

of jet deflection experiments, with the aim of investigating

the dynamics of jet deflection in a regime that is more suited

to some astrophysical jets, namely in a configuration which

allows shocks to propagate across the jet whilst the jet is still

under the influence of the side-wind. To increase the axial

extent of the wind the size of the foil is increased, however

to ensure that the jet is propagating through a near-constant

wind density it is necessary to angle the foil with respect to

the initial jet axis (Fig. 1c). This alteration to the foil angle

also changes the position of the stagnation point (the point

where the velocities of the of jet and wind are perpendicular)

so it can be better diagnosed. The jet and wind parameters are

expected to be broadly similar to those discussed by Lebedev

et al. (2004).

3 Dynamics of jet propagation in a side-wind

Figure 2a shows a schlieren image of the deflection of a jet

in this modified configuration. In the image the jet is seen

propagating vertically from array, which is below the base of

the image. The side-wind is produced by photo-ablation of

the CH foil and propagates right to left (away from the foil),

with a small downward component. As the jet is subjected

to the side wind the jet is steadily deflected in the direction

of the wind motion, as drawn on Fig. 2b (see Lebedev et al.,

2004 for a more detailed discussion of the basic deflection).

The plasma jet in these experiments is highly supersonic,

hence any perturbation to it, such as the ram pressure due

to the side wind should generate strong shocks in the flow

(as was observed by (Lebedev et al., 2004)). The schlieren

diagnostic used in Fig. 2a is sensitive to density gradients

in the plasma, such as those produced by these strong

shocks. Correlation of these structures with increased XUV

emission (Fig. 2c) is consistent with the thermalization of

kinetic energy in these shocks.

The interaction of the jet is much more complex than was

seen in the previous study using a shorter wind (Lebedev

et al., 2004), with numerous structures now present between

the jet and foil. For clarity this image has been repeated in

Fig. 2b, with the many different features that will be discussed

drawn and labelled. The axial position of the tip of the curved

portion of the jet (at the left of the interaction) corresponds

to the expected axial position of the tip of a jet propagating

in vacuum.

At the base of the target we expect a downward component

to the wind (due to the angle of the foil and divergence). On

the schlieren image (Fig. 2a) two shocks are present where

the expanding wind meets the upwards travelling halo plasma

surrounding the jet as it exits the wire array (labelled Halo

shocks in Fig. 2b). The lower of these two shocks is a shock in

the halo and the upper is a reverse shock in the wind (they are

marked Halo shock and Wind shock respectively in Fig. 2d).

In the next three sections we will describe the other struc-

tures observed in the interaction.

3.1 Internal oblique shock formation

On the high magnification image (Fig. 2d) we see that there

is an internal shock in the centre of the jet (labelled OS1).

Fig. 2 Shocks within the jet shown in both (a) low and (d) high mag-nification schlieren images (both at 343 ns). (b) is a repeat of (a) withlabels on the image which are discussed in the text. (c) shows an XUVemission image (Bland et al., 2004) at 380 ns

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As with the earlier experiments with a shorter interaction

this is likely to be the oblique internal shock responsible

for the initial deflection. We see that this internal shock

is not straight, but instead bends each way by a few de-

grees. It is interesting to note that the most pronounced of

these bends coincides with a continuation of the shock in

the halo flow. Thus it is likely that this bend in the inter-

nal shock is associated with a change in the wind density

and hence ram pressure – further experiments would be re-

quired to investigate the effect of variations in the ambient

density.

Further along the jet-wind interaction on the low mag-

nification image we see that more structure is present; one

obvious shock is labelled OS2. In this image it is unclear

what the significance of this shocks is, however we can un-

derstand this better if we look at XUV emission. Figure 2c

shows a gated XUV emission image from the same exper-

iment, however 40 ns after the schlieren image. This image

was taken at 22.5 from the plane containing the laser probe

beam and foil, hence some emission from the surface of the

foil can be seen in the XUV image. The structure seen in the

XUV image is broadly similar to that in the schlieren image,

however these shock features have developed slightly. Again

we see the shock previously labelled OS2; it appears that this

is static in time, and remains almost parallel to the jet, so is

likely to be a second internal oblique shock in the jet (OS2),

further deflecting the jet.

3.2 Formation of a new working surface

The nature of the shock WS2 becomes clear if we look at

simulations of a jet in a side wind. Figure 3 shows a 2D slice

taken from a 3D HD simulation of a jet propagating in a side-

wind. For simplicity this simulation has a constant mass flux

in the jet, constant jet injection velocity and uniform wind

density and velocity. In these simulations we see that as the

jet propagates the upwind surface becomes unstable and a

second (and in the last frame a third) working surface begins

to form. This is similar to what is observed in Fig. 2a and c –

the feature labelled WS2 is likely to be the formation of this

secondary working surface (the first working surface being

at the head of the jet, labelled WS1). The development of

this structure with time can be seen experimentally in Fig. 4,

which shows a series of gated XUV images (for a different

experiment). The development of a second working surface

has also been observed for a different setup using a conical

wire array (Ampleford et al., 2005).

Fig. 3 Simulations of a jet in propagating in a side-wind. 2D slice from a 3D HD simulation (Chittenden et al., 2004) with uniform jet and wind(i.e. different from the experiments)

Fig. 4 Development of the jet-wind interaction with time is shown experimentally by time resolved XUV emission (hν > 30 eV)

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Fig. 5 (a) Low and (b) high magnification schlieren images showingthe interaction of the low density, un-collapsed tip of the jet (from adifferent experiment to all other images)

3.3 Interaction of a low density jet with a denser wind

If we look above the tip of the jet on the earlier schlieren

image (Fig. 2a) we see that more shocks have formed. The

axial position of this material implies that it was ejected be-

fore the well formed jet that has previously been discussed

(Lebedev et al., 2002b), and instead consists of material that

reached the axis before the conical shock was well formed

(Bott et al., 2006).

On a different experiment we can see this interaction in

more detail on a high magnification schlieren image (Fig. 5).

This image shows the low density jet through shadowgra-

phy, and shocks through schlieren effect. We see that there

are actually two shocks present. The shock furthest from the

foil is an internal shock in the jet, producing yet another re-

gion of deflection. The closest shock to the foil is a reverse

shock forming in the wind. It is believed that when this jet

material passed through the lower area of wind the ambi-

ent material was of sufficiently low density that either there

was not enough momentum in the wind at that time or the

mean free path of the jet was too long to be deflected (i.e. a

particle effect that cannot be modeled using a hydrodynamic

simulation). Also on this experiment the low magnification

schlieren image shows a well defined reverse shock in the

wind near the first deflection of the jet.

4 Conclusions

We have discussed experimental data for the deflection of

highly supersonic jets by a cross wind where the cross wind is

effectively continuous in relation to the typical spatial scales

of the jet. Such a configuration could be of interest in mod-

eling the propagation of a jet in a side-wind that is neither

clumpy or gusty (experiments that reach the inverse regime

were discussed in Lebedev et al. (2004)). The data has shown

that many different shocks are formed in the interaction. It is

interesting to note that experiments utilizing two very differ-

ent ambient configurations (here and Ampleford et al. (2005))

both lead to the formation of secondary working surfaces in

the jet. A laboratory 3D HD code has recovered many of the

features of the present experiments; the data should provide

a useful testbed for astrophysical computer simulations of

such a case. Future experiments will aim to follow the evolu-

tion of shocks more closely and attempt to evaluate the shock

jump conditions.

Acknowledgements This research was sponsored by the NNSA underDOE Cooperative Agreement DE-F03-02NA00057 and in part by theEuropean Communitys Marie Curie Actions – Human resource andmobility within the JETSET (Jet Simulations, Experiments and Theory)network under contract MRTN-CT-2004 005592. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the US DOE’s National Nuclear Security Administrationunder contract DE-AC04-94AL85000.

References

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Ampleford, D.J., Lebedev, S.V., Bland, S.N., Ciardi, A., Sherlock, M.,Chittenden, J.P., et al.: Deflection of Supersonic Plasma Jets byIonised Hydrocarbon Targets. In: Davis, J., Deeney, C., Pereira,N.R. (eds.) AIP Conf. Proc. 651: Dense Z-Pinches, pp. 321–324(2002)

Ampleford, D.J., Lebedev, S.V., Ciardi, A., Bland, S.N., Bott, S.C.,Chittenden, J.P., et al.: Formation of working surfaces in radiativelycooled laboratory jets. Astrophys. Space Sci. 298, 241–246 (2005)

Bally, J., Reipurth, B.: Irradiated Herbig-Haro jets in the Orion Nebulaand near NGC 1333. Astrophys. J. 546, 299–323 (2001)

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Bland, S.N., Ampleford, D.J., Bott, S.C., Lebedev, S.V., Palmer, J.B.A.,Pikuz, S.A., et al.: Extreme ultraviolet imaging of wire array z-pinch experiments. Rev. Sci. Instrum. 75, 3941–3943 (2004), doi:10.1063/1.1787927.

Bott, S.C., Lebedev, S.V., Beg, F., Bland, S.N., Chittenden, J.P., Cia-rdi, A., et al.: Dynamics of cylindrically converging precursorplasma flow in wire array Z-pinch experiments. Phys. Rev. E. DOI10.1103/PhysRevE.74.046403 (2006)

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Canto, J., Raga, A.C.: The steady structure of a jet/cloud interaction – I.The case of a plane-parallel stratification. MNRAS 280, 559–566(1996)

Canto, J., Tenorio-Tagle, G., Rozyczka, M.: The formation of interstel-lar jets by the convergence of supersonic conical flows. Astron.Astrophys. 192, 287–294 (1988)

Chittenden, J.P., Lebedev, S.V., Jennings, C.A., Bland, S.N., Ciardi, A.:X-ray generation mechanisms in three-dimensional simulations ofwire array Z-pinches. Plasma Phys. Control. Fusion 46, B457–B476 (2004)

Ciardi, A., Lebedev, S.V., Chittenden, J.P., Bland, S.N.: Modeling ofsupersonic jet formation in conical wire array Z-pinches. LaserPart. Beams 20, 255–261 (2002)

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Frank, A., Blackman, E.G., Cunningham, A., Lebedev, S.V., Ampleford,D., Ciardi, A., et al.: A HED Laboratory Astrophysics TestbedComes of Age: Jet Deflection via Cross Winds. Astrophys. SpaceSci. 298, 107–114 (2005)

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Lebedev, S.V., Ampleford, D., Ciardi, A., Bland, S.N., Chittenden, J.P.,Haines, M.G., et al.: Jet deflection via crosswinds: laboratory as-trophysical studies. Astrophys. J. 616, 988–997 (2004)

Lebedev, S.V., Beg, F.N., Bland, S.N., Chittenden, J.P., Dangor, A.E.,Haines, M.G.: Snowplow-like behavior in the implosion phase ofwire array Z pinches. Phys. Plasmas 9, 2293 (2002a)

Lebedev, S.V., Chittenden, J.P., Beg, F.N., Bland, S.N., Ciardi, A., Am-pleford, D., et al.: Laboratory astrophysics and collimated stellaroutflows: the production of radiatively cooled hypersonic plasmajets. Astrophys. J. 564, 113–119 (2002b)

Lim, A.J., Raga, A.C.: 3D numerical simulations of a radiative Herbig-Haro jet in a supersonic side wind. MNRAS 298, 871–876 (1998)

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DOI 10.1007/s10509-006-9283-9


Hypersonic Swizzle Sticks: Protostellar Turbulence, Outflowsand Fossil Outflow Cavities

A. Frank

Received: 17 July 2006 / Accepted: 20 November 2006C© Springer Science + Business Media B.V. 2007

Abstract The expected lifetimes for molecular clouds has

become a topic of considerable debate as numerical simula-

tions have shown that MHD turbulence, the nominal means

of support for clouds against self-gravity, will decay on short

timescales. Thus it appears that either molecular clouds are

transient features or they are resupplied with turbulent en-

ergy through some means. Jets and molecular outflows are

recognized as a ubiquitous phenomena associated with star

formation. Stars however form not isolation but in clusters

of different density and composion. The ubiquity and high

density of outflows from young stars in clusters make them

an intriguing candidate for the source of turbulence energy in

molecular clouds. In this contribution we present new studies,

both observational and theoretical, which address the issue

of jet/outflow interactions and their abilityto drive turbulent

flows in molecular clouds. Our studies focus on scales as-

sociated with young star forming clusters. In particular we

first show that direct collisions between active outflows are

not effective at stirring the ambient medium. We then show

that fossil cavities from “extinct” outflows may provide the

missing link in terms of transferring momentum and energy

to the cloud.

Keywords Hydrodynamics . Methods: Laboratory . ISM:

Herbig-Haro objects . Stars: Winds . Outflows

1 Introduction

Star formation occurs within Molecular Clouds (MCs), com-

plex structures whose physical evolution is still not clearly

A. FrankDepartment of Physics and Astronomy and Laboratory for LaserEnergetics, University of Rochester, Rochester NY 14627–0171

understood (Ballesteros-Parades et al., 2006). MCs are hi-

erarchical structures with smaller substructures known as

clumps and cores. Star formation is believed to occur in

cores with larger clusters forming from more massive cores.

The expected lifetimes for molecular clouds has become a

topic of considerable debate as numerical simulations have

shown that MHD turbulence, the nominal means of support

for clouds against self-gravity, decays on a crossing timescale

(Goldreich and Kwan, 1974; Arons and Max, 1975; Stone

et al., 1998; MacLow et al., 2004). In light of this result the

traditional view that MCs are long-lived, quasi-static equilib-

rium structures has been challenged by a paradigm in which

star formation occurs on a timescale comparable to the free-

fall time (Ballesteros-Paredes et al., 1999; Hartman, 2003).

In the former case turbulence in the cloud is an important

source of support and regulation of the Star Formation Ef-

ficiency (SFE) and it must be re-supplied over time. In the

latter case turbulence is produced with the cloud (Yamaguchi

et al., 2001) or only needs to be driven up to the point that a

cloud is disrupted.

Feedback from protostars forming within a MC has

been cited by many authors as a principle means of ei-

ther re-energizing turbulence or disrupting clouds (Bally and

Reipurth, 2001). When massive stars form their ionization

fronts, strong stellar winds and eventual supernova blast-

waves are expected to be the major contributor to feedback

(Krumholtz, 2005). In lower mass cluster environments and

environments where the effects of massive stars have not been

felt protostellar outflows will likely be the dominant form of

feedback. In these cases, even if energy is re-supplied from

supra-cluster scales, at some wavenumber the outflow injec-

tion may come to dominate global dynamics. In fact, Ener-

getic outflows associated with low and moderate mass young

stellar objects are known to exert a strong effect on their par-

ent clouds (for a recent review see Bally et al., 2006). The idea

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Fig. 1 Left: Volume renderedimage of density from AMRcolliding jet simulations. 90degree collision of 2 outflowsdisplaced by an impactparameter of b = 1r j . Rightcomparison of Log10 Mass vs.Log10 Velocity for 3 impactparameters: b = 0 (solid line),b = 1r j (dashed line), b = 8r j

(dotted line). Note that thecolliding jet is least effective ataccelerating ambient material(Cunningham et al., 2006a)

that feedback from TT winds could lead to a self-regulating

state of star formation dates back at least as far as Norman and

Silk (1980). Since that time measurements have backed up

the idea. Young stellar outflows in settings such as NGC 1333

have been shown to contain sufficient kinetic energy to excite

a significant fraction of supersonic turbulence in their sur-

roundings and/or unbind and disperse portions of their parent

cloud (Bally and Reipurth, 2001; Knee and Sandell, 2000;

Warin et al., 1996; Matzner, 2002). In spite of the importance

of this process, the global dynamics of multiple outflow/cloud

interactions has yet to be explored in detail. In particular the

nature of the coupling between multiple outflows and the

cloud and their efficacy for generating turbulence and/or dis-

rupting their parent clouds, remains to be characterized.

While invoking jets and outflows to drive turbulent mo-

tions appears attractive, the principle means of energy trans-

fer from jet to cloud appears to come via shock waves, the

so-called “prompt entrainment” mechanism (Chernin et al.,

1994). Thus the effect of a single supersonic outflow is

bounded by the shock wave which defines it. Only those

regions of a cloud which have been swept over by an outflow

will gain energy. Given such a localization of energy and

momentum deposition, the action of multiple, overlapping,

randomly oriented outflows may be required to drive the ran-

dom motions associated with isotropic turbulence. Somehow

the energy and momentum in the localized region engulfed

by a jet or outflow must be randomized and distributed over

many scales. This may occur via MHD waves but the link-

ing of outflows to wave generation has not been shown in

detail as yet (Koduh and Basu, 2006). To explore the role of

outflows in generating random motions we first carried out a

study of colliding active outflows.

2 Collisions between active outflows

We first estimate the probability that two protostellar outflows

interact as a function of protostellar density in the cloud. We

consider a volume V that contains an average outflow density

N and assume that each protostar emits a bipolar outflow. We

approximate the volume of these bipolar outflows as that of

a cylindrical column of length L and radius R. Assuming that

the production frequency of outflows in the cloud is constant,

we can cast the density of outflows active at any given instant

in terms of the stellar density N∗ as N = N∗(toutflow/tcloud).

The probability that two active outflows occupy the same

region of space in the cloud at the same time is then P ≈(Voutflow/Vcloud)2. Solving for N∗ we have

N∗(P) =√

P

π R2L

tcloud

toutflow

(1)

We define Ncritical as the protostellar density that achieves

a volume fill ratio of 10% bowshock overlap: Ncritical ∼N (0.1). Above this intersection probability we expect the

effect of collisions to become appreciable. Assuming typical

values for the protostellar outflow size, bow shock radius, out-

flow lifetime and cloud lifetime we find a Ncritical = 500 pc−3.

This is comparable to the protostellar density of many star

forming regions. Outflow interactions of some form are there-

fore statistically likely to occur in a typical star forming

region.

Based on this conclusion the efficacy of active outflow

interactions in stirring the ambient medium was explored

in a 3-D AMR study of jet collisions (Cunningham Frank

and Blackman, 2006a). Our study focused on hydrodynamic

simulations of the interaction of two orthogonal outflows.

The simulations included the effect of radiative energy loss on

the flow and we investigated the role of the impact parameter

and degrees of collimation. The simulations were carried

out in 3D using the AstroBEAR adaptive mesh refinement

(AMR) code (Fig. 1).

If the collision of outflow streams from adjacent YSO’s

contribute to the turbulent energy budget of their parent

cloud, it would do so by increasing the rate at which the flow

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Fig. 2 Bottom Left: Simulation of fossil cavity driven by a wide anglewind which is active for 104 years. Image taken at 105 years. Note thatbackfilling via rarefaction waves has occurred. Top Left: Cavity drivenby active wide angle wind shown for comparison with frame taken at

t = 3 × 104 y. Right: fractional difference between analytical scalingrelation shown in Equation (2) and results of simulation. Vertical lineshows point when the wind shuts off. After this point the scaling relationand simulation differ by at most a factor of 2 due to geometrical effects

Fig. 3 2-D slices of 3-D simulations of jets with different decay timesdriven into turbulent media. Left Top: Control case in which a constantjet is driven into a quiescent media. Top Right: Constant jet driven intoturbulent media. Lower Left: Slowly decaying jet driven into turbulent

media. Lower Right: Rapidly decaying jet driven into turbulent media.Note effect on bow shock in turbulent simulations. In rapidly decayingjet simulations the outflowing material has been completely subsumedby the turbulence.

imparts momentum into the surrounding molecular gas. This

could occur if the redirected outflow has a volume greater

than the individual outflows. Also if the redirected flow gen-

erates more “splatter” in the sense that a wider range of scales

become energized though vortices generated during the col-

lision then the increased rate of momentum deposition into

the ambient molecular gas would result in an increased rate

of generation of turbulent energy and could thereby provide

support for the parent cloud against gravitational collapse and

star formation. To test this idea we ran simulations of collid-

ing outflows for impact parameters b = (0, 1.0r j , 5.33r j ) for

fully collimated jets and b = (0, 1.0r j , 8.0r j ) for “wide angle

jets” with an opening angle of θ = 15. We then examined

different measures of the acceleration of ambient material by

the outflows including the outflows ability to initiate turbu-

lent motions. These included exploration of the mass velocity

plots M(v) and measures of average vorticity in the grid as a

function of time.

Surprisingly, our results indicated that the high degrees

of compression of outflow material, achieved through radia-

tive shocks near the vertex of the interaction, prevent the

redirected outflow from spraying over a large spatial region.

Furthermore, the collision reduces the redirected outflow’s

ability to entrain and impart momentum into the ambient

cloud. Thus combining the results of our simulations with

consideration of the probabilities of outflow collisions for

direct collisons led us to conclude that individual low ve-

locity fossil outflows, interacting only on long timescales,

are the principle coupling between outflows and the

cloud.

Springer


3 Observations and simulations of fossil protostellar

outflows (NGC 1333)

Many authors have used “0-D” estimates of the energy

present in active outflows and compare these with cloud

turbulent energy (Bally and Reipurth, 2001). These initial

efforts were important but studies of individual objects ex-

press a more complicated picture. In particular the explicit

time-dependent nature of the coupling between cloud mate-

rial and outflows is not addressed in these estimates. Out-

flow power evolves rapidly in time in the strongest phases

(Class 0 sources) lasting a fraction of the star formation

timescale. Thus outflows will continually be turning on and

fading across the history of an active star forming region.

The pitfalls of ignoring the temporal domain was high-

lighted in a recent study of NGC1333 (Quillen et al., 2005).

In this work it was found that velocity dispersions, mea-

sured in 13CO, did not vary across the cloud. There was

no link between active outflows and turbulence. Instead a

new class of outflow signature was identified in the form of

fossil cavities. These fossil cavities proved to be a smok-

ing gun showing strong coupling between outflows and the

molecular cloud. 20 cavities were identified with typical di-

ameter of about 0.1–0.2 pc, and velocity widths 1–3 km/s.

Cavities at a range of sizes and velocities were seen in the

cloud. If these cavities were simply empty regions in the

cloud, the timescale for them to fill in would be less than

a million years implying that they were created relatively

recently.

To make an explicit link with outflow models, cavity prop-

erties were compared with scaling relations for momentum

injection, Po, derived from similarity conditions.

Po ≈ 0.8 Mokms−1

(

n

104cm−3

)(

R

0.1pc

)3

×(

l

0.4pc

)(

t

2 × 105 yr

)

(2)

Using this relation one can estimate the total momentum

injection required to excavate a given cavity. Quillen et al.

(2005) compared the number of cavities and their momentum

requirements with those measured from currently active out-

flows. From this it could be estimated that much of the total

momentum flux from outflows is fed back into the molecular

cloud via the fossil cavities.

In a recent, more detailed study, AMR simulations of fos-

sil cavity evolution using full H2 chemistry and cooling were

performed (Fig. 2, Cunningham, 2006b). In this work jets

and wide angle winds were simulated with an injected mo-

mentum flux that decreased in time. These simulations where

compare with runs with constant momentum flux. The decay-

ing flux models exhibited deceleration of the outflow head

and backfilling via expansion off of the cavity walls. They

also showed lower density contrasts and reached constant

aspect ratios. Most importantly the simulations recover the

basic properties of observed fossil cavities including verify-

ing the scaling relations (Equation (2)). This work also pro-

vided synthetic observations in terms of P-V diagrams which

demonstrate that fossil cavities from both jets and wide angle

outflows are characterized by linear “Hubble-law” expan-

sions patterns superimposed on “spur” patterns indicative of

the head of a bow shock. These should prove useful in future

observational work.

4 Conclusions and future work

We have shown that fossil cavities, rather than active out-

flows, may be the direct link between stellar injection of

mechanical energy and turbulence on scales of young clus-

ters. We note the useful study by Li and Nakumura (2006)

who explored outflow collisions within a collapsing turbulent

cloud and concluded that outflow activity could re-energize

turbulence. These studies do not contradict our results as we

agree with their main conclusion however their simulations

were of lower resolution (1283 for the entire cluster) and

could not resolve the interactions of individual outflows or

include the outflow power evolution (a topic we will explore

in the next section). Thus while turbulent energy may be

supplied at larger scales to the clouds as a whole (via super-

nova or gravitational collapse) which then cascades down,

there is also a separate injection of energy at smaller scales

which constitutes a feedback from the stars within the clus-

ter. Future work will need to make the interplay between

turbulence and energy injection from jets more explicit. Fig-

ure 3 shows initial work in this direction in the form of

simulations of jets into fully turbulent media and the subse-

quent evolution of the jet driven cavity (Cunningham et al.,

2007)

Acknowledgment We acknowledge support for this work from the JetPropulsion Laboratory Spitzer Space Telescope theory grant 051080–001, Hubble Space Telescope theory grant 050292–001, NationalScience Foundation grants AST-0507519, AST-0406799, AST 00–98442 & AST0406823, DOE grant DE-F03–02NA00057, the NationalAeronautics and Space Administration grants ATP04–0000-0016 &NNG04GM12G issued through the Origins of Solar Systems Program,and the Laboratory for Laser Energetics.

References

Arons, Max.: ApJ 196, L77 (1975)Ballesteros-Parades, H., Vazquez-Seandeni.: ApJ 527, 285 (1999)Ballesteros-Paredes, J., Klessen, R., Mac Low, M.-M., Vazquez-

Semadeni, E.: Protostars and Planets V. in: Reipurth, B., Jewitt, D.,Keil, K. (eds.) University of Arizona Press, Tucson, in press (2006)

Bally, J., Reipurth, B.: ARAA 39, 403 (2001)

Springer


Bally, J., Reipurth, B., Davis, C.: Protostars and Planets V. in: Reipurth,B., Jewitt, D., Keil, K. (eds.) University of Arizona Press, Tucson,in press (2006)

Chernin, L., Masson, C., Gouveia dal Pino, E.M., Benz, W.: ApJ 426,204 (1994)

Cunningham, A., Frank, A., Blackman, E.: ApJ, in press (2006a)Cunningham, A., Frank, A., Blackman, E., Quillen, A.: ApJ, in press

(2006b)Cunningham, A., Frank, A., Blackman, E., Quillen, A.: ApJ, in prepa-

ration (2007)Goldreich, P., Kwan, J.: ApJ 189, 441 (1974)Hartmann, L.: ApJ 585, 398 (2003)Krumholtz, M., McKee, K., Klein, R.: Nature 438, 333 (2005)

Knee, L.B.G., Sandell, G.: A&A 361, 671 (2000)Kodu, T., Basu, S.: ApJ 642, 270 (2006)Li, Z.-Y., Nakamura, F.: ApJL 640, L187 (2006)Matzner, C.D.: ApJ 566, 302 (2002)MacLow, M.-M., Klessen: Rev. Mod. Phys. 76, 125 (2004)Norman, C., Silk, J.: ApJ 238, 158 (1980)Quillen, A.C., Thorndike, S.L., Cunningham, A., Frank, A., Gutermuth,

R.A., Blackman, E.G., Pipher, J.L., Ridge, N.: ApJ 632, 941(2005)

Stone, J., Ostriker, E., Gammie, C.: ApJ 508, L99 (1998)Yamaguchi, et al.: PASJ 53, 985 (2001)Warin, S., Castets, A., Langer, W.D., Wilson, R.W., Pagani, L.: AAP

306, 935 (1996)

Springer


DOI 10.1007/s10509-006-9218-5


Plasma Jet Studies via the Flow Z-Pinch

U. Shumlak · B. A. Nelson · B. Balick


Abstract The ZaP sheared-flow Z-pinch produces high den-

sity Z-pinch plasmas that are stable for up to 2000 times the

classical instability times. The presence of an embedded ra-

dial shear in the axial flow is correlated with the observed

stability, and is in agreement with numerical predictions of

the stability threshold. The case is made that using a higher-

Z working gas will produce supersonic plasma jets, consis-

tent with dimensionless similarity constraints of astrophysi-

cal jets. This would allow laboratory testing of some regimes

of astrophysical jet theory, computations, and observations.

Keywords Z-pinch . Plasma jets . Herbig-Haro . Planetary

nebulae

1. Introduction

Astrophysical jets arise from many sources such as planetary

nebulae (PNe), massive black holes, active galactic nuclei

(AGN), Herbig-Haro (HH) objects, and other young stellar

objects (YSO). PNe (Balick and Frank, 2002) and HH ob-

jects (Reipurth and Bally, 2001) both produce supersonic yet

non-relativistic jets,v ∼100–300 km/s, but over differing du-

rations. PNe produce episodic jets and shock fronts, typically

less than 1000 years old. HH objects produce jet outflows last-

ing up to 105 years. Massive black holes and other objects

produce highly relativistic collimated jets that can be mega-

parsecs in length. Most jets are a fraction of a few parsecs

in length and 20–50 times their diameter. Internal flowing

clumps are observed, and the flows are seen to interact with

surrounding subsonic plasma producing bow shocks. Little

U. Shumlak () . B. A. Nelson . B. BalickUniversity of Washington, Washington, USA

is known about the production and stability of astrophysical

jets, nor about details of interactions with their background.

Opportunities to simulate (scaled) astrophysical flow con-

ditions in a stable plasma jet are rare, and almost unparalleled.

Astrophysical jets can be compared with laboratory

plasma through scaling laws and by the use of a variety

of gases for interaction with laboratory plasma jets. For

example background gas entrainment and shock formation

can be studied by using an argon jet impinging upon a helium

background, and using filters to separately record spectra of

the jets and shocks using Ar and He emission lines. Using

working gases of different mass also allows, within the scal-

ing laws, simulations of differing velocities and varying mass

ratios between the jets and the background. Modification of

operating conditions and monitoring the time evolution of

jets is an obviously impossible task with actual astrophysical

jets.

The ZaP flow Z-pinch experiment at the University of

Washington produces plasma jets with remarkable similar-

ities to plasma jets from HH and PNe objects; stable long-

lived high aspect ratio jets, propagating “knots”, and sheared-

flow. By varying the jet and background working gases, ZaP

can serve as a powerful testbed for background interaction,

shock formation and evolution, and scaling studies. This pa-

per explores the possibility of using ZaP to produce and study

plasma jets of interest to astrophysics.

2. Astrophysical jets

While magnetic fields are generally accepted to be involved in

the origin of HH jets (Reipurth and Bally, 2001), and likely in

PNe jets (Gardiner and Frank, 2001; Matt and Balick, 2004),

they prove difficult to measure. Values in the range of 0.03 –

0.1 mG have been reported in the region of the bow shock of

Springer


Fig. 1 Schematic of the ZaP experiment showing the relevant fea-tures. Green cross-hatched region represents the magnetically-confinedand expansion plasma. The inner and outer electrodes are separated

by an insulator at z = −125 cm, the electrode end wall is locatedat z = 75 cm, and the neutral gas injection plane is located atz = −75 cm

HH111, (Morse et al., 1993), fields approaching a few Gauss

have been cited (Ray et al., 1997) in an HH jet from T Tau S.

Surface fields on the order of kG have been detected in the

nuclei of a few PNe by Zeeman splitting in their atmospheric

lines (Jordan et al., 2005), 1–10 G fields are implied in small

regions associated with water and OH masers in PN winds

through the circular polarization of the narrow maser lines

(Miranda et al., 2001). These magnetic fields are often posited

to be azimuthal (toroidal) relative to the jet flow. This requires

an axial current, (with its return path through the surrounding

plasma) not unlike a Z-pinch. Figure 1 of Chevalier and Luo’s

(1994), model of a PN jet is indistinguishable from a Z-pinch

equilibrium, and Contopoulos (1995) makes an analogy of

jet formation with an “astrophysical plasma gun”. However,

pure hydrodynamics may also play a role in shaping these

jets.

In both PNe and HH flows (and even in simulations of

them), velocity shear is observed (Balick and Frank, 2002;

Reipurth and Bally, 2001; Hardee, 2004), e.g. large flow ve-

locity (approximately 300 km/s) is observed in the main

portion of HH47 moving through a slower velocity (35–

120 km/s) background gas (Heathcote et al., 1996). PNe

flows also exhibit a great range of velocities in the same

object (Balick and Frank, 2002; Frank, 1999).

3. The ZaP flow Z-pinch experiment

The ZaP experiment (Shumlak et al., 2001, 2003), shown

in Fig. 1, produces a hydrogen Z-pinch column (100 cm

long and approximately 1 cm radius) with flow velocities

on the order of 100–200 km/s (see Table 1 for more in-

formation). These pulses are seen to be stable up to 2000

times the ideal growth period (corresponding to several flow-

through times). This stability is correlated with the pres-

ence of high flow velocity, radial shear of the axial ve-

Table 1 ZaP machine and operating parameters (for hydro-gen working gas)

Parameter Value

Inner electrode radius rinner 5 cm

Outer electrode radius router 10 cm

Assembly region length zassembly 100 cm

Capacitor bank energy Wbank 30–60 kJ

Peak plasma current Ip 250 kA

Injector voltage (sustainment) Vinj 2 kV

Average density 〈ne〉 2 × 1016 cm−3

Total temperature Te + Ti 200 eV

locity, and continual flow of plasma from the acceleration

region.

3.1. Sheared-flow plasma production and stability

ZaP flow Z-pinches are produced as follows: Neutral gas

is puffed in the coaxial electrode (“acceleration”) region

where it is ionized by high voltage applied between the elec-

trodes. J × B forces accelerate the annulus of plasma from

the coaxial region to the “assembly” region where a Z-pinch

is formed. The 1/r2 magnetic pressure in the acceleration

region assures a radial variation in the axial flow, which is

embedded in the final Z-pinch assembly. The virulent m = 1

“kink” mode is stabilized by a sufficient radial shear of axial

velocity, vz , (Shumlak and Hartman, 1995)

dvz/dr

kvA

0.1 (1)

where k is the axial wavenumber of the instability, and vA is

the Alfven velocity. Remnant plasma (not shown) in the ac-

celeration region near the outer wall continuously flows into

the assembly region, which aids maintenance of a sheared-

flow.

Springer


I p (

kA

)

0

50

100

150

200

250B

1 /<B

0 >

0.0

0.2

0.4

0.6

0.8

1.0Ip

B1/<B0>

Vgap (

kV

)

-1

0

1

2

3

<n

e>

(1

021 m

-3)

0

2

4

6

8

z = -65 cm

z = -25 cm

Time (µs)

0 20 40 60 80

Bw

all

(T)

0.0

0.1

0.2

0.3

0.4

0.5

0.6z = -120 cm

z = -65 cm

z = -25 cm

z = 0 cm

Pulse 21014009Fig. 2 Typical ZaP hydrogenpulse: (a) Plasma current andnormalized m = 1 azimuthalmode data at midplane, (b) gapvoltage, (c) line-averagedelectron density at z = −65(earlier peak) and z = −25 cm,and (d) wall magnetic field atseveral axial locations

Waveforms for a typical ZaP pulse are shown in Fig. 2.

The m = 1 component of the azimuthal field at midplane

(normalized to the azimuthally-averaged field, Fig. 2(a)) is

typically seen to settle to a low value for a period of 20–40

(or more)µs. During this “quiescent” period, density remains

in the coaxial region at z = −65 cm (Fig. 2(c)) and a radial

current flows in the coaxial region, as seen by the variation in

Bθ with z in Fig. 2(d). When the coaxial region density goes

towards zero, the radial current (proportional to d Bθ/dz)

decreases, and the m = 1 component at midplane increases.

During peak plasma current, a Rowogski loop at the endwall

shows up to 5 kA of net current (waveform not shown) leaving

through the hole in the endwall.

Fast camera images with a Balmer Hα filter are shown in

Fig. 3 for a similar pulse. The circular aperture is a 5 cm hole

in the outer electrode, viewed at an oblique angle. (The dark

object on the other side of the plasma is a similar outer elec-

trode hole on the opposite side.) The images show a stable,

well-centered pinch until later in the pulse, after the quiescent

period, where the light emission is seen to twist and break up.

Many characteristics of the pulse are seen to change as the

normalized m = 1 component approaches the approximate

value of 0.2 (corresponding to a radial displacement of

∼1 cm): oscillations of the m = 1 magnitude are not only

larger, but the phase is more erratic; the presence of flow

and shear disappears; and Imacon images no longer show

well-formed light emission.

Fig. 3 Simultaneous view through two ports, z = 0 (lower 8) and z =17 cm, (upper 8); Z -axis vertical (pulse 10830012). Frames, 1 µs apart,advance in time from the far lower left pair to the pair above, thenadvance to the next lower right pair to above, as indicated by the arrows

4. Relevance of laboratory plasma jets

to astrophysical jets

Laboratory plasmas can be of great relevance for testing

theory and observations of astrophysical jets, and for

benchmarking computational studies. Many researchers

have examined the relevant scaling parameters (Ryutov

Springer


Table 2 Comparison of ZaPoperation with hydrogen andargon (estimated for Z = 4)with high and low B

astrophysical jets (AJ). (Valuesare from Refs. (Reipurth andBally, 2001; Heathcote et al.,1996; Ray et al., 1997)

Parameter ZaP (H) ZaP (Ar, est.) AJ (High B) AJ (Low B)

n (cm−3) 1016 5 × 1017 102–104 102–104

B (G) 25×103 140×103 5 3 × 10−5

Te (eV) 100 1 0.1–1 0.1–1

Ti (eV) 100 1 0.1–1 0.1–1

L/r 100 300 10–>100’s 10–>100’s

vflow (km/s) 80–120 32 300 300

vflow/cs 0.75–≈ 1 8 10–30 10–30

B/√

p 8 63 1.6 × 104–5 × 105 0.09–3

and Remington, 2001, 2002; Kane et al., 1999) and have

performed or proposed relevant experiments (Lebedev et al.,

2005; You et al., 2005; Lebedev et al., 2004; Raga et al.,

2001; Remington et al., 2000). To cover the physics of

astrophysical jets, the ideal laboratory experiment would be

able to produce supersonic collimated jets (both magnetized

and unmagnetized) and associated shocks, with sufficient

diagnostics for comparison with theory and computation.

4.1. Similarities of ZaP plasmas with astrophysical jets

Light emission from ZaP at certain times looks remarkably

like those from HH objects, viz., columns of high flow with

propagating clumps. Note the motion of the clumps in Fig. 3.

Although the clumps may be the result of local instabilities,

the global jet structure remains.

These ZaP plasmas are highly magnetized flows, while

plasma flowing past the end wall is mostly unmagnetized.

The addition of background gas will produce HH-like bow

shocks and PNe-like bubbles, allowing detailed laboratory

study of MHD and hydrodynamical formation and evolution

of shocks.

4.2. Similarity constraints

A very lucid derivation of the relevant “similarity criteria” is

given by Ryutov and Ramington (2001) Using ideal MHD

equations, it is shown that for similar spatially varying initial

conditions, two systems will have similarity in their time

evolution when the following terms are held invariant in both

systems:

B∗/√

p∗ and v∗/√

p∗/ρ∗, (2)

where B is the magnetic flux density, v is the fluid flow, p

is the pressure, and ρ is the mass density, evaluated at some

characteristic point. The first constraint is proportional to

the inverse square root of β ≡ p/(

B2/8π)

, and the second

constraint is proportional to the Mach number M ≡ v/cs =v/

√γ p/ρ, where γ = 5/3 for polytropic fluids.

4.3. Plasma jet parameters in ZaP

Table 2 shows parameters for ZaP experimental results with

hydrogen and estimates for operation with argon and are

compared with parameters for jet HH47 (similar values are

assumed for PNe jets) using estimated high and low values

of B. For ZaP, the first constraint of Equation (2) will remain

around the value required by force balance, B∗/√

p∗ ≈ 8–

63 (depending on Z ) for all conditions and is bracketed by

estimates for HH jets. The higher value of B∗/√

p∗ results

from using a higher-Z working gas. Radiative losses cause

the plasma radius to decrease while maintaining the plasma

current. The magnetic field correspondingly increases to high

values to maintain equilibrium. An applied axial field is not

required for either equilibrium or stability, since the required

flow shear does not change if vA is constant. The second con-

straint, high Mach number, requires a colder plasma, which

can be achieved by radiative cooling of high mass working

gases. “Radiative collapse” from bremsstrahlung is not re-

quired, nor is it predicted for ZaP current levels. Many experi-

mental “knobs” on ZaP allow exploration of these constraints

(e.g. working gas, gas puff timings, current waveforms, etc.),

producing a variety of subsonic, supersonic, magnetized, and

unmagnetized jets.

5. Summary

The production and evolution of shocks produced in ZaP by

plasma jets can be studied in a variety of ways with ZaP’s

diagnostic suite. The spectrometer can be placed at angles of

0, 35, and 90 to the Z -axis and can discriminate between

the jet and the shock through their distinct emission lines,

to allow both spatial resolution of velocities and their time

evolution. The bow shock itself can be imaged with Imacon

framing or streak photographs (using appropriate line filters),

ruby laser holography, or schlieren imaging. Internal jet and

shock densities can be time-resolved with multi-chord inter-

ferometry, PDA tomography (with 2 of the PDAs at 90 to

the Z -axis) and with Langmuir probes. Internal magnetics

can be time-resolved with magnetic field probes.

Springer


These diagnostics could provide detailed information of

the bow shock and jet structure, and be compared with and

used to refine computations. These results for the specific as-

trophysical jet configuration under study will be compared to

the astrophysical jet observations, theory, and modeling, to

identify further laboratory experiments and astrophysical jet

observations to be made. This creates a “feedback loop” of

astrophysical jet observations, theory, and modeling leading

to laboratory experiments (and computational comparisons)

which then suggest new observations to be made, modifica-

tions and refinement of the theory, and further modeling.

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Shumlak, U., Hartman, C.W.: Sheared flow stabilization of the m =1 kink mode in Z pinches. Phys. Rev. Lett. 75(18), 3285–3288(1995)

Shumlak, U., Nelson, B.A., Golingo, R.P., Jackson, S.L. et al. Shearedflow stabilization experiments in the ZaP flow Z-pinch. Phys. Plas-mas 10(5), 1683–1690 (2003)

You, S., Yun, G.S., Bellan, P.M.: Dynamic and stagnating plasma flowleading to magnetic-flux-tube collimation. Phys. Rev. Lett. 95,045002–1–4 (2005)

Springer


DOI 10.1007/s10509-006-9206-9


Double-Pulse Laser-Driven Jets on OMEGA

S. Sublett · J. P. Knauer · I. V. Igumenshchev ·

A. Frank · D. D. Meyerhofer


Abstract A double-pulse laser drive is used to create

episodic supersonic plasma jets that propagate into a low

density ambient medium. These are among the first laser

experiments to generate pulsed outflow. The temporal laser-

intensity profile consists of two 1-ns square pulses separated

by 9.6 ns. The laser is focused on a truncated conical plug

made of medium Z material inserted into a high-Z washer.

Unloading material from the plug is collimated within the

cylindrical washer hole, then propagates into the low-Z foam

medium. The resulting jet is denser than the ambient medium.

Double-pulse jet evolution is compared to that driven by a

single laser pulse. The total drive energy is the same for both

jets, as if a source with fixed energy generated a jet from

either one or two bursts. Radiographs taken at 100 ns show

that a single-pulse jet was broader than the double-pulse jet,

as predicted by hydrodynamic simulations. Since the initial

shock creating the jet is stronger when all the energy arrives in

a single pulse, the jet material impacts the ambient medium

with higher initial velocity. Detailed comparisons between

S. Sublett ()Laboratory for Laser Energetics, Rochester, NY 14623;Department of Physics and Astronomy, University of Rochester,14627e-mail: [email protected]

J. P. Knauer · I. V. IgumenshchevLaboratory for Laser Energetics, Rochester, NY 14623

A. FrankDepartment of Physics and Astronomy, University of Rochester,14627

D. D. MeyerhoferLaboratory for Laser Energetics, Rochester, NY 14623;Department of Physics and Astronomy, University of Rochester,14627; Department of Mechanical Engineering, University ofRochester, 14627

single- and double-pulsed jet rheology and shock structure

are presented. 2-D hydrodynamic simulations are compared

to the experimental radiographs.

Keywords Laser experiments . Plasma jet . Episodic

outflow

PACS: 52.30.−q, 41.75.Jv, 42.62.−b, 42.68.Sq, 47.40.−x,

47.56.+r

1. Introduction

Episodic outflows are nearly as ubiquitous as outflows them-

selves in astronomical observations. Comparing double-

pulsed plasma jets with the single-pulsed plasma jets in

the laboratory provides insight into these observations. The

University of Rochester’s OMEGA laser (Boehly et al., 1997)

produces plasma jets with much higher densities than young

stellar objects (YSO) or planetary nebulae (PN), but the jet-

to-ambient density ratio (a dimensionless parameter) is 1 in

experiments, as observed in some astronomical jets. The ex-

perimental Mach numbers are approximately 3, approaching

a range relevant to astronomical jets. Table 1 shows some

ranges of density and velocities, with dimensionless param-

eters in boldface.

2. Laser setup

OMEGA laser beams are used to launch a strong shock into a

medium-Z conical plug set inside a high-Z washer. Material

unloads off the plug, flows through the washer, and forms a

jet once it gathers enough force to penetrate a low-Z ambi-

ent medium. Figure 1 shows a schematic of the target. An-

other set of OMEGA laser beams hits a backlighter target to

Springer


Table 1 Hydrodynamic similarity

OMEGA YSO PN

Jet density cm−3 1021 108 103–106

Ambient density cm−3 1021 107–109 10−4–104

Density ratio 1 0.1–10 1–1000

Jet velocity km/s 6–12 102–103 10–100

Sound speed km/s 3 10–100 1–100

Mach number 2–4 10–40 10

Fig. 1 Schematic of target showing the conical plug in gray, the washerin magenta, and fiducial grids in green and the circular field of view inred. Color images available online

radiograph the jets 100 ns after the initial laser drive. A shield

cone prevents x-rays from the region where the laser hits the

conic plug from illuminating the CCD. Figure 2 shows a

scaled 3-D image of the target.

The same total energy is deposited on the plug in both the

single- and double-pulse cases. Either 7 drive beams impact

the plug at the same time, or the beams are split into 2 pulses:

first 3 beams, then 4 beams 9.6 ns later. The pulse separation

time is large compared to the pulse duration of 1 ns. Laser

ablation pressure launches a shock wave into the plug. A

rarefaction wave is created after the laser pulse is over, and

the rarefaction overtakes the shock wave 1.8 ns later, when a

single-pulse has traveled 38 percent of the way through the

plug or the first of two pulses has traveled 29 percent of the

way through the plug. Then the shock decays as it transits

the rest of the plug.

Material unloading from the plug travels through the re-

mainder of the washer hole, and impacts the foam ambient

medium, forming a jet. In the double-pulse case the first shock

reaches the back of the plug and begins to unload in under

4 ns, well before the second shock enters the plug. A region

Fig. 2 Target rendering by VISRAD. The shield cone appears on goldmesh, the foam medium in turquoise, and the fiducial grids in green.Color images available online

of the ambient medium between fiducial grids is backlit with

a point-projection x-ray source whereby the jet is imaged

onto a non-gated time-integrated vacuum x-ray CCD detec-

tor inside a Spectral Instruments series 800 camera (Spectral

Instruments, www.specinst.com).

3. Jet images

Figure 3 shows radiographs of single- and double-pulsed jets

created from the same type of target 100 ns after the first

laser pulse. The 7-beam single laser pulse is more than twice

as intense as the 3-beam initial laser pulse of the double-

pulse jet. Since ablation pressure is proportional to the 2/3

power of intensity (Lindl, 1998), the initial ablation pressure

of 43 Mbar is 1.5 times stronger for the single pulse jet. The

strong shock propagated through the plug is therefore over

20 percent faster for the single pulse jet. The single-pulse jet

is also broader than the double-pulse jet.

Figure 4 shows another double-pulse jet, also at 100 ns, but

with a lower energy backlighter that probes less deeply into

the core of the jet. The entire length of the jet is not visible,

but the radius, bow shock, and head of the jet is similar to

the jet in Fig. 3(b).

Figure 5 shows simulations for the single- and double-

pulse jets produced in the experiments. The concave mor-

phology of the single-pulse jet head is distinguishable

in Figs. 3(a) and 5(a) from the more lenticular head in

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Fig. 3 Experimentalradiographs of single-pulse jet(a) and double-pulse jet (b) at100 ns

Fig. 4 Double-pulse jet at 100 ns backlit with lower energy than thejet in Fig. 3(b)

Figs. 3(b), 4, and 5(b). The shocks inside the cocoons are

not do differ in simulations and experiments. The rela-

tive jet lengths were not established since the full lengths

of the experimental jets could not be seen past the lip

of the shield cone, visible in Fig. 2. The single-pulse jet

has the greatest density in its Mach disk in both exper-

iments and simulations. The double-pulse jet has a re-

gion as dense as the Mach disk at the base of the jet in

both simulations and experiments, evidence of the second

pulse.

4. Image analysis

Subtracting a dark image from each jet image removes expo-

sure time dependent dark noise and DC offset pixel values.

Flatfielding removes the spatially dependent CCD response.

Taking the flatfield, dark image, and jet image all at the same

exposure time means that the bias is built into the dark im-

age, and no scaling has to take place before the dark image

is subtracted from the flatfield or jet image. Each jet im-

age has the known transmission through the foam removed

so that only the jet areal density remains. Wiener filter-

ing with a Lucy-Richardson algorithm removes large scale

features and makes shock fronts more clear, as shown in

Fig. 6.

5. Conclusions

Episodic hydrodynamic outflows generated by the OMEGA

laser can be used to refine simulations. Two laser drive con-

ditions generated single- and double-pulse jets by unload-

ing material from a moderate-Z plug in a high-Z washer

into a low-Z foam ambient medium. The ratio of jet ve-

locity to the speed of sound in the ambient foam gives

a Mach number of approximately 3. 2-D axisymmetric

DRACO simulations (Keller et al., 1999) predict the shape

of the head of the jet well. The simulated deposition of

laser energy will be refined to better reproduce experi-

mental results, especially to reduce the overestimated ra-

dial expansion of the jets. Targets will be refined so that

the full length of each jet can be observed and the rela-

tive lengths can be compared to predictions. The observ-

able features of these jets will add to the parameter space

Springer


Fig. 5 Simulated radiographsof single-pulse jet (a) anddouble-pulse jet (b) at 100 ns

Fig. 6 Wiener-filtered image of the single-pulse jet showing enhance-ment of shocks compared to Fig. 3. Note the phase contrast visible onthe outer shock

achieved by experimental jets and refine the scaling of ex-

periments to observed astronomical objects. Future work

can study the effects of 3-D structure on the stability of the

jet.

Acknowledgements This work was supported by the U.S. Departmentof Energy Office of Inertial Confinement Fusion under CooperativeAgreement No. DE-FC52-92SF19460, the University of Rochester, andthe New York State Energy Research and Development Authority. TheSupport of the DOE does not constitute an endorsement by DOE of theviews expressed in this article.

References

Boehly, T.R., Brown, D.L., Craxton, R.S., Keck, R.L., Knauer, J.P.,Kelly, J.H., Kessler, T.J., Kumpan, S.A., Loucks, S.J., Letzring,S.A., Marshall, F.J., McCrory, R.L., Morse, S.F.B., Seka, W.,Soures, J.M., Verdon, C.P.: Initial performance results of theOMEGA laser system. Opt. Comm. 133, 495–506 (1997)

Keller, D., Collins T.J.B., Delettrez, J.A., McKenty, P.W., Radha, P.B.,Town, R.P.J., Whitney, B., and Moses, G.A.: DRACO – A newmultidimensional hydrocode. Bull. Am. Phys. Soc. 44, 37 (1999)

Lindl, J.D.: Inertial confinement fusion. Springer-Verlag, New York(1998)

Smalyuk, V.A., Yaakobi, B., Marshall, F.J., Meyerhofer, D.D.: X-rayspectroscopic measurements of areal density and modulations ofcompressed shells in implosion experiments on OMEGA. LLERev. Q. Rep. 83, 124–129 (2000)

Spectral Instruments, Tuscon AZ 85745 (www.specinst.com)

Springer


DOI 10.1007/s10509-006-9258-x


Laboratory Modeling of Standing Shocks and Radiatively CooledJets with Angular Momentum

D. J. Ampleford · S. V. Lebedev · A. Ciardi · S. N. Bland · S. C. Bott · G. N. Hall ·

N. Naz · C. A. Jennings · M. Sherlock · J. P. Chittenden · A. Frank · E. Blackman

Received: 16 May 2006 / Accepted: 19 September 2006C© Springer Science + Business Media B.V. 2006

Abstract Collimated flows ejected from young stars are be-

lieved to play a vital role in the star formation process by

extracting angular momentum from the accretion disk. We

discuss the first experiments to simulate rotating radiatively

cooled, hypersonic jets in the laboratory. A modification of

the conical wire array z-pinch is used to introduce angular

momentum into convergent flows of plasma, a jet-forming

standing shock and into the jet itself. The rotation of the jet

is evident in laser imaging through the presence of discrete

filaments which trace the rotational history of the jet. The

presence of angular momentum results in a hollow density

profile in both the standing conical shock and the jet.

Keywords Herbig-haro objects . Laboratory . Stars .

Winds . Outflows

1 Introduction

Convergent flows and plasma jets are ubiquitous in as-

trophysics; active galactic nuclei, protostars and planetary

D. J. Ampleford () · C. A. JenningsSandia National Laboratories, Albuquerque, NM 87123-1106,USAe-mail: [email protected]

S. V. Lebedev · S. N. Bland · S. C. Bott · G. N. Hall · N. Naz ·M. Sherlock · J. P. ChittendenBlackett Laboratory, Imperial College, London SW7 2BW, UK

A. CiardiObservatoire de Paris, LUTH, Meudon, 92195, France

A. Frank · E. BlackmanDepartment of Physics and Astronomy, University of Rochester,Rochester, NY, USA; Laboratory for Laser Energetics, Universityof Rochester, Rochester, NY, USA

nebulae each have associated outflows (Begelman et al.,

1984; Reipurth and Bally, 2001; Balick and Frank, 2002).

Many of these jets are likely to contain angular momentum

– recent observations of the DG Tauri and other protostellar

jets (Coffey et al., 2004; Bacciotti et al., 2002) have indi-

cated azimuthal velocities ∼10 km/s (compared to an axial

velocity vz ∼ 300 km/s). The presence of angular momen-

tum in these jets is likely to be significant to the dynamics

of the jets and the source object. Specifically for the case of

protostars, the extraction of angular momentum by a jet is

widely believed to be necessary to allow the accretion of ma-

terial from the disk onto the central star. The ability to model

these rotating jets in the laboratory can aid in understanding

these jets provided that the laboratory experiment can reach

the appropriate regime of scaled parameters to make astro-

physical connections (Ryutov et al., 1999; Remington et al.,

2005).

In this paper we discuss a laboratory technique to pro-

duce rotating convergent plasma flows and we will discuss

experimental evidence for rotation of the standing shocks

and highly supersonic jets produced. We will also make a

preliminary comparison between the results presented and

protostellar jets.

2 Experimental configuration

To produce rotating convergent flows, standing shocks and

jets we use a modification of the conical wire array z-pinch

(Lebedev et al., 2002; Ciardi et al., 2002; Lebedev et al.,

2005a). The jets produced in these arrays have been shown

to have certain characteristic dimensionless parameters in the

regime needed for the scaling of protostellar jets. Specifically,

they are highly supersonic and radiatively cooled (Lebedev

et al., 2002), and experiments can be designed such that the

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Fig. 1 Time gated XUV (hν > 30 eV) self emission images taken end-

on to the array (a–d, at ∼210 ns) and interferometry images (e–f, at344 ns and 319 ns respectively). Conical arrays of 16, 18µm W wireswith 30 wire inclination angles are shown with (a, e) no twist presentand with a twists of (b) 2π /64 (c) 2π /32 and (d,f) 2π /16 between theelectrodes. The wires are seen in the outer portion of each image, the

precursor streams are seen between these wires and the central conicalshock. The insets in images (a–d) are soft x-ray emission (hν > 220 eV)from the conical shock for the same twist angles, on the same scale. (g)shows the size of conical shock measured on (b–d), with error barsindicating the imaged thickness of the wall of the shock

jet propagates in an ambient medium (Lebedev et al., 2004;

Frank et al., 2005; Ampleford et al., 2005a). The current pulse

from the MAPGIE pulsed-power generator (1MA, 250 ns,

Mitchell et al., 1996) drives a conical array of fine tungsten

wires. The self-generated global azimuthal field and the pres-

ence of axial and radial components of current through the

wires results in continuous ablation and acceleration of pre-

cursor plasma flows by the J × B force in the radial (inward)

and axial (upward) directions (see Fig 1e). To introduce ro-

tation into this system the two electrodes are twisted with

respect to each other (see Fig. 1 and f). The azimuthal cur-

rent that is now present introduces an axial magnetic field.

This results in an additional component to the Lorentz force

in the azimuthal direction (Fθ = Jr Bz). The magnitude of

angular momentum introduced into the system can be di-

rectly controlled by adjusting the rotation angle between the

electrodes.

The precursor plasma flows meet on the axis of the wire

array producing a standing conical shock. This shock ther-

malizes the kinetic energy associated with radial motion, re-

sulting in an essentially axial flow. A pressure gradient at

the top of the shock accelerates the material; radiative cool-

ing enables the production of a highly supersonic, highly

collimated flow. There is no mechanism to remove the net

angular momentum from the material as it enters the conical

shock and jet; hence it is expected that the shock and jet will

rotate.

3 Rotating standing shock

Figure 1 shows experimental data for conical wire arrays

without and with a twist present (i.e. where angular momen-

tum is not present and where it is expected to be present

respectively). The upper images (Fig. 1a–d) are XUV emis-

sion images taken end-on to the array (looking down the

array axis) for no twist, and with twist angles between the

electrodes = 2π/64, 2π /32, 2π /16. The lower images

(Fig. 1e–f) are shadowgraphy images taken side-on to the

array, with no twist and with a twist of = 2π/16.

In each of the XUV images there is a bright area at the

centre of the image corresponding to the thermalization of

kinetic energy in the standing shock. The insets in these im-

ages are soft x-ray emission (hν > 220 eV), which show that

only this central region is emitting at these higher energies.

This shock is seen on the shadowgraphy images as a dark

region on the array axis where the laser beam is refracted

out of the imaging system by the electron density gradients

(some beam penetrates the centre of the shock in the twisted

case). All the diagnostics show that with increased twist an-

gle, and hence angular momentum, the shock becomes hol-

low and the size of this shock increases (see Fig. 1g). This is

consistent with an increase in the centrifugal force as the an-

gular momentum is increased. It can be shown (Ampleford,

2006a) that the diameter of the hollow shock is consistent

with the balance between the centrifugal force on the rotating

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Fig. 2 Schlieren images of jetsfrom conical wire arrays. (a)shows filament like structures,which rotate around the jet for atwist angle of ∼2π /64. (b–d) areschlieren images of the base ofjets with no twist (b) and withtwists of (c) 2π /64 and (d)2π /16. All images are att = 315–340 ns after start ofcurrent

column and ram pressure from the continuing convergent

flows. This mechanism is also consistent with hybrid code

simulations performed by Sherlock (2003). Experiments us-

ing twisted cylindrical wire arrays, where no angular mo-

mentum is present, demonstrate that the pressure associated

with the magnetic field present on the axis of the array has

a negligible effect on the shock dynamics (Ampleford et al.,

2006a).

Analytic estimates of the shock parameters, MHD simula-

tions and hybrid code simulations each indicate a rotation ve-

locity in the setups shown in Fig. 1 & f of v > 40 km/s (simi-

lar to the axial velocity of the material in the shock). The max-

imum estimated temperature in the shock of 50 eV and charge

state Z ∼ 13 indicates a sound speed of ∼20 km/s, hence the

rotating flow within the standing shock is supersonic. The

density of material in the shock is expected to increase with

time as mass is accumulated from the incident flows.

4 Rotating jet

The standing shock in a conical array produces a highly su-

personic jet, which is surrounded by a lower density halo of

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Fig. 3 Interferograms of thebase of jets with no twist (a) andwith twists of (b) ∼2π /64 and(c) 2π /16. All images are att = 315–340 ns after start ofcurrent. (d) shows radialelectron density profiles fortwists of 2π /64 (dotted) and2π /16 (solid), obtained by Abelinversion of the interferometerimages (the error margin is largeinside the sharp density jumpdue to performing the Abelinversion). Abel inversion of theuntwisted case is not possiblydue to the large density gradientpresent; on the scale of the otherimages this profile is effectivelya δ-function

plasma (Lebedev, 2005a; Ampleford, 2005b). If the shock

that produces this jet is rotating and there is no mechanism

for the plasma to dissipate this angular momentum, then it

is expected that the jet which emerges will contain angular

momentum. Rotation within the central portion of such a jet

can be seen experimentally in Fig. 2a, which is a shadowg-

raphy image of a jet produced by an array with a relatively

small twist angle ( = 1/64, the same setup as Fig. 1b).

In the image two filament-like structures are seen, which

form a double-helix shape. This filamentary structure might

be linked to the discrete nature of the convergent plasma

streams, or an instability in the flow. When the jets from

untwisted conical arrays exhibit similar filamentary struc-

ture no rotation is seen. If it is assumed that these filaments

have a source point at the top of the conical shock which is

static in time, the angle of the filaments can be used to es-

timate the rotation rate of the jet. Where these two streams

cross at the centre of the image (marked point ‘A’) we mea-

sure the pitch angle to be θ = 10 deg. This implies a ratio

vθ/vz = 0.18, or assuming that the axial flow velocity at this

point is 200 ± 50 km/s (Lebedev et al., 2002; Ciardi et al.,

2002; Ampleford, 2005b) we estimate a rotation velocity

vθ ∼ 35 ± 9 km/s.

Figures 2b–d shows jets produced by arrays with three

different twist angles; the jet becomes wider as the angular

momentum in the system is increased. Comparing the un-

twisted case and the largest twist angle (Fig. 2b and d) it is

clear that the introduction of angular momentum has caused

the main jet to become divergent.

Results with this larger twist angle indicate that the halo

around the jet is likely to be rotating. Fig. 3c shows an

interferometer image of the jet for a setup with a twist an-

gle of 2π /16. At the right of the jet near the base of the

image a filament is again present illustrating the rotation of

the central jet (also seen on the shadowgraphy image, Fig

2d); the angle of this filament indicates a rotation velocity

of ∼75 km/s. From comparison to a pre-shot reference in-

terferogram (i.e. without plasma present) the position of the

Springer


outermost boundary of the plasma can be determined. The

approximate position of this plasma-edge as the flow propa-

gates upwards has been marked on the image (white line). As

there is no material outside the halo which surrounds the jet,

the halo should freely expand at its rotation velocity. From

the image we find that the outermost plasma is at a radius of

3.3mm and the velocity of this expansion vexp ∼ 60 km/s (the

thermal velocity cs ∼ 5 km/s is negligible in comparison).

From the interferometer images in Fig. 3 it is possible to

reconstruct the electron density profile of the jet. Figure 3d

shows Abel inversions of the phase difference through the ax-

ial positions indicated on Fig. 3b and c. The density profile

shows that the jet produced is in fact hollow. Furthermore,

with the larger twist there is a very sharp outer edge to the

dense portion of the flow. This shock-like structure is likely

to be formed as the expanding central portion of the jet snow-

plows material as it expands. Additional work is required to

investigate this further.

5 Conclusions and connections to astrophysical jets

Angular momentum is believed to be an integral part of the

dynamics of proto-stellar jets. We have demonstrated a new

technique to introduce angular momentum into radiatively

cooled, highly supersonic jets in the laboratory. The exper-

imental data indicates that the presence of rotation signif-

icantly affects the dynamics of both the jet forming shock

and the jet, most notably producing a hollow density profile.

The ability to introduce angular momentum in laboratory

jet experiments allows the investigation of a large parameter

space; however the connection to astrophysical jets requires

certain scaling criteria to be satisfied, such as the hydrody-

namic scaling relations described by Ryutov et al. (1999).

Providing that the portion of the jet of interest is far from the

source star, the effects of the magnetic field can be neglected

(Hartigan et al., 2003). For radiatively cooled, supersonic jets

some of the criteria can be summarized by the Mach number,

the cooling parameter and the density ratio between the jet

and the ambient medium (Blondin et al., 1990). Angular mo-

mentum is represented in the scaling relations developed by

Ryutov et al. (1999) within the velocity vector. If it is known

that the axial velocity is scaling correctly between the labora-

tory and protostellar jet (as defined in the Mach number of the

flow), then the ratio of the axial to azimuthal velocities is suf-

ficient to define the angle of the flow, and hence the angular

momentum. In the experiments described here the ratio of

azimuthal and axial velocities measured is ∼0.1–0.4, which

is of a similar magnitude to that observed in astrophysical

jets (Coffey et al., 2004; Bacciotti et al., 2002). That being

said, we emphasize that because the flows in our jets are su-

personic and essentially hydrodynamic, the rotating jets of

our experiments are most relevant for the propagation regime

in astrophysical jets rather than the launch regime where the

magnetic fields likely dominate the flows (Blackman, 2006).

Varying the array parameters in the experiments (e.g. diame-

ter, length, and twist angle) allows control of the ratio of the

axial and azimuthal velocities. The laboratory data shows

that different radial positions of the jet are rotating; it may

be possible to modify the array configuration to allow some

control of this radial angular momentum profile. Such con-

trol would be required to attempt a laboratory representation

of the azimuthal velocity profile thought to be present in an

astrophysical jet.

The experiments described here have been performed with

no ambient medium surrounding the jet. Previous experi-

ments have demonstrated the ability to propagate jets from

conical wire arrays (without angular momentum) in an am-

bient medium that is static (Ampleford et al., 2005a) or has

transverse momentum (Lebedev et al., 2004, 2005a; Ample-

ford et al., 2006b). Interesting topics that could be studied

in the future include the angular momentum exchange be-

tween a jet and an ambient medium, and the effect of an-

gular momentum on the deflection of jets. Similarly, the ca-

pability to use an equivalent technique to introduce angular

momentum into laboratory jets where magnetic fields are

dynamically significant will allow an investigation of the ef-

fect of rotation on magnetically driven launch mechanisms

(for example using either radial wire arrays (Lebedev et al.,

2005b) or imploding conical wire arrays (Ampleford et al.,

2006c).

Future experiments would greatly benefit from directly

measuring the radial distribution of the azimuthal velocity

(along with the mass density) in the shock and jet, for example

using spectroscopic techniques, with the aim of developing

a fuller understanding of the angular momentum distribution

within the system.

Acknowledgements The authors would like to thank Dr D.D. Ryutovfor useful discussions on the scaling of rotating jets, Prof M.G. Hainesfor useful discussions on the nature of the filaments observed in thejet and Dr G.S. Sarkisov for assistance with one of the Abel inversionsshown in Fig. 3d. This research was sponsored by the NNSA underDOE Cooperative Agreement DE-F03-02NA00057 and in part by theEuropean Community’s Marie Curie Actions – Human resource andmobility within the JETSET (Jet Simulations, Experiments and Theory)network under contract MRTN-CT-2004 005592. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the US DOE’s National Nuclear Security Administrationunder contract DE-AC04-94AL85000.

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vergent plasma flows and radiatively cooled jets, in preparation(2006a)

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Ampleford, D.J., et al.: Jet deflection by a quasi-steady-state side windin the laboratory. Astrophys. Space Sci. DOI 10.1007/s10509-006-9238-1 (2006b)

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of Astrophysical Jets and the Role of Experiments’ in this issue(2006)

Blondin, J.M., et al.: Astrophys. J. 360, 370 (1990)Ciardi, A., et al.: Laser Part. Beams 20, 255 (2002)Coffey, D., et al.: Astrophys. J. 604, 758 (2004)Frank, A., et al.: Astrophys. Space Sci. 298, 107 (2005)

Hartigan, P.: Astrophys. Space Sci. 287, 111 (2003)Kato, Y., et al.: Astrophys. J. 605, 307 (2004)Lebedev, S.V., et al.: Astrophys. J. 564, 113 (2002)Lebedev, S.V., et al.: Astrophys. J. 616, 988 (2004)Lebedev, S.V., et al.: Plas. Phys. Contr. Fus. 47, B465 (2005a)Lebedev, S.V., et al.: MNRAS 361, 97 (2005b)Lynden-Bell, D.: MNRAS 341, 13 (2003)Mitchell, I.H., et al.: Rev. Sci. Instruments 67, 1533 (1996)Reipurth, B., Bally, J.: Ann. Rev. Astron. Astrophys. 39, 403 (2001)Remington, B.A., et al.: Plas. Phys. Contr. Fus. 47, A191 (2005)Ryutov, D.D., et al.: Astrophys. J. 518, 821 (1999)Sherlock, M.: Ion Collisional Effects in Z-Pinch Precursor Plasma and

Laboratory Astrophysical Jets, PhD thesis, University of London(2003)

Springer


DOI 10.1007/s10509-006-9268-8


Numerical Simulations and Astrophysical Applicationsof Laboratory Jets at Omega

R. F. Coker · B. H. Wilde · J. M. Foster · B. E. Blue ·

P. A. Rosen · R. J. R. Williams · P. Hartigan · A. Frank ·

C. A. Back

Received: 28 April 2006 / Accepted: 17 October 2006C© Springer Science + Business Media B.V. 2006

Abstract We have conducted experiments on the Omega

laser at the University of Rochester that have produced jets

of supersonic Ti impacting and being deflected by a ball of

high density plastic. These mm-sized jets of dense plasma are

highly complex, have large Reynolds numbers, and, given

sufficient time and shear, should produce a fully turbulent

flow. The experiments are diagnosed with a point-projection

backlighter, resulting in a single image per shot. Simulations

of the 3D hydrodynamics capture the large-scale features

of the experimental data fairly well while missing some of

the smaller scale turbulent-like phenomena. This is to be ex-

pected given the limited characterization of the targets as well

as the finite resolution of the 3D simulations. If Euler scaling

holds, these experiments should model larger astrophysical

jets in objects such as HH 110 where an outflow can be seen

colliding with a molecular cloud. However, Euler scaling de-

mands that not only the isothermal internal Mach numbers of

the two systems be similar but also that any dissipative mech-

J. M. Foster · P. A. Rosen · R. J. R. WilliamsAWE, Aldermaston, Reading, UK

R. F. Coker () · B. H. WildeLos Alamos National Laboratory, Los Alamos, NM

B. E. BlueLawrence Livermore National Laboratory, Livermore, CA

P. HartiganDepartment of Physics and Astronomy, Rice University, Houston,TX

A. FrankDepartment of Physics and Astronomy, University of Rochester,Rochester, NY

C. A. BackGeneral Atomic, Inertial Fusion Group, San Diego, CA

anisms, such as radiative cooling or viscous dissipation, be

of equal importance relative to each other. Similar equations

of state are required as well. We discuss such issues in the

context of these experiments and simulations.

Keywords Hydrodynamics . ISM: Herbig-Haro objects .

ISM: Jets and outflows . Methods: Laboratory

1 Background

HH 110 (the left image in Fig. 1) is an astrophysical jet

roughly a parsec in size. Our experiments on Omega (the

right image in Fig. 1) are a few mm in size. What can we learn

about the former from the latter? Strict scaling arguments (see

below) show that under certain conditions the two systems

behave the same way although they are nearly 20 orders

of magnitude different in size. Thus, behavior seen in one

should be applicable to the other. In addition, validation of

codes used to model one system successfully should result

in codes that can be applied to the other equally well; the

better the scaling, the more confidence one has in applying

a code to the other problem. As such, these experiments are

part of a large validation program spanning many national

laboratories and the astrophysical community. In this work,

we focus on simulations of the LANL/SAIC code RAGE,

a radiative hydrodynamics Eulerian code with continuous

adaptive grid refinement that uses a Godunov scheme with

implicit 2T hydrodynamics. The experiments have been very

successful in that they have shown clearly what the codes

can and cannot do. For example, the experiments pointed

out a temporary code issue with shocks converging at r = 0

in RZ co-ordinates. Without detailed knowledge of initial

conditions or a turbulence model, the codes do not reproduce

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Fig. 1 Images of deflected jets.The left image is an HSTobservation of HH110 (Reipurthet al., 1996) while the rightimage is a Zn radiograph fromour experiments on Omega

Fig. 2 The experimental setup.Lasers enter the hohlraum frombelow, resulting in a pressuredrive on the Ti disc. A jet of Tiforms and enters the RF foam. Abacklighter (not shown)perpendicular to the target isused to illuminate the foam. Atransmission radiograph is thencaptured on film

some of the small-scale details of the experiments. However,

the larger scale features are captured quite well (see below).

Figure 2 shows a schematic of the experimental setup. We

use indirect drive, where a number of laser beams (here, 12

beams at 450 J each) are directed into a gold hohlraum. The

laser beams radiatively ablate the gold and plasma accumu-

lates in the hohlraum. The resulting pressure build-up in the

hohlraum drives the target, a cap and washer of TiAlV alloy.

A shock enters the Ti cap, breaks out into the ‘free-run’ region

(the hole in the Ti washer), and produces a jet of material that

enters a foam. A few additional laser beams are directed to-

wards a backlighter pinhole target made of Zn (or Fe in some

cases). The pinhole focuses the resulting 4 to 7 keV photons

so that they illuminate the cylindrical 100 mg/cc RF target

foam. Finally, a transmission radiograph image is captured

on film by a camera. Each experiment or ‘shot’ results in a

single image. A gold grid with a plastic coating is included

on each shot for target registration and resolution modeling.

A gold shield is used to minimize the number of photons

from the drive side of the target that reach the film while a

gold washer or ‘cookie cutter’ is included to minimize direct

shocking of the Ti washer. In some shots, a normal density

plastic ball is placed as in obstacle in the foam. Codes that

include laser drive physics (e.g. NYM at AWE) were used

to match DANTE time-resolved observations of the apparent

brightness temperature of the hohlraum wall from previous

experiments with a very similar drive profile (Foster et al.,

2002). The profile used by RAGE in the models presented

here is shown in the left plot in Fig. 3. The profile corresponds

to the modeled “air” temperature (although the targets are

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Fig. 3 A plot of the temperature profile used to drive the target (left).The nominal laser pulse is a square 1 ns pulse with a peak drive temper-ature of ∼180 ev. Note that the temperature source (used in the RAGEsimulations) is not quite the same as the profile seen by DANTE forreasons such as hole closure (Foster et al., 2002). The right figure shows

the KH instability growth rate using typical values for the experiments.Both linear (dotted) and nonlinear (solid) growth rates are shown. Thewavelengths of perturbations in the targets correspond to wavenum-bers (k) that are larger than the maximum unstable wavenumber so KHgrowth is not likely to occur

shot in a vacuum, the simulations use low-density air since

RAGE requires non-zero starting densities) in the middle of

the hohlraum that is required by RAGE to get the observed

hohlraum wall brightness temperature.

2 Instabilities

Kelvin-Helmholtz (KH) instabilities do not appear in these

experiments. If one assumes a finite velocity gradient across

an interface such that over a scale δ there is a velocity

change V, one gets a KH growth rate like that shown in

the right figure in Fig. 3. The figure shows the growth rate,

in revolutions of growth per unit time, for δ = 5 µm and

V = 5 × 106 cm/s, typical values for these experiments.

Also shown is the linear growth rate that corresponds to

an infinite velocity gradient (Chandrasekhar, 1961). The

minimum wavelength (λ = 2π/k) required for KH growth

is λmin = 50 µm while the most rapid growth occurs at

λpeak = 80 µm. Thus, with λinitial ∼1 µm, KH is not initially

important for these experiments. Note that the wavenumber

corresponding to the most rapid KH growth (∼32 revolutions

of growth over the 100 ns of the experiment) is kpeak = 0.64/

δ. However, the rate of growth for Rayleigh-Taylor (RT)

instabilities goes as√

(kg) ∼ 1 rev/ns for λ = 1 µm. Since

the target surface roughness is ∼1 µm, RT may be important

over the hundreds of ns duration of the experiments; RT is

quite possibly the cause of the ‘smoke’ seen in the right hand

image in Figs. 1 and 5. We have run simulations that included

target perturbations to investigate instability growth. To

simulate burrs on the target of roughly 1 µm, since we

need ∼8 cells across a feature, we require sub-micron

resolution. Other 3D features, such as the offset of the Au

washer relative to the free-run region, need to be included as

well. Such high-resolution 3D models are not yet feasible.

However, coarser 3D simulations with large (> 10µm)

perturbations show that, as expected, the macroscopic

features of the flow are sensitive to the initial conditions.

3 Scaling

These experiments will scale to arbitrary dimensions if

the polytropic Euler equations encompass all the relevant

physics:

ρ

(

∂v

∂t+ v · ∇v

)

= −∇ p

∂ρ

∂t+ ∇ · (ρv) = 0

∂p

∂t− γ

p

ρ

∂ρ

∂t+ v · ∇ p − γ

p

ρv · ∇ρ = 0

For a given adiabatic index, γ , the Euler equations are in-

variant under transformations that preserve the Euler num-

ber, Eu = v√

(ρ/P), where v is a velocity, ρ is density, and

P is pressure. Thus, if one assumes dissipative mechanisms

(thermal diffusion, viscosity, and radiation) are negligible,

the Euler equations contain all the relevant physics and the

experiment can be scaled to arbitrary dimensions (Ryutov

et al., 1999). The conditions for these three mechanisms to

be unimportant are vL/κ = Pe ≫ 1, ρLv/η = Re ≫ 1, and

vτ/L = χ ≫ 1, where κ is the thermal diffusivity, η is the

dynamic viscosity, τ is the radiative cooling time, and L and

v are some length and velocity scale, respectively. Strictly

speaking, the experiment can be shown to scale even if radi-

ation is non-negligible (Ryutov et al., 2001). Table 1 shows

some characteristic values for the experiments as well as for

HH 110, a Herbig-Haro object with a jet extending from a

young star system and being deflected by a large molecular

cloud (Riera et al., 2003). For all regions of the flow that are

important, the local Reynolds number, Re, is more than 105,

so turbulence may develop at late times if there is enough

shear (Robey et al., 2003); it is not clear how fully developed

turbulence affects scaling. In the experiment, which does

not have a fully ionized plasma, viscosity is determined by

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Table 1 List of characteristic values for the jet ‘flute’and bow-shock regions of the Omega experimentsand for HH 110. The local isothermal Mach number,known as the Euler number, is spatially and tempo-rally variable in the experiments, making firm scaling

to HH 110 difficult. If the functional shape of the Eu-ler number, as a function of timescale, is the samebetween the experiments and HH 110, scaling willpersist; otherwise, the experiments scale only for afinite time

Quantity Symbol Omega (core sheath) Omega (bow-shock head) HH 110

Temperature T 0.1 eV 0.06 eV 8000 K

Density ρ 3 g/cc 2 g/cc 1 × 10−21 g/cc

Pressure P 2 × 1010 dyn/cm2 2 × 1011 dyn/cm2 10−9 dyn/cm2

Fluid velocity u 10 km/s 4 km/s 300 km/s

Lengthscale L 20 µm 200 µm 500 AU

Timescale t = L√

(ρ/P) 25 ns 60 ns 50 yrs

Jet sound speed cs 2 km/s 4 km/s 10 km/s

Local Mach number M = u/

cs 5 1 30

Kinematic viscosity v ∼0.1 cm2/s ∼0.05 cm2/s ∼1019 cm2/s

Reynolds number Re ∼105 ∼105 ∼105

Peclet number Pe ∼109 ∼1010 ∼105

Fig. 4 Comparison of simulation (PETRA in blue and RAGE in yel-low) results to the experimental data. In the left figure is displacementof the bow-shock and the ‘pedestal’ (the large rounded feature at the

bottom of the data shown in the right image of Fig. 1). The right figureshows the comparison for the diameter of the bow-shock. Scatter in thedata reflects uncertainty of initial and drive conditions

the degree of coupling between ions (Clerouin et al., 1998).

In the simulations, viscosity is dominated by shock treat-

ments so one can use the sound speed, cs , to estimate Re

by (Landau and Lifshitz, 1987) ρcssh/4. In RAGE the

5 to 95% shock width (sh) is ∼7 cells, regardless of the

details of the problem, so Re = 7/4ρcsx . In that viscosity

is unimportant in both cases, one must have a numerical Re

which is also ≫1 in order for the simulations to correctly rep-

resent the experiments. To get astrophysical scaling, Re for

the experiment, the simulation, and the astrophysical object

all need to be ≫1. This holds true for these Omega exper-

iments and for HH 110 but not for coarse resolution (10 s

of µm) simulations. To get ReRAGE >∼10 in all regions of

interest, we require ∼1 µm resolution. Such high-resolution

simulations do start to capture much of the observed 3D be-

havior (Foster et al., 2005). Of course, such conclusions de-

pend on the choice of length scale; here, we are interested

in the jet ‘flute’ and so we choose the width of that feature

(∼200 µm) as our scale. Re in such a case is large enough in

the simulations to capture the relevant features of the experi-

ments. Note that Euler scaling transforms time as L√

(ρ/P),

so from Table 1 it can be seen that ∼25 ns in the experiments

is equivalent to ∼50 yrs in HH 110. Table 1 also shows that

thermal diffusion is not important in these experiments.

4 Code comparisons

Comparison of simulations by both RAGE and PETRA

to data with no plastic ball is shown in Fig. 4. RAGE

does slightly better than PETRA on the pedestal formation,

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Fig. 5 Images of simulated (left) and experimental (right) radiographs.The circle shows the initial location of the plastic ball. Scale is in units ofmicrons. The stalk holding the ball (not included in this simulation) and

the Au grid can be seen. Due to the coarse 12 µm resolution, the ‘grass’on the pedestal is not clearly seen in the simulations. The Y -shaped‘flute’ (see Section 4) is labeled in the simulated radiograph

Fig. 6 Density contours and velocity vectors for simulations of HH 110(lower left) from de Gouveia Dal Pino (1999), velocity vectors imposedover a scaled color image of density for a simulation of the Omega

experiments (top), and simulated Zn radiographs at 100 (lower middle)and 150 ns (lower right)

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but both codes over-predict the displacement of the bow-

shock, implying the experiment has more late-time dissi-

pation than the codes. The scatter in the experiment data,

reflects shot-to-shot variability; both codes are within the

scatter for the bow-shock diameter but not for the displace-

ment of the bow-shock and the pedestal. For RAGE, new

2D to 3D linking permits better numerical resolution at early

time (during ablation) to capture more of the resulting late

time 3D features (such as the smoke and the grass seen in

Fig. 5). Figure 5 shows a comparison of a simulated RAGE

radiograph to the data. The simulation matches macroscopic

features such as bow-shock location quite well. It of course

does not reproduce small-scale features that are related to

(unknown) details of the initial conditions. We are starting

to use other codes such as FLASH (Calder et al., 2002) to

model the Omega experiments. Initial hydrodynamics-only

FLASH results, using a material energy source in the Ti plug,

with coarse (30 µm) resolution show no fine details at the jet

head but there is apparent KH behavior along the jet that is

not seen in the experiments or RAGE or PETRA simulations.

Figure 6 shows a comparison of a RAGE simulation of

the experiments to a simulation of HH 110 (de Gouveia Dal

Pino, 1999). The images show that the coarse structure of

the flow is similar for the two systems; this is expected since

both are jets colliding with and being deflected by a large

obstacle. However, there has been as yet no attempt to match

the fine details. Figure 6 also shows images of a 3D RAGE

simulation at 100 and 150 ns, illustrating how the jet evolves.

Note the fairly poor resolution (higher resolution simulations

show more 3D ‘smoky’ behavior). Observe the bow-shock

in the plastic ball, the deflection of part of the jet, and the

formation of the pedestal. The ‘flute’ formation and breakup

is particularly hard to model. There is also ‘grass’ on the

pedestal behind (or at – one cannot tell the difference in

the 2D transmission image) the edge of the Mach ring. The

grass is most likely debris being kicked up by the Mach

ring shock traveling perpendicular to the backside of the Ti

washer surface. Such features help point the way to where

better physics models are required.

5 Summary

We have developed a test bed for experiments on Omega that

can be scaled to astrophysical objects. Our present Omega

jet experiments can be well modeled by simulations us-

ing a variety of codes. These particular experiments scale

roughly to HH 110. These types of experiments help vali-

date codes so they can then be used directly on astrophysical

problems. In the future, we hope to move to a higher Mach

number, a higher aspect ratio, and perhaps a radiative jet

(where scaling will still apply). These experiments continue

to drive code improvements as well as quantitative image

analysis.

References

Calder, A.C., et al.: ApJS 143, 201 (2002)Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Ox-

ford Univ. Press, New York (1961)Clerouin, J.G., Cherfi, M.H., Zerah, G.: Europhys. Lett. 42, 37 (1998)de Gouveia Dal Pino, E.M.: ApJ 526, 862 (1999)Foster, J.M., et al.: ApJ 634, L77 (2005)Foster, J.M., et al.: Phys. Plasmas 9, 2251 (2002)Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, Vol. 6, Course of The-

oretical Physics, 2nd ed. (1987)Robey, H.F., et al.: Phys. Plasmas 10, 614 (2003)Reipurth, B., Raga, A.C., Heathcote, S.: A&A 311, 989 (1996)Riera, A., et al.: AJ 126, 327 (2003)Ryutov, D.D., et al.: ApJ 518, 821 (1999)Ryutov, D.D., Remington, B.A., Robey, H.F., Drake, R.P.: Phys. Plas-

mas 8, 1804 (2001)

Springer


DOI 10.1007/s10509-006-9251-4


Laboratory Experiment of Plasma Flow Around Magnetic Sail

Ikkoh Funaki · Hidenori Kojima · Hiroshi Yamakawa ·

Yoshinori Nakayama · Yukio Shimizu


Abstract To propel a spacecraft in the direction leaving the

Sun, a magnetic sail (MagSail) blocks the hypersonic solar

wind plasma flow by an artificial magnetic field. In order

to simulate the interaction between the solar wind and the

artificially deployed magnetic field produced around a mag-

netic sail spacecraft, a laboratory simulator was designed and

constructed inside a space chamber. As a solar wind simula-

tor, a high-power magnetoplasmadynamic arcjet is operated

in a quasisteady mode of 0.8 ms duration. It can generate a

simulated solar wind that is a high-speed (above 20 km/s),

high-density (1018 m−3) hydrogen plasma plume of ∼0.7 m

in diameter. A small coil (2 cm in diameter), which is to simu-

late a magnetic sail spacecraft and can obtain 1.9-T magnetic

field strength at its center, was immersed inside the simulated

solar wind. Using these devices, the formation of a magnetic

cavity (∼8 cm in radius) was observed around the coil, which

indicates successful simulation of the plasma flow of a Mag-

Sail in the laboratory.

Keywords Spacecraft propulsion . Magnetic sail . M2P2 .

Laboratory simulation . Magnetoplasmadynamic arcjet

I. Funaki () · Y. ShimizuJapan Aerospace Exploration Agency, Sagamihara, Kanagawa,229-8510, Japane-mail: [email protected]

H. KojimaUniversity of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan(currently, Ishikawajima-Harima Heavy Industries Co. Ltd.)

H. YamakawaKyoto University, Gokasho, Uji, Kyoto, 611-0011, Japan

Y. NakayamaNational Defense Academy, Yokosuka, Kanagawa, 239-8686,Japan

1 Introduction

In 2005, after a cruising of 28 years, it was reported that

Voyager 1 spacecraft had entered the solar system’s final

frontier where the Sun’s influence ends. The spacecraft is

now passing the termination shock where the solar wind

starts to slow down and reverse due to its first encounters

with pressure from interstellar space. Although the Voyager

successfully unveiled our solar system during its very long

travel, future exploration to the outer planets, or even beyond

the heliosphere should be conducted within several years to

make such explorations attractive.

To drastically shorten the mission trip time to deep space,

some new in-space propulsion systems are proposed. High

priority candidates are: (1) next generation ion thruster which

features high exhaust velocity of more than 50 km/s, (2) sail

propulsion utilizing the energy of the Sun, and (3) aerocap-

turing/breaking systems, which are expected to be used in

combination with high-performance ion thrusters or the sails

if you want to put an orbiter to the outer planets with atmo-

sphere, because the aerocapture system will help the orbiter to

decelerate without fuel consumption. Among the sail propul-

sion systems, solar sails are intensively studied by NASA and

other space agencies targeting at future deep space missions

(Montgomery and Johnson, 2004; Kawaguchi, 2004). Unfor-

tunately, acceleration of the solar sails is usually small due

to heavy materials used for the sail, hence it is difficult to

shorten the mission trip time in particular for the missions

within our solar system. To overcome this difficulty, a mag-

netic sail (usually abbreviated as MagSail) is proposed by

Zubrin because it is expected to achieve high thrust per weight

by capturing the momentum of the solar wind (Zubrin and

Andrews, 1991). When the MagSail is in operation, as shown

in Fig. 1,charged particles approaching the current loop (coil)

are blocked according to the B-field they experience, forming

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Magnetic Field Lines

Loop Current(coil of MagSail Spacecraft)

Magnetospheric Boundary(Magnetopause)

Solar Wind Plasma

PolarCusp

Equatorial Cross-Section

Meridional C

ross-Section

L

Magnetic Cavity

Bow Shock

a

Solar Wind Plasma

Magnetopause

Magnetosphere

B-field

(B=0)

Electron

+-

Cur

rent

InducedE-field

ChargeSeparation Region

B-field

Coil Current

Current

L

Magnetic Cavity (B 0)

rLi

Force

b

Fig. 1 Schematics of magneticsail. (a) Three-dimensionalimage of the plasma flow andmagnetosphere of MagSail. (b)A schematic illustration of thetrajectories of charged particlesat the magnetopause; ions andelectrons incident on a boundarylayer when the polarizationelectric field due to chargeseparation is present

a magnetosphere (or a magnetic cavity) around the coil cur-

rent. The solar wind plasma flow and the magnetic field are

separated by the magnetopause, at which ions entering the

magnetic field are reflected except near the polar cusp region

where the ions can enter deep into the magnetic cavity. Due

to the presence of the magnetosphere, the supersonic solar

wind flow is blocked and decelerated being accompanied by

a bow shock; a wave-drag force is then transferred to the coil

current through electromagnetic processes (Nishida et al.,

2006). Thus the spacecraft is accelerated in the direction of

the solar wind.

The idea of the MagSail, however, did not draw atten-

tion so far because the original MagSail by Zubrin was

unrealistically large spacecraft design with a large hoop

coil of 30 km in radius to form 100 km-radius blocking

area (corresponding to 20-N-class thrust). For such a large

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magnetosphere (or equivalently, a large stand-off distance,

L), fluid-like interactions depicted in Fig. 1 in analogy to the

geomagnetic field are expected. In contrast, ion or electron

kinetic movement becomes important for a smaller L below

the ion Larmor radius (rLi , defined in Fig. 1b). For the

MagSail with a large rLi/L value, coupling between the ion

flow and the magnetic field becomes loose, leading to only

negligible thrust production.

Although many theoretical works are going to study the

effect of magnetospheric size (L) on the thrust of the Mag-

Sail (Fujita, 2004; Akita and Suzuki, 2005) and its deriva-

tives (Winglee et al., 2000; Asahi et al., 2004; Khazanov

et al., 2005), such a complicated plasma flow in a transitional

regime between the MHD scale and the ion kinetic scale will

be more confidently treated by a scale-model experiment.

This paper firstly discusses the most important things, the

scaling considerations that characterize the plasma flow of

the MagSail. Our newly developed experimental simulator

of the magnetic sail is then introduced with some initial test

results, which help understanding the plasma flow structure

of the MagSail.

2 Thrust and scaling parameters of MagSail

2.1 Thrust exerting on MagSail

The force on the current loop depends on the area that

blocks the solar wind. By increasing this blocking area, a

larger thrust is obtained. Therefore, the force exerting on

the coil of the MagSail, F, can be formulated as (Funaki and

Nakayama, 2004),

F = Cd

1

2ρu2

swS (1)

where Cd is thrust coefficient, 1/2ρu2sw the dynamic pressure

of the solar wind, and S = πL2 the characteristic area of

the magnetosphere.

2.2 Definition of magnetospheric size, ion Larmor radius,

and skin depth

Because the density of the solar wind plasma flow around a

Magsail is very small, the charged particles are collision-less

and their movement separates the plasma region outside the

magnetic cavity and the region inside the magnetic cavity.

Simplified picture of this boundary is depicted in Fig. 1b.

When a magnetic dipole Md is located at the center, there

is a balance between the total internal (magnetic) and the

external (plasma) pressures at the boundary:

nmi u2sw = (2Bmp)2

2µ0

(2)

where n is the plasma number density, mi ion mass, uswthe

velocity of the solar wind, 2Bmp the magnetic flux density at

the boundary, µ0 the permeability in vacuum. The magnetic

flux density Bmp at a distance L from the dipole center is

expressed as,

Bmp = Md

4πL3(3)

hence the detachment distance of the boundary from the

dipole center, L, is derived as follows.

L =(

M2d

8µ0π2n mi u2sw

)1/6

(4)

This boundary is usually called a magnetopause, on which the

charged particles, ions and electrons, impinge. The external

space is considered as magnetic field-free. In the idealized

situation, one finds the thickness of the magnetopause is the

order of the plasma skin depth δ as

δ = c/ωp (5)

where c is the light velocity, and ωp the plasma frequency.

The thickness of the magnetopause, however, is considered

larger than δ; it is about the ion gyration radius at the mag-

netopause (Willis, 1971):

rLi = mi usw

e2Bmp

(6)

Because of their heavier mass, the ions tend to penetrate

more deeply into the magnetic field than electrons. This sets

up a charge separation, thus the outward pointing polariza-

tion field restrains the ions. Before the ions can be deflected

by the magnetic field, they are returned by this polarization

field. The electrons, however, experience the Lorentz force

and gain energy in the polarization field. The transverse ve-

locity component of the electrons accounts essentially for the

electric current in the interface, which in case of the magne-

topause is usually referred to as Chapman-Ferraro current.

From Equations (5) and (6), δ ∼ 2 km and rLi = 71 km for

the solar wind flow in Table 1.

2.3 Non-dimensional parameters

The solar wind is a super sonic plasma flow which consists

of collisionless particles. These features are described by

the Mach number, M > 1 as well as the magnetic Reynolds

number, Rm = σµ0uswL ≫ 1. Incorporating typical plasma

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Table 1 Parameters of plasma flow around MagSail

Laboratory

Parameter Space Current experiment Required scaling

Solar wind parameters

Velocity, usw 400 km/s <60 km/s

Plasma number density, n 5 × 106 m−3 1018 m−3

Electron temperature, Te 10 eV 1 eV

Ion temperature, Ti 10 eV (not available yet)

Plasma duration – 0.8 ms

MagSail (coil) parameters

Magnetic moment, Md <108 Tm3 <3×10−5 Tm3

Size of magnetic cavity (stand-off distance), L <70 km <0.1 m 0.1 m

B-field at magnetopause, Bmp 29 nT <1.9 mT

Expected thrust level, F <20 N <0.2 N

Duration of coil exciting current – 0.9 ms

Dimensionless parameters

Mach number, M 8 <3 8

Ratio of ion Larmor radius to L , rLi/L >1 1–10 >1

Ratio of electron skin depth to L , δ/L >0.03 >0.05 <1

Magnetic Reynolds number, Rm = σµ0uswL 108 <15 ≫1

velocity and temperatures of the solar wind, M ∼ 8. In addi-

tion to these two scaling parameters, we defined rLi//L , and

δ/L , hence four non-dimensional parameters in total are in-

troduced. Among them, the parameters Rm, rLi/L , and δ/L ,

are dominated by the size of the magnetosphere, which was

selected as L < 70 km in our preliminary study (see Table 1).

Corresponding non-dimensional parameters are 1 < rLi/L

(the ion gyration radius is comparable to or larger than L,

which is in contrast to the MHD scale requireing rLi/L ≪ 1

in the case of geophysics (Bachynski and Osborne, 1967; Yur

et al., 1995)) and δ/L < 1 (the skin depth is comparable to or

smaller than L). If the thickness of the magnetopause is small

enough in comparison to L, almost all of the incident ions

are reflected at the magnetospheric boundary, hence large

thrust on the coil of the MagSail is expected. Vice versa, if

the thickness of the magnetopause is much larger than L, no

interaction between the plasma flow and the magnetic field is

anticipated. We treat a transitional region between the MHD

scale (thin magnetopause mode) and the ion kinetic scale

(thick magnetopause mode) in this experiment.

3 Experimental simulation of MagSail

Our simulator consists of a high-power magnetoplasmady-

namic (MPD) solar wind simulator and a coil simulating a

MagSail spacecraft, both of which are operated in a pulse

mode. As shown in Fig. 2,the MPD arcjet is mounted on

a flange of a space chamber (2.5-m in diameter), provid-

ing up to 0.7-m-diameter plasma plume at the center of the

space chamber. A coil of 20 mm in diameter was located in

the plume of the MPD arcjet. Into the artificially produced

magnetic field by the coil, a plasma jet from the MPD arcjet

was introduced to observe possible interactions.

3.1 Solar wind and MagSail simulator

The MPD arcjet is the key device which provides a high-

density and high-speed plasma jet necessary for our MagSail

experiment. The discharge chamber of the MPD arcjet is

50 mm in inner diameter and 100 mm in length, which con-

sists of eight anode electrode rods (made of molybdenum)

that are azimuthally equally spaced, and a short 20-mm-

diameter cathode rod (thoriated tungsten); both electrodes

are insulated by boron nitride (Shimizu et al., 2005). Since

the MPD arcjet is operated in a pulse mode, a fast-acting

valve (FAV) is used to feed hydrogen gas. When the FAV is

opened, the gas in the reservoir flows through choked orifices

of 1.2 mm in diameter. The mass flow rate of hydrogen gas

was controlled by adjusting the reservoir pressure, obtaining

a gas pulse of about 11 ms duration in a rectangular wave-

form. After the gas pulse reaches its quasi-steady state, the

ignitron of a pulse-forming network (PFN1, capacitor bank

for the MPD arcjet) is triggered. PFN1 has an L-C ladder

circuit that supplies the discharge current up to 20 kA with a

0.8 ms flat-topped waveform in a quasi-steady mode. In con-

trast, PFN2 for the MagSail simulator (coil) supplies rather

small current (below 3 kA), and a 20-turn-coil is required to

produce up to 1.9 T magnetic field strength at the center of

the 2-cm-diameter coil. These pulsed devices are synchro-

nized; after the gas feed and the magnetic field by the coil

(MagSail) reach their steady states, the discharge of the MPD

arcjet is initiated.

Springer


Anode

Cathode

Vacuum Chamber

MPD arcjet (Solar Wind Simulator,SWS)

φ50mm

Plasma Plume

Coil(φ20mm)

MagSail Simulator (MSS)

φ 2

.5m

H2

GasTank

Controller

Fast ActingValve

600mm

SWS Capacitor Bank(PFN1)25kA/0.8ms

z

r

MSS Capacitor Bank(PFN2) 3kA/0.9ms

Magnetic Field

Plu

me D

iam

ete

r ~

φ0.7

m

Fig. 2 Experimental setup

Operational parameters of these devices are summarized

in Table 1, in which the temperatures and density of a plasma

jet obtained by Langmuir probes as well as the ion velocity

by the time of flight method are provided. One can see that

the MPD arcjet obtains smaller velocity and temperature in

comparison with the solar wind, but its density is quite large;

such large density is necessary to simultaneously meet the

conditions of rLi/L ∼ 1, δ/L < 1, and L ∼ 0.1 m. The last

condition (L ∼ 0.1 m) severely limits the range of the mag-

netic Reynolds number, Rm; so far we obtained Rm ∼ 15.

Rm may be increased by further increasing usw, L, or Te. Such

improvement is also required for the Mach number, because

until now we did not achieve the Mach number of the solar

wind.

3.2 Plasma flow around the coil

Figure 3 shows close-up photos of simultaneous operation

of both the solar wind simulator (MPD arcjet) and the Mag-

Sail (coil). In Fig. 3a (the case without coil current, hence no

magnetic field is present), it was found that the plasma jet

directly impinges on the coil surface and the light emission

is limited to the region only near the coil, whereas the light

emission pattern is away from the coil when the coil current,

hence the B-field, is applied (Figs. 3b and c). Expected field

structure in the case of Fig. 3c is depicted in Fig. 3d. Two light

emission regions can be identified; one is the upstream region

where the field is expected to be magnetic field free; another

is the region near the coil, in which the plasma being trapped

in the deformed magnetic field is observed. The dark region

between the two light emission regions may correspond to

the magnetospheric boundary, where the plasma flow cannot

enter but being reflected. If we assume that boundary 1 is

a bow shock, the dark region between the two boundaries

may be the magnetosheath flow hence the magnetic cavity

is bounded by boundary 2. Before identifying the structure

of the flow around the scale-mode, however, the field and

flow structure should be checked using ether optical or probe

measurement. Anyway, it is confirmed that the radius of the

magnetic cavity observed in Fig. 3c is consistent with L cal-

culated from the plasma data using Equation (4), which was

about 8 cm in this case; hence it is expected that the plasma

flow around a MagSail spacecraft was successfully scaled

down and was demonstrated in our new experimental simu-

lator.

4 Summary

An experimental simulator of a plasma flow around Mag-

Sail was designed and fabricated. A high-density plasma jet

above 1018 m−3 is supplied by a hydrogen MPD arcjet, from

which a high-speed plasma jet (20–60 km/s) is ejected. In

case a 20-mm-radius coil with 1.9 T magnetic field strength

at the center of the coil was inserted into the plasma plume,

a magnetic cavity was observed around the coil, which in-

dicated the plasma flow around the properly scaled MagSail

was experimentally simulated.

The original MagSail by Zubrin required a spacecraft with

a large coil of 30 km in radius to form 100-km-radius block-

ing area to produce 20-N-class thrust. Because the spacecraft

proposed by Zubrin will operate in the MHD scale, the ion

Larmor radius (rLi ) is smaller than the magnetospheric size

(L); such a MagSail spacecraft in the MHD scale requires

unrealistically large magnetosphere, hence huge coil, that is

impossible to build in space. In this experiment, however, we

have attempted the scale-model experiment of a smaller (ion

Springer


Fig. 3 Close-up view of a plasma flow around the Coil; MPD arcjet isoperated at a discharge current of 20 kA and a mass flow rate of 0.4 g/s;also, the coil produces (a) no magnetic field, (b) 0.2-T, and (c) 1.8-Tmagnetic field strength at the coil center. The magnetic dipole axis isnormal to the plasma flow as is expected in (d)

Lamor scale) MagSail, which is at most L = 70 km in space,

hence rLi ∼ L . In this case of moderately sized MagSail, our

preliminary probe diagnostics of the scale-model shows that

thrust coefficient Cd is approximately 0.5, which also indi-

cates that thrust production is possible by the MagSail in the

ion Larmor scale. Direct measurement of thrust exerting on

the coil is off course inevitable to demonstrate the MagSail,

which is the next step of our model experiment.

Acknowledgments We would like to thank the members of the mag-netoplasma sail research group (in particular, Dr. H. Ogawa, Dr. K.Fujita, Dr. I. Shinohara, and Mr. H. Nishida in Institute of Space andAstronautical Science (ISAS), Prof. K. Toki of Tokyo University ofAgriculture and Technology, Dr. H. Otsu of Shizuoka University, andDr. Y. Kajimura of Kyushu University) for their hearty support andvaluable advices. This research is supported by the Grant-in Aid forYoung Scientist (B) (No. 15760594) by Japan Society for Promotion ofScience, and by the Inamori Grant Program of the Inamori Foundation.The supports by the engineering committee as well as the space plasmacommittee of ISAS are also appreciated.

References

Akita, D., Suzuki, K.: Kinetic analysis on plasma flow of solar windaround magnetic sail. 36th AIAA Plasmadynamics and LasersConference, AIAA 2005-4791, Toronto, June (2005)

Asahi, R., Funaki, I., Fujita, K., Yamakawa, H., Ogawa, H., Nonaka,S., Sawai, S., Nishida, H., Nakayama, Y., Otsu, H.: Numericalstudy on thrust production mechanism of a magneto plasma sail.AIAA, 40th AIAA/ASME/SAE/ASEE Joint Propulsion Confer-ence, AIAA 2004-3502, Florida, July (2004)

Bachynski, M.P., Osborne, F.J.F.: Laboratory geophysics and astro-physics. In: Anderson, T.P., Springer R.W., (eds.) Advances inPlasma Dynamics. Northwestern University Press (1967)

Fujita, K.: Particle simulation of moderately-sized magnetic sails. J.Space Technol. Sci. 20(2), 26–31 (2004)

Funaki, I., Nakayama, Y.: Sail propulsion using the solar wind. J. SpaceTechnol. Sci. 20(2), 1–16 (2004)

Kawaguchi, J.: A solar power sail mission for a Jovian Orbiter andTrojan asteroid flybys. 55th International Astronautical Congress,IAC-04-Q.2.A.03, Vancouver, October (2004)

Khazanov, G., Delamere, P., Kabin, K., Linde, T.J.: Fundamentals of theplasma sail concept: Magnetohydrodynamic and kinetic studies.J. Propulsion Power 21(5), 853–861 (2005)

Montgomery, E.E., Johnson, L.: The development of solar sail propul-sion for NASA science missions to the inner solar system.AIAA-2004-1506, 45th AIAA/ASME/ASCE/AHS/ASC Struc-tures, Structural Dynamics & Materials Conference, Palm Springs,April (2004)

Nishida, H., Ogawa, H., Funaki, I., Fujita, K., Yamakawa, H.,Nakayama, Y.: Two-dimensional magnetohydrodynamic simula-tion of a magnetic sail. J. Spacecraft Rockets 43(3), 667–672(2006)

Shimizu, Y., Toki, K., Funaki, I., Kojima, H., Yamakawa, H.: Develop-ment of magnetoplasmadynamic solar wind simulator for Magsailexperiment. 29th International Electric Propulsion Conference,IEPC-2005-201, Princeton, October–November (2005)

Willis, D.M.: Structure of the magnetopause. Rev. Geophys. SpacePhys. 9(4), 953–985 (1971)

Winglee, R.M., Slough, J., Ziemba, T., Goodson, A.: Mini-magnetospheric plasma propulsion: tapping the energy of the solarwind for spacecraft propulsion. J. Geophys. Res. 105(A9), 21067–21077 (2000)

Yur, G., Rahman, H.U., Birn, J., Wessel, F.J., Minami, S.: Laboratoryfacility for magnetospheric simulation. J. Geophys. Res. 100(A12),23727–23736 (1995)

Zubrin, R.M., Andrews, D.G. : Magnetic sails and interplanetary travel.J. Spacecraft Rockets 28(2), 197–203 (1991)

Springer


DOI 10.1007/s10509-006-9213-x


Astrophysical Jets of Blazars and Microquasars

Markus Bottcher


Abstract Some recent developments in the study of rela-

tivistic jets in active galactic nuclei and microquasars are

reviewed. While it has been well established for some time

that extragalactic jets found in radio galaxies, quasars, and

BL Lac objects are the site of ultrarelativistic particle accel-

eration, the recent identification of the Galactic jet source

and microquasar LS 5039 as a source of very-high-energy

gamma-ray emission has underlined the striking similarity

between the two types of astrophysical jet sources. In this

paper, I will present an overview of the dominant radiation

and particle acceleration processes and observational tests to

distinguish between such processes. The wide-ranging analo-

gies between Galactic and extragalactic jets, but also their

distinct differences, in particular those caused by the pres-

ence of the companion star in Galactic microquasar systems,

will be exposed.

Keywords Gamma-rays: theory . Radiation mechanisms:

non-thermal . X-rays: binaries . Stars: winds . Outflows

1. Introduction

High-velocity, collimated outflows are a frequent phe-

nomenon, commonly associated with the accretion of ma-

terial onto central objects. For example, they occur as non-

relativistic jets in young stellar objects, as mildly relativistic

outflows from Galactic X-ray binaries, and as highly rel-

ativistic outflows from active Galactic nuclei (AGN) and

gamma-ray bursts. In this article, I will focus on analogies

and similarities, but also distinct, characteristic differences

M. BottcherAstrophysical Institute, Department of Physics and Astronomy,Ohio University, Athens, OH 45701, USA

between the highly relativistic jets of AGN, in particular the

class of AGN termed “blazars”, and X-ray binaries.

2. Phenomenology of blazars

Blazars (BL Lac objects and γ -ray loud flat spectrum ra-

dio quasars [FSRQs]) are the most extreme class of active

galaxies known. They have been observed at all wave-

lengths, from radio through very-high energy (VHE) γ -

rays. 46 blazars have been identified with high confidence

as sources of >100 MeV emission detected by the EGRET

telescope on board the Compton Gamma-Ray Observatory

(Hartman et al., 1999; Mattox et al., 2001), and 11 blazars

(Mrk 421, Mrk 501, PKS 2155−314, 1ES 2344+514, 1H

1426+428, 1ES 1959+650, PKS 2005−489, 1H 2356−309,

1ES 1101−232, 1ES 1218+304, and PG 1553+113) have

now been detected at VHE γ -rays (>350 GeV) by ground-

based air Cerenkov telescopes. Many of the EGRET-detected

γ -ray blazars appear to emit the bulk of their bolometric lu-

minosity at γ -ray energies. Blazars exhibit variability at all

wavelengths on various time scales. Radio interferometry

often reveals one-sided kpc-scale jets with apparent superlu-

minal motion.

The broadband continuum spectra of blazars are domi-

nated by non-thermal emission and consist of two distinct,

broad components. A sequence of blazar sub-classes, from

FSRQs to low-frequency peaked BL Lac objects (LBLs)

to high-frequency peaked BL Lacs (HBLs) can be defined

through the peak frequencies and relative νFν peak fluxes,

which also seem to be correlated with the bolometric lumi-

nosity (Fossati et al., 1998, see also Fig. 1).

Figure 1 already illustrates that in particular the high-

energy emission from blazars can easily vary by more than

an order of magnitude between different EGRET observing

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νL

ν [

erg

s/s

]3C279

June 1991 vs. Dec. 1992 / Jan. 1993

P1 (June 1991 flare)

P2 (Dec 92 / Jan 93)

FSRQ = Flat Spectrum Radio Quasar

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Oct/Nov 2000 vs. July 1997

July 1997

Nov. 1, 2000

LBL = Low frequency peaked

BL Lac object

c)a) b)

Fig. 1 SEDs of 3C 279 (Hartman et al., 2001), BL Lacertae (Bottcherand Bloom, 2000; Bottcher and Reimer, 2004), and Mrk 501 (Petryet al., 2000). For each object, two simultaneous broadband spectra at

two different epochs are shown. The curves show model fits, using aleptonic jet model

epochs (see also von Montigny et al., 1995; Mukherjee et al.,

1997, 1999). However, high-energy variability has been ob-

served on much shorter time scales, in some extreme cases

less than an hour (Gaidos et al., 1996). BL Lac objects oc-

casionally exhibit particularly interesting X-ray variability

patterns, which can be characterized as spectral hysteresis

in hardness-intensity diagrams (e.g. Takahashi et al., 1996;

Kataoka et al., 2000). This has been interpreted as the syn-

chrotron signature of gradual injection and/or acceleration of

ultrarelativistic electrons into the emitting region, and subse-

quent radiative cooling (e.g. Kirk et al., 1998; Georganopou-

los and Marscher, 1998; Kataoka et al., 2000; Kusunose et al.,

2000; Li and Kusunose, 2000; Bottcher and Chiang, 2002).

Figure 2d shows the results of a recent BeppoSAX observa-

tion of BL Lacertae (Ravasio et al., 2003; Bottcher et al.,

2003). Rapid flux and spectral variability of blazars is also

commonly observed in the optical regime, often character-

ized by a spectral hardening during flares (see, e.g., Fig. 2c,

or Lainela et al., 1999; Villata et al., 2002).

3. Models of blazar emission

The high inferred bolometric luminosities, rapid variability,

and apparent superluminal motions provide compelling evi-

dence that the nonthermal continuum emission of blazars is

produced in light day sized emission regions, propagat-

ing relativistically along a jet directed at a small angle with

respect to our line of sight. It is generally agreed that the low-

frequency component of blazar SEDs might be synchrotron

radiation from nonthermal, ultrarelativistic electrons. Sev-

eral electron injection/acceleration scenarios have been pro-

posed, e.g. impulsive injection near the base of the jet (Der-

mer and Schlickeiser, 1993; Dermer et al., 1997), isolated

shocks propagating along the jet (e.g., Marscher and Gear,

1985; Kirk et al., 1998; Sikora et al., 2001; Sokolov et al.,

2004), internal shocks from the collisions of multiple shells

of material in the jet (Spada et al., 2001), stochastic parti-

cle acceleration in shear boundary layers of relativistic jets

(e.g., Ostrowski and Bednarz, 2002; Rieger and Duffy, 2004),

magnetic reconnection in Poynting-flux dominated jets (e.g.,

Sikora et al., 2005), or hadronically initiated pair avalanches

(Kazanas and Mastichiadis, 1999).

While the electron-synchrotron origin of the low-

frequency emission is well established, there are two fun-

damentally different approaches concerning the high-energy

emission. If protons are not accelerated to sufficiently high

energies to reach the threshold for pγ pion production on

synchrotron and/or external photons, the high-energy emis-

sion will be dominated by ultrarelativistic electrons and/or

pairs (leptonic models). In the opposite case, the high-energy

emission will be dominated by cascades initiated by pγ pair

and pion production as well as proton, π±, and µ± syn-

chrotron radiation (hadronic models). These two approaches

have so far been mostly discussed separately. However, the

recent observation of isolated TeV flares without simulta-

neous X-ray flares in 1ES 1959+650 (Krawczynski et al.,

2004) and Mrk 421 (Blazejowski et al., 2005) may provide

rather strong support for scenarios in which elements of both

classes of jet models might be relevant (hybrid models).

3.1. Leptonic blazar models

In leptonic models, the high-energy emission is produced

via Compton scattering off the same ultrarelativistic elec-

trons which are producing the synchrotron emission at lower

frequencies. Possible target photon fields are the synchrotron

photons produced within the jet (the SSC process: Marscher

and Gear, 1985; Maraschi et al., 1992; Bloom and Marscher,

1996) or external photons (the EC process). Possible sources

of external seed photons include accretion-disk photons

entering the emission region directly (Dermer et al., 1992;

Dermer and Schlickeiser, 1993) or after reprocessing in

Springer


13.514.014.5

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1.4

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1.7

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RXTE PCAHEGRA UL

Oct. 31Nov. 2

STACEE/CELESTE

VERITAS

July 26/27, 2000

MAGIC

c) d)

b)a)

Fig. 2 Spectral variability fitting of BL Lacertae in 2000 (Bottcherand Reimer, 2004): (a) Time-dependent leptonic fits to BL Lac in itshigh state around Nov. 1, 2000; (b) Various hadronic fits to the samedata as represented in panel (a). The different models vary mainly intheir co-moving magnetic-field value and co-moving synchrotron pho-ton energy density. (c) Comparison of the fits from panel (a) and (b) to

the optical color-magnitude correlation; (d) comparison of the fits frompanel (a) to the X-ray hardness-intensity correlation during a short-termflare observed by BeppoSAX on Nov. 1, 2000. Based on count rates inthe three BeppoSAX NFI energy channels LECS [0.5–2 keV], MECS[2–4 keV], and MECS [4–10 keV], and X-ray hardness ratios: HR1 =MECS[2–4]/LECS[0.5–2], HR2 = MECS[4–10]/MECS[2–4]

surrounding material like the broad-line regions of a quasar

(Sikora et al., 1994; Dermer et al., 1997), jet synchrotron

emission reprocessed by circumnuclear material (Ghisellini

and Madau, 1996), infrared emission from circumnuclear

dust (Blazejowski et al., 2000), or synchrotron radiation from

other (earlier/later, faster/slower) emission regions along the

jet (Georganopoulos and Kazanas, 2003). Radiation spectra

may be modified by γ γ absorption internal and external to

the source, with the former leading to the injection of ad-

ditional relativistic electron-positron pairs, and synchrotron

self absorption must be taken into account in a self-consistent

leptonic blazar model. As the emission region is propagat-

ing relativistically along the jet, continuous particle injection

and/or acceleration and subsequent radiative and adiabatic

cooling, as well as particle escape have to be considered. Also

the deceleration of the jets, in particular in HBLs, may have

a significant impact on the observable properties of blazar

emission (Georganopoulos and Kazanas, 2003; Ghisellini

et al., 2005).

Detailed modeling of both spectra and spectral variabil-

ity of blazar spectra with time-dependent leptonic jet models

has been done very successfully for a variety of blazars. Sev-

eral examples for spectral fits to contemporaneous spectral

energy distributions of various blazars are shown in Fig. 1.

However, it has also been demonstrated that spectral fitting

alone is generally insufficient to constrain the multitude of

parameters in leptonic jet models, in particular in cases in

which external photon sources turn out to be non-negligible

(see, e.g. Bottcher et al., 2002). Invaluable additional con-

straints can be obtained when variability information is in-

cluded in time-dependent modeling efforts. Figure 2 illus-

trates recent results of spectral variability modeling of the

broadband emission from BL Lacertae during a broadband

observing campaign in 2000 (Ravasio et al., 2003; Bottcher

Springer


et al., 2003). One of the key results of the leptonic modeling

part of that work (Bottcher and Reimer, 2004) was that lep-

tons would need to be accelerated out to energies of several

hundred GeV, into a power-law distribution with number in-

dex q = 2.3, consistent with first-order Fermi acceleration

at relativistic, parallel shocks.

3.2. Hadronic blazar models

If a significant fraction of the kinetic luminosity in the jet

is converted into the acceleration of relativistic protons

and those protons reach the threshold for pγ pion pro-

duction, synchrotron-supported pair cascades will develop

(Mannheim and Biermann, 1992; Mannheim, 1993). The

acceleration of protons to the necessary ultrarelativistic

energies requires high magnetic fields of at least several tens

of Gauss. In the presence of such high magnetic fields, also

the synchrotron radiation of the primary protons (Aharonian,

2000; Mucke and Protheroe, 2000) and of secondary muons

and mesons (Rachen and Meszaros, 1998; Mucke and

Protheroe, 2000, 2001; Mucke et al., 2003) must be taken into

account in order to construct a self-consistent synchrotron-

proton blazar (SPB) model. Electromagnetic cascades can

be initiated by photons from π0-decay (“π0 cascade”),

electrons from the π± → µ± → e± decay (“π± cascade”),

p-synchrotron photons (“p-synchrotron cascade”), and

µ-, π - and K -synchrotron photons (“µ±-synchrotron

cascade”).

Mucke and Protheroe (2001) and Mucke et al. (2003)

have shown that the “π0 cascades” and “π± cascades” gen-

erate featureless γ -ray spectra, in contrast to “p-synchrotron

cascades” and “µ±-synchrotron cascades” that produce a

double-humped γ -ray spectrum. In general, direct proton

and µ± synchrotron radiation is mainly responsible for the

high energy bump in blazars, whereas the low energy bump

is dominanted by synchrotron radiation from the primary

e−, with a contribution from secondary electrons. Figure 2b

shows a fit to the broadband SED of BL Lacertae in 2000,

using the hadronic SPB model.

3.3. Hybrid blazar models

While standard leptonic SSC models predict a close temporal

flux correlation between the synchrotron and Compton com-

ponents, recent monitoring observations of the TeV-blazars

1ES 1959+650 (Krawczynski et al., 2004) and Mrk 421

(Blazejowski et al., 2005) at X-ray and TeV-energies revealed

TeV flares without accompanying X-ray flares, a behaviour

sometimes referred to as ”orphan TeV flares”. In the case of

1ES 1959+650, this phenomenon was preceded by an ordi-

nary, correlated X-ray and TeV-flare, which can be generally

well understood in the context of leptonic SSC models. This

finding strongly suggests the need for models that explain

flares dominated by leptonic interactions as well as flares

where non-leptonic components must play an important role

within the same system. The recent orphan TeV flare of

1ES 1959+650 led to the development of the hadronic syn-

chrotron mirror model (Bottcher, 2005; Reimer et al., 2005).

In this model, the primary, correlated X-ray and TeV flare is

explained by a standard SSC model while the secondary or-

phan TeV-flare is explained by π0-decay γ -rays as a result of

photomeson production in the Delta(1232)-resonance from

relativistic protons interacting with the primary synchrotron

flare photons that have been reflected off clouds located at

pc-scale distances from the central engine.

Some earlier developments along the lines of hadronic

processes in the context of models with leptonically

dominated blazar emission include the following: (a) The

“supercritical pile” model of (Kazanas and Mastichiadis,

1999) suggests a runaway pair production avalanche initi-

ated by mildly relativistic protons on reflected synchrotron

photons via pγ pair production as the primary pair injection

mechanism in blazar jets. (b) Atoyan and Dermer (2003)

suggest the conversion of ultrarelativistic protons into

neutrons via pγ pion production on external soft photons

as a possible mechanism to overcome synchrotron losses of

protons near the base of blazar jets and, thus, to allow blazar

jets to remain collimated and transport their kinetic energy

out to kpc scales.

4. The jets of microquasars

In the standard picture, the high-energy (X-ray – γ -ray) spec-

tra of X-ray binaries generally consist of two major compo-

nents: A soft disk blackbody with a typical temperature of

kT ∼ 1 keV, and a power-law at higher energies. Neutron-

star and black-hole X-ray binaries exhibit at least two main

classes of spectral states, generally referred to as the high/soft

state, and the low/hard state (for a review see, e.g. Liang,

1998; McClintock Remillard, 2004). The high-energy spec-

tra of X-ray binaries in the soft state are characterized by a

thermal blackbody component, believed to be associated with

thermal emission from an optically thick, geometrically thin

accretion disk (Shakura and Sunyaev, 1973), and a power-

law tail with a photon indexŴ ≥ 2. Generally, no high-energy

cutoff of the high-energy power-law is detected. In the hard

state, the spectrum is dominated by a power-law, with a slope

Ŵ < 2 and a cut-off at ∼ a few hundred keV. Two additional

states, namely the Very High State (VHS) and the Interme-

diate State (IS) share common features of both the high and

the low state, namely a hard powerlaw and a prominent disk

blackbody. In addition to prominent X-ray emission, many

X-ray binaries are also known to be associated with extended

radio jet structures (for a recent review see, e.g., Fender et al.,

2004). In analogy to their extragalactic cousins, those sources

Springer


are generally termed “microquasars”. When the sources are

in the high/soft or low-hard states, one observes quite often a

positive correlation between the radio and hard X-ray emis-

sion, apparently anti-correlated with the soft X-ray emission

(e.g. Corbel et al., 2000, 2001), indicating that the production

of a steady, relatively slow (bulk Lorentz factorŴ j 2) radio

jet is suppressed in the high/soft state. In the IS and VHS,

occasionally intermittent, faster (Ŵ j 2) jets are observed

(e.g., Fender et al., 2004).

While, in the conventional view of X-ray binaries (includ-

ing microquasars), the X-ray and γ -ray emission is attributed

to Comptonized emission (Sunyaev and Titarchuk, 1980;

Titarchuk, 1994) arising from hot thermal (kT ≫ 1keV) or

relativistic, non-thermal electrons close to the black hole

(Liang and Price, 1977; Bisnovatyi-Kogan and Blinnikov,

1977; Shapiro et al., 1976; Narayan and Yi, 1994; Chen et al.,

1995, e.g.), the tentative EGRET detections of at least two

Galactic microquasars at MeV–GeV γ -ray energies, namely

LS 5039 (Paredes et al., 2000) and LSI+61303 (Gregory

and Taylor, 1978; Taylor et al., 1992; Kniffen et al., 1997),

the detection of X-ray jet structures in several microquasars

using Chandra and XMM-Newton (Corbel et al., 2002; Tom-

sick, 2002, e.g.,), and, most recently, the detection of very-

high-energy (VHE) γ -ray emission from LS 5039 (Aharo-

nian et al., 2005) have re-ignited interest in jet models for the

high-energy emission from microquasars, analogous to the

commonly favored models for blazars. A jet origin of the X-

ray emission of microquasars has been suggested by several

authors, e.g., Markoff et al. (2001, 2003a,b), who discussed

the possibility of synchrotron emission from relativistic elec-

trons in the jet extending from the radio all the way into the

X-ray regime.

5. Models of high-energy emission from

microquasar jets

Microquasars now join blazar AGNs as a firmly established

class of very-high energy γ -ray sources. Because of their

apparent similarity with their supermassive AGN cousins,

it has been suggested that Galactic microquasars may be

promising sites of VHE γ -ray production (e.g., Bosch-

Ramon et al., 2005a). High-energy γ -rays of microquasars

can be produced via hadronic (e.g. Romero et al., 2003)

or leptonic processes. In the latter case, the most plausi-

ble site would be close to the base of the mildly relativistic

jets, where ultrarelativistic electrons can Compton upscat-

ter soft photons. Possible sources of soft photons are the

synchrotron radiation produced in the jet by the same ul-

trarelativistic electron population (SSC = synchrotron self-

Compton; Aharonian and Atoyan, 1999), or external pho-

ton fields (Bosch-Ramon and Paredes, 2004; Bosch-Ramon

et al., 2005a). The X-ray – soft γ -ray spectral variability

features expected in such a microquasar jet model have re-

cently been investigated in a detailed parameter study by

Gupta et al. (2006), and a brief summary of the main re-

sults of that work can also be found in these proceedings

(Gupta and Bottcher, 2006).

Up to this point, radiation physics widely analogous to

blazar jets may be considered. However, both LS 5039 and

LSI+61 303 are high-mass X-ray binaries which are rather

faint in X-rays, with characteristic 1–10 keV luminosities

of ∼1034 ergs s−1. This is much lower than the character-

istic bolometric luminosity of the high-mass companions of

these objects, at L∗ 1038 erg/s. Consequently, the domi-

nant source of external photons in LS 5039 and LSI+61

303 is the companion’s optical/UV photon field. The intense

radiation field of the high-mass companion will also lead to

γ γ absorption of VHE γ -rays in the ∼ 100 GeV–TeV pho-

ton energy range if VHE photons are produced close to the

base of the jet (Aharonian et al., 2005; Bottcher and Dermer,

2005).

It is primarily the presence of the prominent azimuthally

asymmetric photon source provided by the companion star,

which leads to substantial differences in the expected spectra

and variability patterns of microquasars, compared to blazars.

The most obvious of these consequences would be a tempo-

ral modulation of the high-energy emission on the time scale

of the orbital period. Both effects of the orbital-phase de-

pendent Compton scattering and γ γ absorption will lead to

characteristic spectral variability patterns. The orbital mod-

ulation patterns due to γ γ absorption patterns have been

investigated in detail in Bottcher and Dermer (2005), while

the isolated orbital-modulation effects on Compton scattered

starlight photons have been worked out in detail by Dermer

and Bottcher (2006). A brief summary of the orbital modu-

lation patterns due to those two effects, with particular focus

on model parameter choices appropriate to LS 5039 (Casares

et al., 2005) can also be found in these proceedings (Bottcher

and Dermer, 2006).

In addition to the orbital modulation effects from γ γ

absorption and starlight Compton scattering, there are at

least two other effects that could potentially lead to a quasi-

periodic temporal variation in the high-energy emission from

microquasar jets:

(a) If the binary orbit has a substantial eccentricity (e.g.,

e = 0.35 for the case of LS 5039, Casares et al., 2005),

the rate of mass transfer from the stellar compan-

ion to the compact object, which is believed to be

dominated by wind accretion, is likely to be peri-

odically modulated. This modulation would also be

expected to appear at radio and X-ray energies and

would be expected to lead to an overall hardening of

the γ -ray spectrum at all energies with increasing γ -

ray flux. In contrast, the orbital-phase dependent γ γ

Springer


absorption trough due to interactions of VHE pho-

tons with companion starlight would lead to a spectral

hardening at E 100 GeV, but a spectral softening

at E 100 GeV at increasing γ -ray flux levels, and

the starlight Compton-scattering signature would pro-

vide an overall softening of the VHE γ -ray flux with

increasing flux level because of the more severe ef-

fect of the Klein-Nishina cutoff at a more favorable

orientation for starlight Compton scattering (Dermer

and Bottcher, 2006; Bottcher and Dermer, 2006).

(b) The orientation of the jet may also be mis-aligned

with respect to the normal of the orbital plane (Mac-

carone, 2002; Butt, 2003) and possibly precessing

about the normal (Larwood, 1998; Torres et al.,

2005), leading to additional modulations, including a

changing Doppler boosting factor. A stationary mis-

alignment of the jet could lead to a slight enhance-

ment of the orbital modulation (if the jet makes a

smaller angle with the line of sight than the orbital-

motion axis) or reduce it (in the opposite case).

A γ -ray flux modulation due to jet precession can

easily be disentangled from the orbital modulation

since the precession period is generally different from

the orbital period, so that its effect would average

out when folding observational data with the orbital

period.

Acknowledgements This work was partially supported by NASAthrough XMM-Newton GO grant no. NNG 04GI50G and NASA IN-GEGRAL Theory grant no. NNG 05GK59G.

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Springer


DOI 10.1007/s10509-006-9243-4


Simulation of the Magnetothermal Instability

Ian J. Parrish · James M. Stone

Received: 14 April 2006 / Accepted: 22 August 2006C© Springer Science + Business Media B.V. 2006

Abstract In many magnetized, dilute astrophysical plas-

mas, thermal conduction occurs almost exclusively paral-

lel to magnetic field lines. In this case, the usual stability

criterion for convective stability, the Schwarzschild crite-

rion, which depends on entropy gradients, is modified. In

the magnetized long mean free path regime, instability oc-

curs for small wavenumbers when (∂P/∂z)(∂ ln T/∂z) > 0,

which we refer to as the Balbus criterion. We refer to the

convective-type instability that results as the magnetother-

mal instability (MTI). We use the equations of MHD with

anisotropic electron heat conduction to numerically simu-

late the linear growth and nonlinear saturation of the MTI

in plane-parallel atmospheres that are unstable according to

the Balbus criterion. The linear growth rates measured from

the simulations are in excellent agreement with the weak

field dispersion relation. The addition of isotropic conduc-

tion, e.g. radiation, or strong magnetic fields can damp the

growth of the MTI and affect the nonlinear regime. The in-

stability saturates when the atmosphere becomes isothermal

as the source of free energy is exhausted. By maintaining

a fixed temperature difference between the top and bottom

boundaries of the simulation domain, sustained convective

turbulence can be driven. MTI-stable layers introduced by

isotropic conduction are used to prevent the formation of un-

resolved, thermal boundary layers. We find that the largest

component of the time-averaged heat flux is due to advective

motions as opposed to the actual thermal conduction itself.

Finally, we explore the implications of this instability for a

J. M. Stone: Program in Applied and Computational Mathematics,Princeton University, Princeton, NJ 08544

I. J. Parrish () · J. M. StoneDepartment of Astrophysical Sciences, Princeton University,Princeton, NJ 08544e-mail: [email protected]

variety of astrophysical systems, such as neutron stars, the

hot intracluster medium of galaxy clusters, and the structure

of radiatively inefficient accretion flows.

Keywords Accretion . Accretion disks . Convection .

Hydrodynamics . Instabilities . MHD . Stars: neutron .

Turbulence . Galaxy clusters

1 Introduction

In many dilute, magnetized astrophysical plasmas, the elec-

tron mean free path between collisions can be many orders

of magnitude larger than the ion gyroradius. In this regime,

the equations of ideal magnetohydrodynamics (MHD) that

describe the fluid plasma must be supplemented with

anisotropic transport terms for energy and momentum due

to the near free-streaming motions of particles along mag-

netic field lines (Braginskii, 1965). Thermal conduction is

dominated by the electrons compared to ions by a factor of

the square root of the mass ratio. In a non-rotating system,

it is sufficient to neglect the ion viscosity (Balbus, 2004). A

fully collisionless treatment may be done using more com-

plex closures such as Hammett and Perkins (1990).

The implications of anisotropic transport terms on the

overall dynamics of dilute astrophysical plasmas is only be-

ginning to be explored (Balbus, 2001; Quataert et al., 2002;

Sharma et al., 2003). One of the most remarkable results ob-

tained thus far is that the convective stability criterion for a

weakly magnetized dilute plasma in which anisotropic elec-

tron heat conduction occurs is drastically modified from the

usual Schwarzschild criteria (Balbus, 2000). In particular,

stratified atmospheres are unstable if they contain a temper-

ature (as opposed to entropy) profile which is decreasing up-

ward. There are intriguing analogies between the stability

Springer


properties of rotationally supported flows (where a weak

magnetic field changes the stability criterion from a gradi-

ent of specific entropy to a gradient of angular velocity), and

the convective stability of stratified atmospheres (where a

weak magnetic field changes the stability criterion from a

gradient of entropy to a gradient of temperature). The for-

mer is a result of the magnetorotational instability (MRI;

Balbus and Hawley, 1998). The latter is a result of anisotropic

heat conduction. To emphasize the analogy, we will refer to

this new form of convective instability as the magnetother-

mal instability (MTI). The MTI may have profound implica-

tions for the structure and dynamics of many astrophysical

systems.

In this paper, we use numerical methods to explore

the nonlinear evolution and saturation of the MTI in two-

dimensions. We adopt an arbitrary vertical profile for a strat-

ified atmosphere in which the entropy increases upward

(and therefore is stable according to the Schwarzschild cri-

terion), but in which the temperature is decreasing upwards

(and therefore is unstable according to the Balbus criterion,

(∂P/∂z)(∂ ln T/∂z) < 0). We confirm the linear growth rates

predicted by Balbus (2000) for dynamically weak magnetic

fields and numerically measure the growth rates for stronger

fields. We find that in the nonlinear regime vigorous convec-

tive turbulence results in efficient heat transport. Full de-

tails are published in Parrish and Stone (2005), hereafter

PS.

These results may have implications for stratified atmo-

spheres where anisotropic transport may be present. The most

exciting potential application is to the hot X-ray emitting gas

in the intracluster medium of galaxy clusters (Peterson and

Fabian, 2005; Markevitch et al., 1998). The hot plasmas are

magnetized and have very long mean free paths along the

magnetic field lines; thus, they are a prime candidate for this

instability. For example, the Hydra A cluster has T ≈ 4.5

keV and a density of n ≈ 10−3 − 10−4 cm−3 giving a mean

free path that’s almost one-tenth of the virial radius. Other

applications are to the atmospheres of neutron stars with

moderate magnetic fields and radiatively inefficient accretion

flows.

2 Physics of the MTI

2.1 Equations of MHD and linear stability

The physics of the MTI is described by the usual equations of

ideal MHD with the addition of a heat flux, Q, and a vertical

gravitational acceleration, g.

∂ρ

∂t+ ∇ · (ρv) = 0, (1)

∂(ρv)

∂t+ ∇ ·

[

ρvv +(

p + B2

8π

)

I − BB

4π

]

+ ρg = 0, (2)

∂B

∂t+ ∇ × (v × B) = 0, (3)

∂E

∂t+ ∇ ·

[

v

(

E + p + B2

8π

)

− B(B · v)

4π

]

+∇ · Q + ρg · v = 0, (4)

where the symbols have their usual meaning with E the to-

tal energy. The heat flux contains contributions from electron

motions (which are constrained to move primarily along field

lines) and isotropic transport, typically due to radiative pro-

cesses. Thus, Q = QC + QR , where

QC = −χC bb · ∇T, (5)

QR = −χR∇T, (6)

where χC is the Spitzer Coulombic conductivity (Spitzer,

1962), b is a unit vector in the direction of the magnetic

field, and χR is the coefficient of isotropic conductivity.

Some progress can be made analytically in the linear

regime of the instability. I introduce two useful quantities,

χ ′C = γ − 1

PχC and χ ′ = γ − 1

P(χC + χR) . (7)

With WKB theory, one is able to obtain a dispersion rela-

tion. The details of this process are to be found in Section 4

of Balbus (2000). The most important result of the linear

analysis is the instability criterion,

k2v2A − χ ′

C

ρχ ′∂P

∂z

∂ ln T

∂z< 0, (8)

where v2A = B2/4πρ is the Alfven speed. In the limit of in-

finitesimal wavenumber, the instability criterion shows that

any atmosphere with the temperature and pressure gradients

in the same direction is unconditionally unstable, i.e.

∂P

∂z

∂ ln T

∂z> 0. (9)

We refer to the instability criterion Eq. (9) as the Balbus

criterion. The instability criterion of the magnetorotational

instability (Balbus and Hawley, 1998) can be written

k2v2A + d

d ln R> 0, (10)

where is the angular velocity. The similarity between the

MRI and the MTI is self-evident. Strong magnetic fields are

Springer


capable of stabilizing short-wavelenth perturbations in both

instabilities through magnetic tension.

2.2 Computational method

We use the 3D MHD code ATHENA (Gardiner and Stone,

2005) with the addition of an operator-split anisotropic ther-

mal conduction module for our simulations. Our initial state

is always a convectively stable state (d S/dz > 0) in hydro-

dynamic equilibrium. We implement two different bound-

ary conditions for exploring the nonlinear regime. The first

boundary condition is that of an adiabatic boundary con-

dition at the upper and lower boundaries, i.e. a Neumann

boundary condition on temperature. This situation is ideal for

single-mode studies of linear growth rates. In the Neumann

boundary condition the magnetic field is reflected at the upper

and lower boundaries, consistent with the adiabatic condition

on heat flow. The second boundary condition fixes the the

temperature at the upper and lower boundaries of the atmo-

sphere, i.e. a Dirichlet boundary condition on temperature.

This setup is useful for driven simulations where we wish to

study the effects of turbulence. In this boundary condition,

the magnetic field is again reflected at the boundaries of the

box, but heat is permitted to flow across the boundary from

the constant temperature ghost zones. With these choices of

boundary conditions, the net magnetic flux penetrating the

box is constant in time as there is zero Maxwell stress at

the boundary. If the net magnetic flux penetrating the box is

initially zero, then in two dimensions the magnitude of the

magnetic field must decay in time as a result of Cowling’s

anti-dynamo theorem, but otherwise the saturated state is not

affected.

3 Results

3.1 Single-mode perturbation and qualitative

understanding of the MTI

By examining a single mode-perturbation to the background

state, we can gain an intuitive understanding of the phys-

ical mechanism of the instability. We begin by perturbing

a convectively-stable, but MTI-unstable atmosphere with a

very weak sub-sonic and sub-Alfenic sinusoidal velocity per-

turbation in a box with adiabatic boundary conditions. The

evolution of the magnetic field lines is shown in Fig. 1 for

several different times. Notice in the upper right plot that at

x ≈ 0.03 that a parcel of fluid (as traced by the frozen-in

field line) has been displaced upward in the atmosphere. As

this parcel of fluid comes to mechanical equilibirum with the

background state, it adiabatically cools. The magnetic field

line now is partially aligned with the background tempera-

ture state, thus thermally connecting this parcel of fluid with a

Fig. 1 Snapshots of the magnetic field lines for a single-mode pertur-bation with adiabatic boundary conditions at various times during theevolution of the instability. (upper left) Inititial condition; (upper right)Linear phase; (lower left) Non-linear phase; (lower right) Saturated state

t = 0

t = 320

t = 640

t = 2880

t = 1280

z

Fig. 2 Vertical profile of the horizontally-averaged temperature profilein the single mode case at various times. The initial state is a mono-tonically decreasing temperature profile with respect to height, and thefinal state is isothermal

hotter parcel deeper in the atmosphere. As a result, heat flows

along the field, causing the higher parcel to become buoyant.

This buoyant motion causes the field line to be more aligned

with the background temperature, increasing the heat flux,

and generating a runaway instability.

It is instructive to examine the behavior of the temperature

profile of the atmosphere. Figure 2 shows the horizontally

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Fig. 3 Solutions of the MTI dispersion relation in the weak field limitfor an atmosphere with d ln T/d ln S = −3. The axes are normalizedto the local Brunt-Vaisala frequency, N . The crosses are growth ratesmeasured from simulations

averaged temperature profile of this run at various times (nor-

malized to the sound crossing time). The initial state is a linear

temperature profile decreasing with height. As the instability

progresses, the temperature profile becomes more and more

isothermal. Saturation occurs, as one would expect, when the

temperature profile is almost completely isothermal, since the

source of free energy has been depleted. More details are to

be found in Section 4 of PS.

3.2 Linear growth rates

By following Section 4.3 of Balbus (2000) and using the

Fourier convention, exp(σ t + ikx), one can derive a weak-

field dispersion relation for the MTI as

(

σ

N

)3

+ 1

γ

(

σ

N

)2(χ ′T k2

N

)

+(

σ

N

)

+d ln T

d ln S

(

χ ′cT k2

N

)

= 0, (11)

where N is the Brunt-Vaisala frequency, the natural fre-

quency of adiabatic oscillations for an atmosphere. With the

single-mode perturbation simulations we are able to mea-

sure the growth rate for a variety of situations. Figure 3 plots

the nondimensionalized growth rate versus wavenumber for

theory (solid line) and the measured values from simulations

(crosses). As can be seen these are in very good agreement.

There are essentially two ways to suppress the growth of

the MTI. First, strong magnetic fields can exert tension that

limits the growth and saturation of the instability. Second,

isotropic conduction, as would result from radiative trans-

port, can effectively short-circuit the thermal driving along

field lines necessary for this instability to occur. For more

detailed analysis, we refer the reader to Section 3 of PS.

3.3 Nonlinear regime and efficiency of heat transport

In order to assess the efficiency of heat transport in the magne-

tothermal instability, one needs to examine multimode sim-

ulations seeded with Gaussian white noise perturbations and

conducting boundary conditions at the top and bottom of the

Fig. 4 Snapshots of the magnetic field in the run with stable layers. (far left) Early linear phase; (middle left) early non-linear phase. (middle right)The MTI drives penetrative convection into the stable layers, and at late times (far right) magnetic flux is pumped into the stable layers

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Fig. 5 Time evolution of thehorizontally-averaged heat fluxat the midplane and 80% heightof the simulation domain in therun with stable layers. The totalheat flux (thick solid line) issubdivided into Coulombic (thinsolid line), radiative (dashedlined), and advective (dottedline) components. Theinstantaneous total heat flux isdominated by advective motions

domain. The simplest such set-up results in narrow, unre-

solved boundary layers at the upper and lower boundaries,

thus, making the heat flux difficult to measure accurately.

As an alternative, we utilize a more physically relevant sim-

ulation. This setup involves an atmosphere that is convec-

tively stable throughout, but MTI unstable only in the cen-

tral region. The surrounding regions are stabilized to the

MTI through the addition of isotropic conducitivity. As a

result, the central unstable region is well-resolved. Figure 4

shows the evolution of magnetic field lines as this instability

progresses. The magnetic field lines shown essentially track

the central unstable region; however, the third panel clearly

shows a plume of fluid that is penetrating into the stable

layer as a result of convective overshoot. This phenomenon

is well-known in the solar magnetoconvection literature (To-

bias et al., 2001; Brummell et al., 2002). In three dimensions

this type of behavior greatly amplifies the magnetic field in

a local magnetic dynamo.

More quantitative measurements can be made by compar-

ing the time- and horizontally-averaged vertical heat fluxes

and breaking it down into Coulombic, radiative, and isotropic

components. Figure 5 shows these quantities plotted as a

function of time. At the midplane, the oscillatory advective

flux is clearly dominant at any given instant in time; how-

ever, averaged in time the advective heat flux contributes

roughly 23

of the total heat flux. The Coulombic flux, which

is relatively constant in time, contributes the remaining 13. To

determine the heat conduction efficiency of this instability,

we compare it to the expected vertical heat flux across the

simulation domain for pure uniform isotropic conductivity,

namely, Q0 ≈ 3.33 × 10−5. The time-averaged heat conduc-

tion at the midplane is⟨

Qtot,50%

⟩

≈ 3.54 × 10−5, which in-

dicates that the instability transports the entire applied heat

flux efficiently.

4 Conclusions and application

The most important conclusion of this work is that atmo-

spheres with d S/d Z < 0 are not necessarily stable to con-

vection. In fact, dilute atmospheres with weak to moderate

magnetic fields can be convectively unstable by the Balbus

criterion resulting in an instability that we call the magne-

tothermal instability. We have verified using MHD simula-

tions that the measured linear growth rates agree with ana-

lytic WKB theory as predicted by Balbus (2000). For adia-

batic boundary conditions, we find the saturated state is an

isothermal temperature profile, corresponding to the exhaus-

tion of the free energy in the system. For a driven instability

with conducting boundary conditions, we find that the MTI

efficiently transports heat, primarily by advective motions of

the plasma in the vigorous convection that results.

The most promising application of this instability is to

clusters of galaxies. Structure formation calculations assum-

ing CDM cosmologies predict monotonically decreasing

temperature profiles of the intracluster gas (Loken et al.,

2002). Observations of clusters with Chandra, such as Hydra

A (DeGrandi and Modlendi, 2002), however, indicate essen-

tially flat temperature profiles. The intracluster medium is

dilute, magnetized, and has a mean free path that could be as

high as one-tenth of the cluster virial radius. It may be that

these temperature profiles are representative of the saturated

state of the MTI. This possibility will be explored in future

work.

References

Balbus, S.A., Hawley, J.F.: Rev. Mod. Phys. 70, 1 (1998)Balbus, S.A.: ApJ 534 420 (2000)

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Balbus, S.A.: ApJ 562, 909 (2001)Balbus, S.A.: ApJ 616, 857 (2004)Braginskii, S.I.: in: Leontovich, M.A. (ed.), Reviews of Plasma Physics,

Consultants Bureau, New York, Vol. 1, p. 205 (1965)Brummell, N.H., Clune, T.L., Toomre, J.: ApJ 570, 825 (2002)DeGrandi, S., Molendi, S.: ApJ 567, 163 (2002)Gardiner, T., Stone, J.: J. Comp. Phys. 205, 509 (2005)Hammett, G.W., Perkins, F.W.: Phys. Rev. Lett. 64, 3019 (1990)

Loken, C., et al.: ApJ 579, 571 (2002)Markevitch, M., et al.: ApJ 377, 392 (1998)Parrish, I.J., Stone, J.M.: ApJ 633, 334 (2005)Peterson, J.R., Fabian, A.C.: astro-ph0512549 (2005)Quataert, E., Dorland, W., Hammett, G.W.: ApJ 577, 524 (2002)Sharma, P., Hammett, G., Quataert, E.: ApJ 596, 1121 (2003)Spitzer, L.: Physics of Fully Ionized Gases. Wiley, New York (1962)Tobias, S.M., et al.: ApJ 549, 1183 (2001)

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DOI 10.1007/s10509-006-9250-5


Sheared Flow as a Stabilizing Mechanism in Astrophysical Jets

Lucas F. Wanex · Erik Tendeland


Abstract It has been hypothesized that the sustained narrow-

ness observed in the asymptotic cylindrical region of bipolar

outflows from Young Stellar Objects (YSO) indicates that

these jets are magnetically collimated. The jz × Bφ force

observed in z-pinch plasmas is a possible explanation for

these observations. However, z-pinch plasmas are subject

to current driven instabilities (CDI). The interest in using

z-pinches for controlled nuclear fusion has lead to an exten-

sive theory of the stability of magnetically confined plasmas.

Analytical, numerical, and experimental evidence from this

field suggest that sheared flow in magnetized plasmas can

reduce the growth rates of the sausage and kink instabil-

ities. Here we propose the hypothesis that sheared helical

flow can exert a similar stabilizing influence on CDI in YSO

jets.

Keywords Astrophysical jets . Linear analysis . Sheared

flow

1 Collimation of astrophysical jets

The sustained narrowness of the asymptotic cylindrical re-

gion of many YSO bipolar outflows spawns the hypothesis

that an intrinsic collimating mechanism is present in the jet

plasma. Here we use the term “asymptotic cylindrical region”

to mean the narrow cone between the disk-wind initial com-

pression region and the jet termination region in the ambient

medium. A typical example of this comes from Hubble Space

Telescope Observations of HH30 (Burrows et al., 1996). The

apparent opening angle of this jet becomes narrower in the

asymptotic cylindrical region of the outflow when compared

L. F. Wanex () · E. TendelandUniversity of Nevada, Reno, NV, USA

to the opening angle near the jet source (Mundt et al., 1990).

Similar examples of recollimation in other jets have also

been observed (Konigle and Pudritz, 2000). Observations

of this nature have lead to the concept that the jet narrowness

is maintained by self-collimation (Shu et al., 2000). Self-

collimation can be caused by hoop stresses from a toroidal

magnetic field in the jet plasma.

This collimating mechanism is similar to the jz × Bφ force

observed in z-pinch plasmas. If this mechanism is to be con-

sidered as the cause of the sustained narrowness of some

YSO jets the possibility that these jets carry current far from

the accretion disk must be admitted. It is well known that

a cylindrical current-carrying plasma column with a heli-

cal magnetic configuration is subject to MHD instabilities

(Bateman, 1978; Chandrasekhar, 1961). The key instabili-

ties are the interchange (sausage) and kink modes (Freidberg,

1987). The sausage m = 0 and kink m = 1 instabilities have

been invoked to explain the observation of knots, wiggles,

and filamentary structures in astrophysical jets (Nakamura

and Meier, 2004; Reipurth and Heathcote, 1997). It has also

been suggested that the disruption caused by the kink in-

stability discredits the magnetic collimation model (Spruit

et al., 1997). However, it is an open question whether MHD

instabilities will disrupt YSO jet collimation (Konigle and

Pudritz, 2000).

Two mechanisms for reducing z-pinch plasma instabili-

ties may explain why this is so. Z-pinch plasmas will be sta-

ble if the ratio of axial to azimuthal magnetic field strength

is greater than the Kruskal-Shafranov limit (Kruskal and

Schwarzschild, 1956; Shafranov, 1954). Some stellar evo-

lution models predict that low-mass protostars form in an in-

terstellar medium that is supported by a magnetic field (Ray,

2004; Konigl and Pudritz, 2000). This interstellar field could

supply the instability reducing mechanism in these jets; how-

ever the efficiency of the pinch effect is reduced because the

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axial magnetic field must be compressed as well as the jet

plasma itself (Shumlak and Hartman, 1995). Analytical, ex-

perimental and numerical results show that sheared flow in

z-pinch plasmas can reduce the growth of MHD instabil-

ities (Arber and Howell, 1995; Bateman, 1978; DeSouza-

Machado et al., 2000; Golingo et al., 2005; Shumlak and

Hartman, 1995; Sotnikov et al., 2002, 2004; Ruden, 2002;

Winterberg, 1985, 1999; Wanex et al., 2004; Wanex, 2005a).

Velocity shear stabilization is primarily a phase mixing pro-

cess that disrupts the growth of unstable modes (DeSouza-

Machado et al., 2000; Wanex et al., 2005b).

There are sound theoretical reasons why Keplerian shear

could be present in jets originating from an accretion disk

(Bacciotti et al., 2004; Bally et al., 2002; Volker et al., 1999).

The disk-wind model of jet formation in protostars postulates

that the source of the jet comes from a wide radial band

of the accretion disk (Konigl and Pudritz, 2000). Since the

rotational motion in the disk varies with distance from the

central object the disk-wind driven jet velocity profile may

also vary with distance from the central object. Axial and

rotational sheared flows have been observed in astrophysical

jets (Bally et al., 2002; Bacciotti et al., 2002, 2003, 2004;

Coffey et al., 2004).

In this paper we will focus on the stabilizing effects of

sheared helical flow. Due to space limitations a brief sum-

mary of the evidence that sheared flow reduces instabilities

in z-pinch plasmas will be presented and the possibility that

sheared flow could be responsible for the sustained narrow-

ness in YSO jets will be considered.

2 MHD model of an astrophysical jet

The MHD jet is modeled as perfectly conducting cylindrical

plasma. The jet is then considered as a region in space where

supersonic plasma and an electric current flow. The electric

current I is balanced by a return current Ir of equal size. This

return current can be modeled as a diffuse flow in the ambient

medium or as a sheet on the jet surface (Lery and Frank,

2000). Jet models with return current have been referred to

as “cocoon jets” (Appl and Camenzind, 1992; Lesch et al.,

1989; Nakamura and Meier, 2004). MHD instabilities grow,

but do not propagate in stationary z-pinch plasmas (Appl

et al., 2000). The kink mode in a plasma column moving

with uniform velocity would simply move with the flow and

thus grow at the same rate as a stationary kink (Shumlak and

Hartman, 1995). By transforming to a frame that moves with

the jet, the jet plasma becomes analogous to a stationary

cylindrical z-pinch plasma. The velocity of the jet in this

frame is modeled with

v = ⌊0, v0φ/√

a + r , v0z(1/√

a − 1/√

a + r )⌋ (1)

(in cylindrical coordinates) where v0φ and v0z are constants

of proportionality and a is a small number to prevent the ve-

locity from going to infinity at r = 0. The functional form of

both velocity components is approximately Keplerian. In a

frame that moves with the axial velocity at the center of the

jet the axial plasma flow appears to be increasingly swept

back with increasing radius. At the origin the axial velocity

is zero and increases to a maximum at the edge of the jet. Both

the azimuthal and axial components of velocity have Kep-

lerian shear because the disk-wind is modeled as the source

of plasma in the jet. This model is intended to simulate the

asymptotic jet cylinder far from the surface of the accretion

disk and termination region.

3 Stability criteria

3.1 Kelvin–Helmholtz instability

The form of the Kelvin–Helmholtz instability (KHI) con-

sidered here occurs at the tangential boundary between the

edge of the jet and the ambient medium. The KHI can be ex-

cited by the velocity discontinuity that exists at this boundary

(Keppens et al., 2005). For astrophysical jets this instability

will be excited if the velocity discontinuity is greater than

the Alfven velocity (Nakamura and Meier, 2004). For the

Keplerian velocity profile considered here the velocity at the

outer edge of the jet is lower than the velocity at the cen-

ter of the jet (relative to the ambient medium). Sub-Alfvenic

discontinuities are in general not subject to KHI so our anal-

ysis will focus on velocity profiles that meet this stability

criterion.

3.2 Sausage instability

The sausage instability is an interchange mode that can cause

axisymmetric pinches or bulges to grow exponentially in the

jet plasma. It has been shown analytically that the sausage

instability can be stabilized in z-pinch plasmas with sheared

axial flow V ′ meeting the following criteria

V ′ > γ√

ln (R), (2)

where γ is the growth rate and R is a dimensionless parameter

analogous to the Reynolds number (DeSouza-Machado et al.,

2000). In this caseγ ∼ vT /r0 where vT is the average thermal

velocity and r0 is the radius of the plasma. If we let V ′ ∼∇V/r0, where ∇V is the difference in the axial velocity

between the center of the jet and the edge of the jet, the

stability requirement is

∇V >√

ln (R) vT . (3)

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Thus we see that ∇V must be above a threshold to prevent

the sausage instability.

3.3 Kink instability

A global stability analysis of the kink mode using non-

relativistic compressible ideal MHD with gravity neglected

will demonstrate the stabilizing influence of sheared flow in

current-carrying jets. The MHD equations are made dimen-

sionless by normalizing the scales to the average radius of the

jet in the asymptotic cylindrical region, the average thermal

velocity of the jet plasma, the ambient pressure, and the ambi-

ent density. The linearized equations are solved numerically

with a generic two-step predictor-corrector, second-order ac-

curate space and time-centered advancement scheme (see

Wanex et al., 2005b for details). The problem is treated by

introducing perturbations into the plasma equilibrium state

and following their linear development in time. All perturbed

plasma variables (magnetic field, density, pressure, and ve-

locity) have the form ξ (r )ei(kz z+mφ−ωt). Initially the growth

rates of the perturbed plasma variables are uncorrelated, how-

ever, after several growth times the solution converges to the

fastest growing unstable mode. For this analysis the jet is

considered to be in equilibrium with the surrounding medium

across its boundary at r = r0. The use of fixed boundary con-

ditions allows a global stability analysis of internal unstable

modes (Arber and Howell, 1995; Appl et al., 2000).

It has been shown that sheared azimuthal flow is effective

at reducing the growth rate of the kink instability in z-pinch

plasmas but has little effect on axisymmetric modes (see

Wanex et al., 2005b for a detailed explanation). Sheared axial

flow is effective for stabilizing the sausage mode (DeSouza-

Machado et al., 2000). This suggests that the growth of both

the sausage and kink instabilities can be reduced by com-

bining axial and azimuthal velocity components to produce

helical sheared flow. For this reason sheared helical flow will

be used in this analysis of the kink instability.

Here we present the results of this analysis for two equi-

librium profiles. The parabolic profile is obtained by using

the magnetic field produced by a constant current density in

the jet. We also present results for the constant electron ve-

locity (Bennett) equilibrium profile with field maximum at

2r0/3.

4 Results

Figure 1 shows the results of the linear analysis on the kink in-

stability for the parabolic equilibrium profile with azimuthal

velocity 0.3/√

0.1 + r . The instability growth rates are re-

duced to zero for v0z > 1. Figure 2 shows the results for

the Bennett equilibrium profile with the same velocity as in

Fig. 1. The instability growth rates are reduced to zero for

Fig. 1 This is a 3D plot of the kink instability growth rates for theconstant current density equilibrium profile with v0φ = 0.3 and a = 0.1.The growth rate is shown on the z-axis (in units of vT /r0), the wavenumber is shown of the y-axis (in units of 1/r0) and the value of thecoefficient v0z on the x-axis (in units of vT ). As an example of how tointerpret the plot consider the kink instability growth rate for v0z = 0.3,the growth rate for axial wave numbers 2 and 3 are zero, the growth ratethen increases with increasing wave number to a maximum of ∼0.25for wave numbers 8 and 9 and then decreases to zero for wave numbersabove ∼14. Instability growth rates are zero for wave numbers 2–20 forv0z > 1

Fig. 2 This is a 3D plot of the kink instability growth rates for theconstant electron velocity equilibrium profile with v0φ = 0.3 and a =0.1. The growth rate is shown on the z-axis (in units of vT /r0), the wavenumber is shown on the y-axis (in units of 1/r0) and the value of thecoefficient v0z on the x-axis (in units of vT ). As an example of how tointerpret the plot consider the kink instability growth rate for v0z = 0.9,the growth rate for axial wave numbers 1 to 3 are zero, the growth ratethen increases with increasing wave number to a maximum of ∼0.1 forwave number 5 and then decreases to zero for wave numbers above ∼7.Instability growth rates are zero for wave numbers 1–10 for v0z > 1.2

v0z > 1.2. This can be interpreted to mean that sheared heli-

cal flow can stabilize the kink instability for the parameters

and profiles considered here if

v0z > 1.2. (4)

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Using (1) and (3) one finds that

v0z >

√ln (R)

2.2vT (5)

is the stability criterion for the sausage instability (r = 1 and

a = 0.1). Thus if (4) and (5) are satisfied the growth rates for

both the sausage and kink instability can be reduced to zero

for both of these examples. The Kelvin-Helmholtz stability

condition can also be satisfied if the velocity at the edge of

the jet is sub-Alfvenic in a frame at rest with respect to the

ambient medium.

5 Conclusion

The results of this analysis suggest that the Kelvin-

Helmholtz, sausage and kink instabilities in current carrying

jets can be suppressed by Keplerian helical sheared flow for

some equilibrium profiles. These results are sufficiently posi-

tive to motivate further analysis of the hypothesis that sheared

helical flow can stabilize YSO jets. More work is required to

extend the investigation to a larger range of parameters and

equilibrium profiles.

References

Appl, S., Camenzind, M.: Astron. Astrophys. 256, 354 (1992)Appl, S., Lery, T., Baty, H.: Astron. Astrophys. 355, 818 (2000)Arber, T.D., Howell, D.F.: Phys. Plasmas 3, 554 (1995)Bacciotti, F., Ray, T.P., Mundt, R., Eisloffel, J., Solf, J.: ApJ 576, 222

(2002)Bacciotti, F., Ray, T.P., Eisloffel, J., Woitas, J., Solf, J., Mundt, R., Davis,

C.J.: Astrophys. Space Sci. 287, 3 (2003)Bacciotti, F., Ray, T.P., Coffey, D., Eisloffel, J., Woitas, J.: Astrophys.

Space Sci. 292, 651 (2004)Bally, J., Heathcote, S., Reipurth, B., Morse, J., Hartigan, P., Schwartz,

R.: Astron. J. 123, 2627 (2002)Bateman, G., Schneider, W., Grossman, W.: Nucl. Fusion 14, 669 (1974)Bateman, G.: MHD Instabilities. The MIT Press, Cambridge (1978)Burrows, C.J., Stapelefeldt, K.R., Watson, A.M., et al.: ApJ 473, 437

(1996)Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability.

Clarendon Press, Oxford (1961)

Coffey, D., Bacciotti, F., Woitas, J., Ray, T.P., Eisloffel, J.: Ap&SS 292,553 (2004)

Coker, R., Wilde, B., Keiter, P., et al.: Bull. Am. Phys. Soc. 49(33), 255(2004)

Coppins, M., Bond, D.J., Haines, M.G.: Phys. Fluids 27, 2886 (1984)DeSouza-Machado, S., Hassam, A.B., Sina, R.: Phys. Plasmas 7, 4632

(2000)Fendt, C., Cemeljic, M.: A&A 395, 1045 (2002)Freidberg, J.P.: Ideal Magnetohydrodynamics. Plenum, New York

(1987)Golingo, R.P., Shumlak, U., Nelson, B.A.: Phys. Plasmas 12, 062505

(2005)Keppens, R., Baty, H., Casse, F.: Space Sci. Rev. 121, 65 (2005)Konigl, A., Pudritz, R.E.: In: Mannings V., Boss A.P., Russell S.S. (eds.)

Protostars and Planets IV. The University of Arizona Press, Tucson(2000)

Kruskal, M.D., Schwarzschild, M.: Proc. R. Soc. London Ser. A 223,348 (1956)

Lery, T., Frank, A.: ApJ 533, 897 (2000)Lesch, H., Appl, S., Camenzind, M.: A&A 225, 341 (1989)Mundt, R., Ray, T.P., Buhrke, T., Raga, Z.C., Solf, J.: A&A 232, 37

(1990)Nakamura, M., Meier, D.L.: AIP Conf. Proc. 703, 308 (2004)Ray, T.Muxlow, T.W.B., Axon, D.J., Brown, A., Corcoran, D., Dyson,

J., Mundt, R.: Nature 385, 415 (1997)Ray, T.: Bull. Amer. Phys. Soc. 49, 70 (2004)Reipurth, B., Heathcote, S.: In: Reipurth B., Bertout C. (eds.) Herbig-

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Ruden, E.: IEEE Trans. Plasma Sci. 30, 611 (2002)Shafranov, V.D.: At. Energ. 5, 38 (1954)Shu, F.H., Najita, J.R., Shang, H., Li, Z.: In: Mannings, V., Boss, A.P.,

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Shumlak, U., Hartman, C.W.: Phys. Rev. Lett. 75, 3285 (1995)Sotnikov, V.I., Paraschiv, I., Makhin, V., Bauer, B.S., Leboeuf, J.N.,

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DOI 10.1007/s10509-006-9222-9


How to Produce a Plasma Jet Using a Single and Low EnergyLaser Beam

Ph. Nicolaı · V. T. Tikhonchuk · A. Kasperczuk ·

T. Pisarczyk · S. Borodziuk · K. Rohlena · J. Ullschmied


Abstract Under suitable conditions on laser intensity, focal

spot radius and atomic number a radiative jet was launched

from a planar target. This jet was produced using a relatively

low energy laser pulse, below 500 J and it presents similar-

ities with astrophysical protostellar jets. It lasts more than

10 ns, extends over several millimeters, has velocity more

than 500 km/s, the Mach number more than 10 and the den-

sity above 1018 cm−3. The mechanism of jet formation was

inferred from the dimensional analysis and hydrodynamic

two-dimensional simulations. It is related to the radiative

cooling while the magnetic fields play a minor role.

Keywords Jets . Outflows . Laser-plasma

PACS numbers: 98.38.Fs, 52.50.Jm, 95.30.Qd

1. Introduction

The jets are ubiquitous in the Universe, from active galactic

nuclei (Cecil et al., 1992; Bride and Perley, 1984) to Young

P. Nicolaı () · V. T. TikhonchukCentre Lasers Intenses et Applications, UMR 5107 CEA – CNRS– Universite Bordeaux 1, 33405 Talence cedex, Francee-mail: [email protected]

A. Kasperczuk · T. Pisarczyk · S. BorodziukInstitute of Plasma Physics and Laser Microfusion, ul. Hery 23,00-908 Warsaw 49, Poland

K. RohlenaInstitute of Physics AS CR, Na Slovance 2, 182 21 Prague 8,Czech Republic

J. UllschmiedInstitute of Plasma Physics AS CR, Za Slovankou 3, 182 00Prague 8, Czech Republic

Stellar Objects (Zinnecker et al., 1998; Reipurth et al., 1986)

(YSO). The physics involved in jets formation is compli-

cated and covers a large range of subjects. Consequently

the numerical simulations require multidimensional codes

accounting for hydrodynamics, ionization, radiation trans-

port, equations of state and magnetic fields. The complexity

of the phenomenon makes it challenging to devise labora-

tory experiments which are needed to benchmark the codes

and to model certain aspects of large scale astrophysical

phenomena. Recent experiments, carried out with Z-pinches

(Lebedev et al., 2002; Ampleford et al., 2005) or with high

energy lasers (Farley et al., 1999; Shigemori et al., 2000;

Foster et al., 2002; Rosen et al., 2005) showed the interest

and the relevance of the laboratory jets to some astrophysical

jets. In these experiments, jets were produced from a radia-

tive collapse of a convergent plasma flow or a shock wave,

at a stagnation point. In terms of the experiment geometry,

this method of jet production requires a high energy and/or

a multiple laser beams.

The present study addresses the problem of jet formation

using a single and low energy laser beam. The experiments

were carried out at the Prague Asterix Laser System

(PALS) iodine laser facility (Jungwirth et al., 2001). The

experimental images showed a jet formation under certain

conditions (Borodziuk et al., 2004; Kasperczuk et al.,

2006). Although the radiative effects are important, the

jet creation is neither induced by plasma collision nor by

shock convergence. The numerical simulations carried

out with a multi-physics, two-dimensional (2D) radiative

magneto-hydrodynamic code (Buresi et al., 1986; Drevet,

1997; Nicolai et al., 2000). It was found out that the main

mechanism which transforms the ablated plasma in a plasma

jet is the radiative cooling of expanding plasma. Under

appropriate conditions on the laser energy and the focal spot

radius, the jet is formed having a Mach number larger than

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10, temperatures around 50 eV and electron densities above

1018 cm−3. Moreover, these jets verify the scaling criteria,

detailed by Ryutov et al. (1999), of the laser experiment to

astrophysical conditions.

2. Experiment

The experiment was performed at the PALS iodine laser

facility (Jungwirth et al., 2001; Borodziuk et al., 2004;

Kasperczuk et al., 2006). A laser beam was focused on a flat

massive target by an aspherical lens at the first (1.315 µm)

and the third (0.438 µm) harmonic. The essential part of

experiments was carried out with the laser energy on target

EL = 100 J for both harmonics. The laser pulse duration of

∼ 0.4 ns was kept constant in all shots. To study the expansion

of ablated plasma, a three-frame interferometric system with

an automatic image processing was employed. The diagnos-

tic system was illuminated by the third harmonic of the iodine

laser. Each of the interferometric channels was equipped with

its own independent interferometer of the folding wave type

with a 250 mm focal length and a wedge angle of 3 degrees.

The images were recorded by high resolution 10 bits dy-

namics Pulnix TM-1300 CCD cameras, with the matrix of

1300 × 1300 pixels. The temporal resolution of the interfer-

ometer was determined by the probe pulse duration, which,

in the case of the third harmonic was approximately 0.3 ns

full width at half maximum (FWHM). The spatial resolution

of 20 µm was deduced from the target edge washers. The

delay between subsequent frames was set to 3 ns.

A typical sequence of electron density distributions is pre-

sented in Fig. 1. The first three lines correspond to three ma-

terials: aluminium, copper and silver. In all cases, a plasma

jet is present and it remains visible during a long time. Dis-

tinctly, the jet becomes narrower as the atomic number in-

creases. This observation indicates that radiative processes

play an important role in the experiment. The last sequence

shows the effect of a laser focal spot radius change and a laser

wave length decrease. It was shown in Ref (Borodziuk et al.,

2004; Kasperczuk et al., 2006) that a transition to shorter

wavelength improves the jet parameters. Also the jet for-

mation depends strongly on the focal spot size. As the spot

radius decreases, but the laser energy is kept constant, the ab-

lated plasma ejection becomes quasi-isotropic, whereas, as

the spot radius increases, the plasma extends along the laser

axis. Last, for this radius, 100 µm, a complex multi-bubble

structure can be seen in the figures (pointed out with arrows).

It corresponds to an annular plasma ejection at angles ∼40–

45. This plasma ejection also appears in the simulation.

3. Interpretation of the jet formation

The simulations of the plasma dynamics were per-

formed with the laser-plasma interaction hydrodynamic code

Fig. 1 Experimental sequence of the electron density isolines at instantsof 2, 5, 8 ns for RL = 300 µm, EL = 100 J and the wavelength of0.438 µm for three target materials: Al (a), Cu (b) and Ag (c). ForRL = 100 µm and Al target (d). The densities are in 1018 cm−3

FCI2 (Buresi et al., 1986; Drevet, 1997). This code is cur-

rently used to simulate laser experiments. It has been com-

pared with other codes (Lindl et al., 2004) and successfully

reproduced experiments (Nicolai et al., 2000). The code in-

cludes two-dimensional hydrodynamics, ion and classical

or nonlocal electron conduction, thermal coupling and de-

tailed radiation transport. Ionization, equations of state and

opacity data are tabulated, assuming a local thermodynamic

equilibrium (LTE) or a non-LTE depending on the plasma

parameters. The laser propagation, refraction and collisional

absorption are treated by a ray tracing algorithm. A resistive

MHD package accounting for the azimuthal magnetic fields

generated by the thermal sources (crossed gradients of the

density and temperature) was also included.

Our numerical study started with the following parame-

ters: the pulse duration was τL = 0.4 ns with the temporal

shape I (t) ∼ sin2(π t/2τL ) and the laser energy EL = 100 J,

which leads to the maximum laser power PL = 0.2 TW. For a

given spot radius RL = 300µm, the maximum laser intensity

was I ≃ 8.8 × 1013 W/cm2. The laser beam was focused on

the surface of a massive copper solid target. The intensity pro-

file on the target was not well-characterized and we approxi-

mate it by a super-Gaussian function I (r ) ∼ exp[−(r/RL )n]

with n = 8.

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In order to identify the physical origin of the jet forma-

tion, we performed several runs where certain physical ef-

fects were artificially switched off. The first simulation was

carried under the following conditions: (i) without magnetic

fields, (ii) without the radiation transport and (iii) using a

flux limiter equal to 8% for the electron heat conduction.

Figure 2a presents iso-densities, at 5 ns, well after the end

of the laser pulse. This simulation obviously does not repro-

duce the experimental result. The plasma flow is too broad.

This result has to be compared with those shown in Fig. 2b.

In this simulation, the radiation transport is turned on. The

results are much closer to the experiment. Initially spherical,

the plasma plume is elongated after a few ns. Its radius is two-

three times smaller than in the previous case and the height is

about 3 mm is also in agreement with the experiment. Due to

the radiative cooling, the plasma temperature decreases 2.5

times to the value around 50 eV. The electron heat flux, which

scales as T 7/2, induces, if the radiation transport is switched

off, a more homogeneous density distribution. The intensity

of the X-ray emission is proportional to the square of the

plasma density, it cools more efficiently the dense plasma

and reduces the internal plasma pressure. Comparing both

simulations, we can see that the radiation cooling reduces

the pressure by a factor 30. Consequently, with the radiation

transport, the characteristic expansion velocities are smaller,

specifically in the radial direction. This latter simulation is

considered as the reference. All the following simulations

were performed with the radiation transport switched on.

The magnetic field generation and the nonlocal heat con-

duction are the competing processes that may operate un-

der the present conditions and could be partly responsi-

ble of the jet formation. Indeed, the density gradients are

rather steep and non-collinear with the temperature gradients.

Therefore the nonlocal electron heat conduction (Luciani

et al., 1983; Schurtz et al., 2000) and self-generated magnetic

fields (Nicolai et al., 2000; Glenzer et al., 1999) could be op-

erational. To check on these effects, we analyzed the temporal

behavior of the magnetic field. It reaches a maximum, about

1 MG, during the pulse duration and then it decreases quickly

after the pulse end. Its maximum at t = 5 ns is only 8.6 kG

and its effect on the plasma motion is completely negligible

(Fig. 2c).

In order to address the effect of a tighter laser focusing, for

the same energy and pulse duration, the focal spot size was

reduced three times (RL = 100 µm), so the laser intensity

was nine times higher. The smaller energy deposition volume

leads to a higher temperature, a smaller absorption and a

higher plasma pressure. The expansion from a smaller focal

spot size is faster and the plasma expansion becomes more

spherical (Fig. 2d). In addition, the density falls below the

resolution limit of a few 1017 cm−3 at the distance less than

1 mm. Note that one observes as in the experiment, a ring-

like structure in the direction of about 45 degrees (marked

–0.1

0.0

0.1

(cm)

–0.1

0.0

0.1

(cm)

–0.1

0.0

0.1

(cm)

–0.1

0.0

0.1

(cm)

–0.1

0.0

0.1

(cm)

–0.1

0.0

0.1

(cm)

0.1 0.2 0.3 0.4 (cm) 0.1 0.2 0.3 0.4 (cm)

2 : 1.49

1 : 0.90

3 : 2.46

4 : 4.06

5 : 6.71

6 : 11.1

7 : 18.3

8 : 30.3

2 : 1.49

1 : 0.90

3 : 2.46

4 : 4.06

5 : 6.71

6 : 11.1

7 : 18.3

8 : 30.3

(e)

(c)

(a) (b)

(d)12

3

4

5

(f)

2 1356 4

1234

1234

123456

123

Fig. 2 Electron density distributions at t = 5 ns, for EL = 100 J, τL ∼400 ps, λL ∼ 0.438 µm and RL = 300 µm. Without the radiationtransport (a), the reference simulation (b), with the magnetic fields (c),with a smaller laser spot radius (d), with a different energy distributionin the the focal spot (e) and with an aluminium target (f). See text fordetails. The densities are in 1018 cm−3

by an arrow). This is due to collision of a hot central plasma

with a cold material at the crater border.

The laser energy distribution in the focal spot has an im-

pact on the jet formation. For comparison, we present in

Fig. 2e, the density profile obtained using a gaussian shape

(n = 2). The jet still appears, but an energy distribution

change modifies the form of the jet.

Last, in order to test the target atomic number (Z) depen-

dence, we replaced the copper by aluminium in the simula-

tion (Fig. 2f). As expected, the jet becomes wider and more

isotropic, which is consistent with a less efficient radiation

cooling.

The computational results can be confirmed from a di-

mensional analysis. The characteristic hydrodynamic time

depends on the focal spot radius RL and on the ion acoustic

velocity cs = ((Z + 1)T/mi )1/2.

th = RL/cs = 0.1 RL (A/Z )1/2T −1/2 ns, (1)

where A is the atomic mass in units of proton mass, RL is

in µm and T in eV. The radiative cooling time is the ra-

tio of the plasma thermal energy and radiated power. For

simplicity here we neglect the line emission which is not

too important in light materials. Then the power of the

bremsstrahlung emission is given by (Book, 1980): Pbr =1.7 × 10−32 Zn2

e T 1/2 W/cm3, where the electron plasma

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density is in cm−3. The plasma energy density is defined as

E p = 1.5 neT = 2.4 × 10−19neT J/cm3. Consequently, the

radiation cooling time reads

tr = E p/Pbr = 1.4 × 1022T 1/2/Zne ns. (2)

For the typical parameters of this experiment: RL = 300 µm,

A = 63, Z = 15, T = 100 eV, and ne = 1022 cm−3 (be-

tween the critical density and the ablation front, that is,

around the launching zone), we find the hydrodynamic time

∼ 6 ns and the radiation time ∼ 0.9 ns. The radiative cooling

makes an important effect in agreement with the observa-

tions and the simulations. In the contrary, for the tighter fo-

cusing RL = 100 µm, the temperature is higher T ∼ 300 eV

(Z = 20) and consequently the hydrodynamic time is shorter

∼ 1 ns, while the radiation time becomes longer ∼ 1.2 ns.

In that case the radiative losses are less important and the

plasma expansion is more symmetric. However, one should

keep in mind that these estimates are rather qualitative. From

one hand, we certainly underestimate the radiative losses by

neglecting the line emission. From the other hand, the hy-

drodynamic motion leading to the jet formation is in reality

two-dimensional. Equations (1) and (2) just point out the

main qualitative dependencies of the jet formation mecha-

nism. The jet is better formed if the the spot radius and the

atomic number increase, and if the temperature and the laser

wavelength decrease.

In order to present interest in the astrophysical context, the

laboratory plasma jet needs to have the same dimensionless

parameters as the astrophysical jets. In addition, indepen-

dently of the objet size and its time evolution, the hydrody-

namics is scalable if the dissipative processes are negligible.

These processes may be expressed in terms of dimensionless

parameters: the Peclet and the Reynolds numbers (Ryutov

et al., 1999). The first one mesures the convective transport

relative to the conduction: Pe = Ujet Rjet/κ , where Ujet and

Rjet are the jet characteristic velocity and radius. The thermal

diffusivity reads

κ = 2 × 1021T 5/2/(Z + 1)ne cm2/s, (3)

where is the Coulomb logarithm. The Reynolds num-

ber defines the ratio of inertial force to viscous force: Re =Ujet Rjet/ν, where the kinematic viscosity reads:

ν = 2 × 1019T 5/2 A−1/2 Z−3/ne cm2/s. (4)

Experiments in which Pe ≫ 1 and Re ≫ 1 could be scaled

to the astrophysical system. In laser experiment, the jet

typical parameters are T ∼ 50 eV, ne ∼ 5 × 1018 cm−3,

Ujet ∼ 500 km/s, Rjet ∼ 0.5 mm, Z ∼ 10 and ∼ 5. Un-

der these conditions one obtains κ ∼ 105 cm2/s, ν ∼ 2 cm2/s

and so Pe ∼ 20 and Re ∼ 106. Therefore, our experi-

ment satisfies the conditions for the hydrodynamic scal-

ing with nevertheless a particular attention for the thermal

conduction.

4. Conclusions

We have studied the formation of a plasma jet using a sin-

gle laser beam and a simple planar massive target. The ex-

periment and the simulations indicate that this jet may be

launched using a relatively low laser energy. Under these

conditions, the dissipative processes can be neglected and

the laser produced jet can be scaled to astrophysical condi-

tions. Moreover, such a jet is rather flexible and could be used

for modeling of interaction between the astrophysical jet and

the ambient clouds. For that, one can place in front of the jet,

a solid foil, a foam or a gas jet (Lebedev et al., 2002; Am-

pleford et al., 2005; Foster et al., 2002; Rosen et al., 2005),

by changing the angle of the laser beam incidence by a few

tens of degrees. In addition, by modifying the pulse duration

and intensity, one can modify the velocity and the density of

the jet. Always using a single laser beam, one could build

a series of pulses with an increasing intensity. Each pulse

induces a jet faster than the previous one. A correct timing of

such a pulse sequence should create a series of plasma jets

interacting one with another.

Acknowledgements This work is partly supported by the AquitaineRegion Council, by the Association EURATOM-IPPLM (contract NoFU06-CT-2004-00081), by the Ministry of Scientific Research and In-formation Technology in Poland (project No 3 T10B 024 273), andby the Ministry of Schools, Youth and Sports of the Czech Republic(project No LC528).

References

Ampleford, D., Lebedev, S.V., Ciardi, A., Bland, S.N., et al.: Astrophys.Space Sci. 298, 241 (2005)

Book, D.L.: NRL Plasma Formulary, (Washington D.C., Naval Re-search Laboratory) p. 52 (1980)

Borodziuk, S., Kasperczuk, A., Pisarczyk, T., Gus’kov, S. et al.: Opt.Appl. 34, 31 (2004)

Bride, A.H., Perley, R.: Annu. Rev. Astron. Astrophys. 22, 319 (1984)Buresi, E., Coutant, J., Dautray, R., et al.: Laser Part. Beams 4, 531

(1986)Cecil, G., Wilson, A.S., Tully, R.B.: Astrophys. J. 390, 365 (1992)Drevet, C.: In: Proceedings of the 16th International Conference Fusion

Energy, Montreal 1996, IAEA, Vienna, vol.3, p. 57 (1997)Farley, D., Estabrook, K.G., Glendinning, S.G. Glenzer, S.H., et al.:

Phys. Rev. Lett. 83, 1982 (1999)Foster, J.M., Wilde, B.H., Rosen, P.A., Perry, T.S.: Phys. Plasmas 9,

2251 (2002)Glenzer, S., Alley, W., Estabrook, K., DeGroot, J. et al.: Phys. Plasmas

6, 2117 (1999)

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Jungwirth, K., Cejnarova, A., Juha, L., Kralikova, B. et al.: Phys. Plas-mas 8, 2495 (2001)

Kasperczuk, A., Pisarczyk, T., Borodziuk, S., Ullschmied, J. et al.: Phys.Plasmas 13, 062704 (2006)

Lebedev, S.V., Chittenden, J.P., Beg, F.N., et al.: Astrophys. J. 564, 113(2002)

Lindl, J.D., Amendt, P., Berger, R.L., Gendeinning, S.G. et al.: Phys.Plasmas 11, 339 (2004)

Luciani, J.-F., Mora, P., Virmont, J.: Phys. Rev. Lett. 51, 1664 (1983)Nicolaı, Ph., Vandenboomgaerde, M., Canaud, B., Chaigneau, F.: Plys.

Plasmas 7, 4250 (2000)

Reipurth, B., Bally, J., Graham, J., Lane, A., et al.: Astron. Astrophys.164, 51 (1986)

Rosen, P.A., Wilde, B.H., Williams, R.J., et al.: Astrophys. Space Sci.298, 121 (2005)

Ryutov, D., Drake, R.P., Kane, J., Liang, E. et al.: Astron. Astrophys.518, 821 (1999).

Schurtz, G., Nicolai, Ph., Busquet, M.: Phys. Plasmas 7, 4238 (2000)and reference therein

Shigemori, K., Kodama, R., Farley, D.R., et al., Phys. Rev. E.: 62, 8838(2000)

Zinnecker, H., McCaughrean, M., Rayner, J.: Nature 394, 862 (1998)

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DOI 10.1007/s10509-006-9286-6


Experimental Design for the Laboratory Simulationof Magnetized Astrophysical Jets

Radu Presura · Stephan Neff · Lucas Wanex

Received: 22 April 2006 / Accepted: 28 November 2006C© Springer Science + Business Media B.V. 2006

Abstract Concepts of several experimental configurations

for the investigation of magnetized jets and their interaction

with magnetized environments are presented. In the planned

experiments, the plasma jets will be created by laser ablation

of shaped targets, while magnetic and electric fields with the

required configurations will be produced independently by

a pulsed power generator. In particular, the recently coupled

Terawatt laser Tomcat and Terawatt pulsed power generator

Zebra will be used for experiments.

Keywords Laboratory astrophysics . Astrophysical jets .

Ablation plasma

1 Introduction

Highly-collimated, oppositely directed jets are observed to

originate from a variety of astrophysical systems at scales

differing by many orders of magnitude, from young stellar

objects (YSO; e.g. Reipurth and Bally, 2001; de Gouveia Dal

Pino, 2005), to neutron stars and black holes (e.g. Mirabel and

Rodriguez, 1998, 1999), to active galactic nuclei (AGN; e.g.

de Gouveia Dal Pino, 2005; Livio, 2002; Massaglia, 2003).

Despite similarities in their appearance, the observed and

inferred properties of these jets are very different (see the

first two lines in Table 1). For example, the AGN jets are

relativistic, less dense than the environment, and emit non-

thermal continuum radiation in a broad spectrum. The YSO

jets are denser than the ambient medium and the line radia-

tion behind the shock regions produces strong cooling that

influences the morphology and dynamics of the jets (Blondin

R. Presura () · S. Neff · L. WanexNevada Terawatt Facility, University of Nevada, Renoe-mail: [email protected]

et al., 1990). However, both types of jets form when mate-

rial from an accretion disk falls onto a central object in the

equatorial plane and is ejected at supersonic speeds along the

rotation axis. Most likely, both are accelerated and collimated

by mechanisms in which magnetic fields play dominant roles

(Blandford and Payne, 1982; Shu et al., 1994; Lovelace et al.,

1987). Such fields have been measured or inferred from re-

cent observations for jets with different origins (Ray et al.,

1997; Zavala and Taylor, 2005). Nevertheless, open questions

remain, mainly regarding the formation of jets; their propa-

gation in different environments with the related effects on

stability, matter entrainment, and turbulence; and their ter-

mination to form Herbig-Haro objects, hot-spots, and radio

lobes. To help address such issues, the laboratory simulation

of astrophysical jets emerged as a very promising comple-

mentary investigation tool. Well-controlled laboratory exper-

iments with scaled-down plasma jets can provide additional

information, valuable for identifying the essential physical

mechanisms, deciding between different models, and bench-

marking simulation codes. The focus of this paper is on the

design of experiments to investigate the properties of magne-

tized plasma jets and their interaction with the environment.

Such experiments at the Nevada Terawatt Facility (NTF), us-

ing two independent Terawatt (TW) sources of plasma and

magnetic field, can explore parameter ranges that become ac-

cessible through the independent control of the plasma and

field parameters.

Considering the vast difference in parameters between as-

trophysical and laboratory plasma jets (Table 1), it is obvi-

ously impossible to achieve a complete scaling down, that

is to design an experiment which is identical in all essential

dimensionless parameters with the natural system. Instead,

the jet dynamics and the role of dissipative processes are ad-

equately simulated if the dimensionless plasma parameters

of the order of unity in the astrophysical system are also of

Springer


Table 1 Typical parameters of plasma jets originating from activegalactic nuclei (AGN) and young stellar objects (YSO), and possibleto produce in the laboratory: from the front and back of a laser irra-diated target and from a previous experiment (Presura et al., 2005).Physical parameters of the jets are compared in the first five columns:size, velocity, electron density, electron temperature, magnetic field. In

the next six columns, the comparison includes dimensionless parame-ters: the Mach number, the Reynolds number, the Peclet number, thedensity contrast, the plasmaβ, and the magnetic Reynolds number. Thenumbers in parentheses are inferred from modeling. The Mach num-ber for the laboratory jet was estimated without taking into account theeffect of radiative cooling

Flow L (cm) v (cm/s) ne (cm−3) Te (eV) B (G) M Re Pe η β ReM

AGN 1018 –1024 1010 NA NA 10−3 –10−5 (10–100) – – (<10−2) (100) –

YSO 1016 – 1019 1–5×107 10 – 105 10−3 – 1 10−4 10 – 50 1013 1011 1–20 1–10 1015

Front 0.2 2–6×107 1018–1019 1–3×102 ∼5×105 ∼10 ∼1000 ∼10 10 1 ∼100

Back 0.2 ∼107 ∼1020 ∼50 – ∼4 1500 1000 100 – –

Presura et al. (2005) 0.02 3×107 1018 150 8 10 40 0.1 10 1 30

the order of unity in the laboratory, and those that are much

larger (smaller) than one in nature are likewise much larger

(smaller) than one in the experiment (Ryutov et al., 2001,

1999; Baranov, 1969). For example, a magnetic Reynolds

number of the order of 10 in the laboratory may have the

same physical significance as a magnetic Reynolds number

many orders of magnitude larger has for an astrophysical jet,

if the magnetic field diffusion is negligible over the time scale

of interest, which is many orders of magnitude shorter in the

laboratory. Once such “scaling” can be justified, the small

scale laboratory simulations of astrophysical phenomena be-

come relevant. For magnetized plasma jets, one additional

layer of difficulty results from the fact that when the plasma

temperature T increases, the hydrodynamic Reynolds num-

ber and the Peclet number decrease (∝ T −1 for magnetized

electrons) while the magnetic Reynolds number increases

(∝ T 3/2). In this case, the particular physical aspect inves-

tigated determines which dimensionless parameters have to

be maximized.

Overcoming scaling difficulties, several well controlled

experiments produced valuable results that boosted the con-

fidence of the astrophysical community in laboratory-based

astrophysics. These experiments investigated a variety of top-

ics including: the effect of radiative cooling on the collimated

propagation of supersonic plasma jets produced with intense

lasers from shaped targets (Farley et al., 1999; Shigemori

et al., 2000) and with conical-array z-pinch experiments

(Lebedev et al., 2002); the hydrodynamic stability and the

termination shock of a laser-produced supersonic plasma jet

launched in a dense ambient medium (Foster et al., 2002);

the supersonic jet deflection and internal shocks produced by

ambient plasma winds (Lebedev et al., 2004); the stabilizing

effect of an axial magnetic field upon the kink instability

of a plasma jet produced in a coaxial gun (Hsu and Bellan,

2003); the generation of rotating jets with twisted wire arrays

(Ampleford et al., this issue; Lebedev et al., 2005a); the for-

mation of magnetic tower outflows from a radial wire array

z-pinch (Lebedev et al., 2005b).

At the NTF, the role played by magnetic fields in the accel-

eration, collimation, and interaction of jets with the ambient

medium is of primary interest. Two prominent jet theories,

the magnetocentrifugal model (Blandford and Payne, 1982;

Shu et al., 1994), and the Poynting-flux-dominated model

(Lovelace et al., 1987) require the astrophysical jets to carry

current and to be collimated by helical magnetic fields. This

configuration is subject to magnetohydrodynamic (MHD)

instabilities. In fact, the sausage m = 0 and kink m = 1 in-

stabilities have been invoked to explain the observation of

knots, wiggles, and filamentary structures in astrophysical

jets (Pearson, 1996; Reipurth and Heathcote, 1997). If such

instabilities develop indeed, they should eventually disrupt

the jet collimation. On the other hand, axial magnetic fields or

radial velocity shears could prevent the disruption (Hsu and

Bellan, 2003; Wanex et al., 2005). Jets from young stellar ob-

jects are most likely threaded by axial magnetic field (Konigl

and Pudritz, 2000). Both axial and rotational sheared flows

have been observed in astrophysical jets (Bacciotti et al.,

2004; Bally et al., 2002; Coffey et al., 2004). In addition, stel-

lar evolution models predict that low-mass protostars form in

an interstellar medium that is supported by a magnetic field

(Ray et al., 1997; Konigl and Pudritz, 2000). This magne-

tized medium could supply the instability reducing mech-

anism in current carrying jets. However, the velocity shear

layer formed at the interface between the jet and the ambient

creates conditions favorable for the growth of the Kelvin-

Helmholtz instability. Three-dimensional MHD simulations

(Ryu et al., 2000) showed that the magnetic fields embedded

in the plasma flow affect the instability differently depend-

ing on their strength, indicated by the Alfven-Mach number.

When MA ≥ 50, the instability remains hydrodynamic in na-

ture, but the dissipation is enhanced through magnetic recon-

nection; when 4 ≤ MA ≤ 50, the magnetic field is amplified

in the Cat’s Eye, leading to the reorganization of the flow

into a stable configuration; when 2 ≤ MA ≤ 4, the magnetic

field tension, enhanced during the linear growth, prevents the

transition to the nonlinear phase, so the flow is nonlinearly

Springer


Table 2 Facilities operational and in different stages of developmentat the Nevada Terawatt Facility. For Zebra as magnetic field generator,the values given are the magnetic energy in the experimental region and

the duration over which the field varies with less than 5%, around thecurrent maximum

Device Implementation E (J) τ (ps) I (W/cm2) B (T) Operational

Zebra pulsed Wire array 5×104 105 1011 >250 1999

power B (Presura et al., 2005) 103 6×104 20 2004

generator B (Martinez et al., this issue) 103 6×104 200 2005

Tomcat Q-switch (Presura et al., 2005) 5 5×103 1014 2004

laser Long pulse 15 8×102 3×1015 2005

Short pulse 10 1 1018 2006

Leopard Long pulse 200 103 3×1016 2007

laser Short pulse 35 0.35 1019 2007

stable; when MA ≤ 2, the instability is not initiated, so the

MHD flow is linearly stable.

2 Experimental facilities at the NTF

Magnetized plasma jet experiments will be performed at the

Nevada Terawatt Facility (NTF) using the facilities opera-

tional and in various stages of development listed in Table 2

together with their main parameters. The synergistic com-

bination of these devices allows independent control of the

plasma flow and magnetic field parameters and thus offers

access to parameter ranges and experimental configurations

unattainable otherwise. A previous experiment (Presura et al.,

2005) based on the Tomcat Q-switch laser (Table 2) coupled

with the Zebra generator (Table 2, line 2) focused on the

interaction of an explosive ablation plasma plume with am-

bient magnetic field, with relevance to active magnetospheric

experiments and to the expansion of supernova remnants in

magnetized interstellar medium. The typical values of the pa-

rameters in the plasma-field interaction region are included

in the last line of Table 1. The 10 TW short pulse laser Tomcat

(Wiewior, to be published) and the 2 TW z-pinch generator

Zebra (Bauer et al., 1997) were recently coupled. Examples

of plasma jet parameters achievable with the coupled de-

vices are given in Table 1. The plasma flow parameters are

based on simulations with the 1-dimensional hydrodynamic

code MULTI (Ramis et al., 1988) for the front/back irradia-

tion of CH thick/thin targets. The effects of a magnetic field

B ≈ 20T were estimated analytically. The values of the di-

mensionless parameters indicate that relevant regimes can

be attained. These facilities are complemented by a mature

suite of laser diagnostics: shadowgraphy, schlieren imaging,

interferometry, and Faraday rotation. They provide informa-

tion regarding the spatial distribution of plasma density and

magnetic field. Spectroscopy in several spectral ranges and

Thomson scattering are under development for plasma tem-

perature estimates and additional density and field measure-

ments.

3 Plasma jet experiments

Although previous experiments demonstrated that both

pulsed power generators and lasers can generate plasma jet

conditions relevant to astrophysical situations, the focus here

will be on laser produced jets and their interaction with mag-

netic and electric fields or magnetized ambient plasma pro-

duced by current carrying conductors. Conical targets are

especially suited to create jets by laser ablation (Farley et al.,

1999; Shigemori et al., 2000). The ablated plasma initially

expands supersonically perpendicular to the concave surface

of the target. Due to the symmetry of the target, this cre-

ates a conical shock which refracts the converging flow to

form a supersonic jet propagating along the symmetry axis

(Tenorio-Tagle et al., 1988).

The velocity shear layer formed at the interface between

protostellar jets and the ambient creates conditions favorable

for the growth of the Kelvin-Helmholtz instability. However,

an axial magnetic field embedded in the plasma flow can

provide stabilization mechanisms (Ryu et al., 2000). An ex-

perimental set-up for the study of the effect of a magnetic

field upon the Kelvin-Helmholtz instability is illustrated in

Fig. 2. A plasma jet created by ablation of a conical tar-

get is directed towards a region where ambient plasma and

magnetic field can be generated independently. For typical jet

density ne ≥ 1018cm−3 and ambient density ne ≥ 1017cm−3,

the density contrast η = ρjet/ρambient ≈ 10, similar to that ob-

served for protostellar jets. Magnetized ambient plasma can

be created by ablation or desorption from a current carrying

conductor. Non-magnetized ambient plasma can be gener-

ated by laser ablation. The magnetic fields can be generated

with coils driven by the Zebra pulsed power generator. Con-

ductor ablation was observed for helical coils, at fields around

200 T (Fig. 1a,b). Magnetic flux densities up to 100 T were

obtained with horse shoe coils with no conductor ablation

(Fig. 1c) (Martinez et al., this issue). In these experiments

the stability of the plasma jet will be monitored with time

gated laser shadow imaging. The jet density and magnetic

field in the boundary layer will be determined with time gated

Springer


Fig. 1 Laser shadow images: (a) helical coil (reference); (b) helical coil (at 150 T); (c) horse shoe coil (at 100 T). All coils are made of 1 mm thickstainless steel wire

Fig. 2 Plasma jet produced by laser ablation of a conical target, prop-agating along a uniform external magnetic field. An ambient plasma,magnetized or not can be created at the location of the coil

interferometry and Faraday rotation. Faraday rotation in glass

probes (Martinez et al., this issue) will be used to monitor

the magnetic field produced by the coil.

Using independent plasma and field sources (Fig. 2) al-

lows the injection of the laser generated jet at an arbitrary an-

gle with respect to the ambient magnetic field. This general

configuration allows the investigation of termination shocks

and of plasma flow penetration across the magnetic field. In

such experiments, the plasma density and temperature, and

the magnetic field distributions are the relevant parameters.

They can be determined with time gated laser imaging, short

wavelength imaging, and Faraday rotation, respectively.

Astrophysical jet models (Blandford and Payne, 1982; Shu

et al., 1994; Lovelace et al., 1987) assume that the jets carry

current. As a result, they are expected to be MHD unstable

and, in fact, the sausage m = 0 and kink m = 1 instabilities

might explain the structures observed in astrophysical jets

(Pearson, 1996; Reipurth and Heathcote, 1997). To investi-

gate in the laboratory the effect of possible stabilizing mech-

anisms such as axial magnetic fields and axial or azimuthal

sheared flows, additional experimental configurations are

considered. These rely on plasma flows created by laser ab-

lation in external electric or magnetic fields produced with

the pulsed power generator. In this case, instead of a shaped

target, a shaping environment is used for the jet formation.

For example, a ring focus on an insulating target surround-

ing the high voltage electrode will produce a plasma plume

threaded by a radial electric field Er (Fig. 3a). This will in-

duce into the axial plasma flow with velocity vz an azimuthal

magnetic field Bϕ = Er/vz . For example, embedding a 25 T

magnetic field in a plasma flow with velocity around 400 km/s

requires an electric field around 100 kV/cm, which is below

the threshold for field electron emission, about 300 kV/cm.

Applying an additional axial magnetic field Bz will produce

jet rotation with vϕ = Er/Bz . In a variation of this set-up

(Fig. 3b), the plasma is produced by laser ablation of a pin tip,

so the expansion, guided by the radial electric field, bridges

the inter-electrode gap. When a radial current is established

through the plasma, the system behaves like a plasma fo-

cus or magnetic tower (Lebedev et al., 2005b) under the ac-

tion of a force with density fz = jr Bϕ , producing a dense

current-carrying column and a large radius cocoon-like dif-

fuse current return plasma. An additional axial magnetic field

Bz in this set-up, such that Bz ≥ Kµ0 Iz/2πr , is expected to

have a stabilizing effect upon the current-carrying jet. Here

K is a constant dependent on the instability mode, and the

rest of the right hand side is the azimuthal magnetic field. A

configuration of wires thick enough to delay ablation can be

used in z-pinch configuration to create an ambient magnetic

field. For instance, the Joule heating of 8 wires with 1 mm

diameter each is insufficient to reach the melting temper-

ature before the Zebra current peak. Such a twisted conical

array (Fig. 3c) will produce an axial magnetic field, similar to

that inferred in previous experiments (Ampleford et al., this

issue; Lebedev et al., 2005a), parallel to the current flow-

ing in the plasma jet. Other quasi-force-free configurations

Springer


Fig. 3 Possible experimentalarrangements for the generationof magnetized and currentcarrying jets using laser ablationin strong external fields. Thecoaxial inner (brown) and outer(grey) cylinders are theelectrodes of the pulsed powergenerator. The plume on the topof the center electroderepresents the laser producedplasma. The straight linesrepresent electric field lines in(a) and (b), and conductors in(c) and (d)

(Furth et al., 1988), including twisted cylindrical wire ar-

rays and helical wire arrays, in which the current and the

magnetic field vector are parallel, can be used to investi-

gate the stabilizing effect of the magnetic field. The set-up

presented in Fig. 3d, with a regular conical wire array, can

be used for null results, without axial magnetic field. In ac-

tual experiments, to diagnose the jet formation phase, the

outer electrode, represented as a cylinder in Fig. 3c and 3d,

consists of individual current return rods, to allow diagnos-

tics access. The jet stability and interaction during its prop-

agation is investigated above the electrode structure. The jet

stability will be monitored with laser and short-wavelength

imaging and the magnetic field configuration with Faraday

rotation.

The radiative cooling of jets by optically thin radiation

emission has a significant effect on the dynamics and mor-

phology of supersonic flows such as the protostellar jets.

This effect was investigated through simulations (Blondin

et al., 1990) and was evidenced in laboratory experiments

with lasers (Farley et al., 1999; Shigemori et al., 2000) and

z-pinch (Lebedev et al., 2002). These papers show that the

radiative cooling is stronger for higher atomic numbers. This

dependence can be exploited to vary the Mach number and

the density contrast in the experiment. The effect can be

evidenced by measuring the density distribution with laser

imaging and by imaging the self-emission in soft x-rays with

multi-frame instruments.

4 Conclusions

The stability and collimation of magnetized jets and their

interaction with magnetized ambient plasma will be investi-

gated experimentally at the NTF. The experiments will take

advantage of the existing coupled TW laser and TW pulsed

power generator, which allow the independent variation of

the plasma flow and magnetic field parameters. A variety of

experimental configurations were identified that address sig-

nificant aspects of the physics of jets and are complementary

to other current efforts.

Acknowledgements The authors thank V. Ivanov and P. Laca for thehelp with the laser diagnostics. The authors are thankful for very usefuldiscussions of some of these concepts with A. Frank, D. Ampleford, S.Lebedev, B. Remington, and R. P. Drake. This work was supported byDOE/NNSA under the UNR grant DE-FC52–01NV14050.

References

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(2004)Bally, J., Heathcote, S., Reipurth, B. et al.: Astron. J. 123, 2627 (2002)Baranov, V.B.: Cosmic Res. 7, 98 (1969)Bauer, B.S., Kantsyrev, V.L., Winterberg, F., et al.: AIP Conf. Proc. 409,

153 (1997)Blandford, R.D., Payne, D.G.: MNRAS 199, 883 (1982)

Springer


Blondin, J.M., Fryxell, B.A., Konigl, A.: Astrophys. J. 360, 370 (1990)Ciardi, A., Lebedev, S.V., Chittenden, J.P., et al.: Laser Part. Beams 20,

255 (2002)Coffey, D., Bacciotti, F., Woitas, J., et al.: Astrophys. J. 604, 758 (2004)de Gouveia Dal Pino, E M.: Adv. Space Res. 35, 908 (2005)Farley, D.R., Estabrook, K.G., Glendinning, S.G., et al.: Phys. Rev. Lett.

83, 1982 (1999)Foster, J.M., Wilde, B.H., Rosen, P.A., et al.: Phys. Plasmas 9, 2251

(2002)Furth, H.P., Jardin, S.C., Montgomery, D.B.: IEEE Trans. Magn. 30,

1467 (1988)Hsu, S.C., Bellan, P.M.: Phys. Rev. Lett. 90, 215002 (2003)Konigl, A., Pudritz, R.E.: in Protostars and Planets IV, edited by

Mannings, V. et al., The University of Arizona Press, Tucson, 2000,p. 759

Lebedev, S.V., Ampleford, D., Ciardi, A., et al.: Astrophys. J. 616, 988(2004)

Lebedev, S.V., Chittenden, J.P., Beg, F.N., et al. : Astrophys. J. 564, 113(2002)

Lebedev, S.V., Ciardi, A., Ampleford, D.J., et al.: Plasma Phys. Control.Fusion 47, B465 (2005a)

Lebedev, S.V., Ciardi, A., Ampleford, D.J., et al.: MNRAS 361, 97(2005b)

Livio, M.: Nature 417, 125 (2002)Lovelace, R.V.E., Wang, J.C.L., Sulkanen, M.E.: Astrophys. J. 315, 504

(1987)Martinez, D., et al.: Astrophys. Space Sci., doi: 10.1007/s10509-006-

9275-9 (2006)

Massaglia, S.: Astrophys. Space Sci. 287, 223 (2003)Mirabel, I.F., Rodriguez, L.F.: Nature 392, 673 (1998)Mirabel, I.F., Rodriguez, L.F.: Ann. Rev. Astron. Astrophys. 37, 409

(1999)Pearson, T.J.: in Energy Transport in Radio Galaxies and Quasars, edited

by Hardee, P.E., et al. (ASP Conf. Series Vol. 100, NRAO, Green-bank, 1996), p. 97

Presura, R., Ivanov, V., Sentoku, Y., et al.: Astrophys. Space Sci. 298,299 (2005)

Ramis, R., Meyer-ter-Vehn, J., Schmaltz, R.: Comput. Phys. Commun.49, 475 (1988)

Ray, T.P., Muxlow, T.W.B., Axon, D.J., et.al.: Nature 385, 415 (1997)Reipurth, B., Bally J.: Ann. Rev. Astron. Astrophys. 39, 403 (2001)Reipurth, B., Heathcote, S.: in Herbig-Haro Flows and the Birth of Low

Mass Stars, edited by Reipurth, B., Bertout, C., IAU Symposium182, 3 (1997)

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Plasmas 8, 1804 (2001)Shigemori, K., Kodama, R., Farley, D.R.: Phys. Rev. E 62, 8838 (2000)Shu, F., Najita, J., Ostriker, E., Wilkin, F.: Astrophys. J. 429, 781 (1994)Tenorio-Tagle, G., Canto, J., Rozyczka, M.: Astron. Astrophys. 202,

256 (1988)Wanex, L.F., Sotnikov, V.I., Leboeuf, J.N.: Phys. Plasmas 12, 042101

(2005)Wiewior, P., et al.: to be published.Zavala, R.T., Taylor, G.B.: Astrophys. J. 626, L73 (2005)

Springer


DOI 10.1007/s10509-006-9244-3


Excitation of Electromagnetic Flute Modes in the Process ofInteraction of Plasma Flow with Inhomogeneous Magnetic Field

V. I. Sotnikov · R. Presura · V. V. Ivanov · T. E. Cowan ·

J. N. Leboeuf · B. V. Oliver


Abstract Laboratory experiments on the interaction of a

plasma flow, produced by laser ablation of a solid target with

the inhomogeneous magnetic field from the Zebra pulsed

power generator demonstrated the presence of strong wave

activity in the region of the flow deceleration. The deceler-

ation of the plasma flow can be interpreted as the appear-

ance of a gravity-like force. The drift due to this force can

lead to the excitation of flute modes. In this paper a lin-

ear dispersion equation for the excitation of electromagnetic

flute-type modes with plasma and magnetic field parameters,

corresponding to the ongoing experiments is examined. Re-

sults indicate that the wavelength of the excited flute modes

strongly depends on the strength of the external magnetic

field. For magnetic field strengths∼0.1 MG the excited wave-

lengths are larger than the width of the laser ablated plasma

plume and cannot be observed during the experiment. But

for magnetic field strengths ∼1 MG the excited wavelengths

are much smaller and can then be detected.

Keywords Flute instability . Laser plasma ablation .

Laboratory astrophysics experiments

1 Introduction

Interaction of plasma flows with magnetic fields plays an

important role in astrophysics and space physics, ranging

V. I. Sotnikov () · R. Presura · V. V. Ivanov · T. E. CowanUniversity of Nevada at Reno, NV 89557

J. N. LeboeufJNL Scientific, Casa Grande, AZ 85222

B. V. OliverSandia National Laboratories, NM 87123

from supernova explosions and interaction of the solar wind

with the magnetopause to the barium release experiments in

the ionosphere. During recent experiments on interaction of

laser ablated plasma flows with the magnetic field created

by the Zebra pulse power generator strong wave activity was

detected in the region of plasma flow deceleration by the

magnetic field (Presura et al., 2006). Similar phenomena can

take place during Novae explosions (Zakharov, 2003) and ar-

tificial magnetospheric releases, similar to the AMPTE mag-

netotail release (Bernhardt et al., 1987; Ripin et al., 1993).

To study its linear excitation and nonlinear evolution, a non-

linear set of equations for electrostatic potential, magnetic

field, and density has been derived in the low frequency limit

(ω ≪ ci ,where ω is the frequency of the excited mode and

ci = ZeB0z/Mi c is the cyclotron frequency of the ion with

charge Z and mass Mi = µMp, Mp being the proton mass)

from two-fluid macroscopic equations which include gyro-

viscosity (Sotnikov et al., accepted by IEEE TPS, 2006). The

experimental set-up is illustrated in Fig. 1.The plasma flow

was created by laser ablation of a massive solid CH2 tar-

get using the “Tomcat” laser (5 J, at 1054 nm, 6 ns). At best

focus, the laser irradiance on target was ≈1014 W/cm2. An

azimuthal magnetic field Bϕ(r ) = µ0 I/2πr was produced

by an axial current I (0.6 MA) flowing in a 14 mm diameter

rod used as z-pinch load of the Zebra pulsed power gener-

ator with 200 ns rise time (see Fig. 1). The magnetic field

generated at the peak of the pulse was 8 T at the laser target

surface and 17 T at the rod surface, measured with mag-

netic probes. The laser was synchronized with the z-pinch so

that the ablation plasma was produced and evolved during a

≈30 ns period of constant magnetic field at the current peak.

The background pressure was <10−5 Torr to assure that col-

lisions with neutrals had a negligible influence on the plasma

evolution. No plasma was generated at the rod surface. The

plasma was probed with a laser operating at 532 nm with a

Springer


Fig. 1 Experimental set-up

0.2 ns pulse width. We fielded two-frame schlieren and two-

frame interferometry diagnostics with 7 ns delay between

the two frames. Schlieren images were recorded either in the

plane x–z containing the magnetic field lines or in the plane

x-y perpendicular to B (Figure 1). Interferometric measure-

ments were performed in the x–y plane, simultaneously with

x–z Schlieren images.

2 Electromagnetic flute mode instability

Two-fluid macroscopic equations will be used to describe

low frequency flute modes (ω ≪ ci ) in a weakly inhomo-

geneous plasma with external magnetic field B0z(x) (Sotnikov

et al., accepted to IEEE TPS, 2006 and references there in).

The following notations were chosen: direction of the plasma

flow along the x-axis, and external magnetic field along the

z-axis. As is customary in flute mode turbulence, oscillations

are taken to be uniform in the direction of the magnetic field,

i.e. the wave vector along the magnetic field k|| = 0. We

consider an inhomogeneous high beta plasma in slab geom-

etry with density n0(x) in the presence of an inhomogeneous

magnetic field B0z(x), where x is the direction of inhomo-

geneity. In Fig. 2 the experimental plasma density profile and

the adjusted magnetic field profile are presented. Enhanced

wave activity was observed in the (x, y) plane along the y-

direction (along the plasma-magnetic field interface) inside

the plasma–magnetic field interface (region A in Fig. 2).

Deceleration of plasma flow in the process of interaction

with the magnetic field causes the gravity-like term g = gex ,

x

nB

A

θ

Fig. 2 Experimental density and adjusted magnetic field radial profiles

which drives the instability of the flute modes. In the finite

beta plasma in the region where plasma flow kinetic energy

is equal to magnetic field energy, the electric field in the flute

oscillations is not irrotational (i.e. ∇ × E = 0), in contrast

with the low beta case, and is written as E = −∇ − 1c∂ A∂t

.

The plasma density N can be expressed as the sum of a

slowly varying component with x, the equilibrium plasma

density n0(x), and a perturbed component due to flute os-

cillations δn(x, y, t). Likewise the magnetic field is writ-

ten as Bz = B0z(x) + δBz(x, y). We also assume the quasi-

neutrality condition Ne = Z Ni . The equilibrium condition

in this case is written as:

κB =(

1 + Z Te

Ti

)

βi

2κN + βi

V 2T i

g, (1)

where

κN = − 1

n0

dn0

dx> 0, κB = 1

B0z

d B0z

dx> 0, βi = 8πn0i Ti

B20

As will be shown later, the characteristic wave lengths of the

flute modes which can be excited for the plasma parameters

observed in the experiment (Fig. 1) are much larger than the

typical size of the region where interaction takes place, when

the strength of the magnetic field, produced by the pulsed

power generator is∼0.1 MG. But for much stronger magnetic

field strength (∼1 MG) typical wavelengths of the excited

flute modes are much smaller than the size of the region

where interaction occurs. The following relation connects

δn and δBz :

δBz

B0z

= −1

2βδn

n0

, where β = βi + βe. (2)

From (2) it follows that in a low beta case we can neglect the

electromagnetic component in the flute mode and consider it

as electrostatic. The dispersion equations for the frequency

and the growth rate of the electromagnetic flute oscillations,

valid for both low and high beta plasma, was derived in Sot-

nikov et al., 2006. In the system where ions are at rest the

dispersion equation is written as:

ω2 + ky Vtotω − 1

4(

1 + 12β)k2

y V(0)

iy V(0)

i Dy

+ 1 + 12βe

1 + 12β

k2y

k2⊥

gκN

Z= 0. (3)

In this equation:

Vtot = V(0)

iy − 1 + 14β

1 + 12β

V(0)

i Dy − 1

1 + 12β

V (0)ey ;

Springer


Fig. 3 Dependence of growth rate γ (in (gκn)1/2 units) of the electromagnetic flute-type modes on the wave vector ky (in units of the inverse ionskin depths ωpi /c). In (a) magnetic field B0 = 0.1 MG and in (b) magnetic field B0 = 1 MG

V(0)

iy = − g

0i

+ V(0)

i Dy ; V (0)ey = κN

V 2T e

0e

; V 2T e = Te

me

;

V(0)

i Dy = −κN

V 2T i

0i

; V 2T i = Ti

Mi

;β= βi + βe; k2⊥ = k2

x + k2y .

We have solved numerically the dispersion relation in Eq. (3)

using plasma parameters typical of the experiments: n0e =1.0 × 1018 cm−3, n0i = 3.0 × 1017 cm−3, Ti ∼ Te = 150 eV,

B0 = (0.1 − 1.0) MG and g ∼ 5 × 1014 cm/s2. Only modes,

propagating along the plasma-magnetic field interface region

(along the y-direction) were considered since during the ex-

periment only perturbations along the y-axis were observed.

The solution is displayed in Fig. 3, where the growth rate

of the flute-like electromagnetic perturbations is plotted as

a function of wave vector ky . As follows from Fig. 3, in the

case when the external magnetic field strength B0 = 0.1 MG,

the smallest possible wavelength of the excited flute modes

along the y-direction is λ∼2 cm. These waves cannot be ex-

cited in the system, because their wavelengths are larger then

the characteristic size of the laser ablated plasma plume in

this direction. But when the magnetic field strength is in-

creased to B0 = 1 MG, the part of the excited wave spec-

trum with large ky corresponds to the wavelengths λ∼1 mm

and these waves can be excited along the y-direction inside

the plasma-magnetic field interface region.. The growth rate

yields a characteristic time for the instability to develop of

∼10 ns.

3 Conclusion

In support of the planned experiments in the NTF labora-

tory to investigate generation of the flute modes in the region

of plasma flow deceleration, we investigated the solutions

of the dispersion relation which describes excitation of the

electromagnetic flute modes in a finite beta plasma. Obtained

growth rates for the plasma and magnetic field parameters

corresponding to the experimental setup show that the width

of the excited wave spectrum of the flute modes strongly de-

pends from the strength of the external magnetic field B0.

For the smaller values of the magnetic field (B0 ∼ 0.1 MG)

the characteristic scale of the flute mode perturbations is of

the order of ∼2 cm and it exceeds the characteristic width

of the region where excitation of the flute modes is ex-

pected to take place. But with increase of the magnetic field

strength (B0 ∼ 1 MG) it is possible to excite much shorter

wavelengths ∼1 mm. This allows to excite flute modes in-

side the deceleration region along the plasma-magnetic field

interface.

Acknowledgements This work was supported by the United StatesDepartment of Energy under the following grants: Grant No. DE-FC52-01NV14050 at the University of Nevada at Reno, Grant No. DE-AC04-94AL85000 at Sandia National Laboratories.

References

Bernhardt, P.A., et al.: J. Geophys. Res. A 92, 5777–5794 (1992)Presura, R., Ivanov, V.V., Esaulov, A., Sentoku, Y., Sotnikov, V.I., Laca,

P., Astanovitskiy, A.L., Chiu, C., Ditmire, T., Horton, W., Keely,S., Le Galloudec, B., Renard-Le Galloudec, N., Cowan, T.E.:Anisotropic expansion of a laser-produced-plasma in a strong mag-netic field. Submitted to PRL (2006)

Ripin, B.H., Huba, J.D., McLean, E.A., Manka, C.K., Peyser, T., Burris,H. R., Grun, J.: Phys. Fluids B 5, 3491 (1993)

Sotnikov, V.I., Ivanov, V.V., Cowan, T.E., Leboeuf, J.N., Oliver,B.V., Coverdale, C.A., Jones, B.M., Deeney, C., Mehlhorn, T.A.,Sarkisov, G.S., LePell, P.D.: Investigation of electromagneticflute mode instability in a high beta Z-pinch plasma. IEEE TPS,accepted for publication in (2006)

Zakharov, Y.P.: Plasma Sci. 31, 1243–125 (2003)

Springer


DOI 10.1007/s10509-006-9249-y


Plasma Jet Experiments Using LULI 2000 Laser Facility

B. Loupias · E. Falize · M. Koenig · S. Bouquet · N. Ozaki · A. Benuzzi-Mounaix ·

C. Michaut · M. Rabec le Goahec · W. Nazarov · C. Courtois · Y. Aglitskiy ·

A. YA. Faenov · T. Pikuz


Abstract We present experiments performed with the

LULI2000 nanosecond laser facility. We generated plasma

jets by using specific designed target. The main measured

quantities related to the jet such as its propagation veloc-

ity, temperature and emissive radius evolution are presented.

We also performed analytical work, which explains the jet

evolution in some cases.

Keywords Astrophysical jet . Laboratory astrophysics .

Self-similar solutions . Laser

B. Loupias () · M. Koenig · N. Ozaki · A. Benuzzi-Mounaix ·M. Rabec le GoahecLaboratoire pour l’Utilisation des Lasers Intenses, UMR7605,CNRS – CEA – Universite Paris VI – Ecole Polytechnique, 91128Palaiseau Cedex, Francee-mail:

E. Falize · S. BouquetCEA/DIF/Departement de Physique Theorique et Appliquee, BP12, 91680 Bruyeres-le-Chatel, France

C. MichautLaboratoire de l’Univers et de ses Theories, UMR8102,Observatoire de Paris, 92195 Meudon, France

W. NazarovUniversity of St Andrews, School of Chemistry, Purdie Building,North Haugh, St Andrews, United Kingdom

C. CourtoisCEA/DIF/DCR, BP 12, 91680 Bruyeres-le-Chatel, France

Y. AglitskiyScience Applications International Corporation, McLean, Virginia22102, USA

A. YA. Faenov · T. PikuzMulticharged Ions Spectra Data Center of VNIIFRTI,Mendeleevo, Moscow Region, 141570, Russia

1 Introduction

Astrophysical jets take place in astronomical systems ex-

hibiting accretion disk such as proto-stars, supernovae, pul-

sars, active galactic nuclei and so forth (de Gouveia Dal

Pino, 2005). Whereas from these systems the launching

could be different, the jets share some common charac-

teristics, as the collimation. Furthermore, more or less

spaced emission knots, which move at high speed away

from the central source, have been observed. Finally the

jets are ended with bow shocks, which can be understood

like a “working surface” with the ambient surrounding

medium.

To understand these phenomena, astrophysical jets have

been the subject of elaborate studies in both theory (Raga

et al., 1990) and observations (Reipurth et al., 2002).

Nevertheless, they still raise problems such as the “jet colli-

mation”. Jet experiments can offer an additional approach

to improve our understanding of the physical processes,

which occur during the jet propagation (Shigemori et al.,

2000; Blue et al., 2005; Lebedev et al., 2005; Farley et al.,

1999). Using for the target a cone filled with foam (Koenig

et al., 1999) or high-Z material doped foam, we generate

plasma jets and we observe its propagation in vacuum. We

present jet characteristic measurements such as its velocity,

Vjet, the jet emissive radius evolution, R(t), and its corre-

sponding temperature, Te(t). For the first time all of these

parameters were obtained simultaneously on a single shot

basis, allowing a complete characterization of the jet. We

also derive a purely hydrodynamic analytical solution for

the evolution of the jet radius, which is in good agreement

with the experimental measurements. This new analytical

result well describes the whole evolution of the expanding

fluid.

Springer


2 Experiments

The target is composed by a three layer pusher (0.16µm

Al/20µmCH/3µm Ti) from laser side and a cone filled with

foam at two different densities, 50 mg/cc and 100 mg/cc. We

also used high-Z material doped foam in order to create a

more radiative jet and to use X-ray shadowgraphy. For some

of the targets, we added a washer at the cone exit to in-

crease the plasma jet collimation. The two LULI2000 beams

(wavelength: 527 nm – total energy: 500 J – pulse duration:

1.5 ns – 500µm spot diameter – IL ∼ 2 × 1014 W/cm2) are

focused onto the pusher to generate by rocket effect a shock

that propagates through the foam. The cone geometry allows

us to drive the plasma flow along the cone axis to create the

jet. Our main goal, regarding the experimental diagnostics, is

measuring the jet characteristics to determine the best target

design to have the most collimated plasma jet. In order to ful-

fill this goal, we implemented several diagnostics as shown

in Fig. 1.

2.1 Transverse diagnostics

We use transverse VISAR (Celliers et al., 2004) to measure

the jet velocity. The front edge of the jet is higher or equal to

the critical density for our probe beam (YAG 2ω). Therefore,

the fringe break slope (Fig. 2(a)) provides Vjet. We can also

notice a shift at the end of the fringes. From this shift, we

can get the jet density gradient evolution ahead the jet but the

data analysis is in progress. The measurements provide Vjet ≈160 km/s for 50 mg/cc foam density and Vjet ≈ 120 km/cc for

100 mg/cc. We produce jets with a mach number M ≈ 8 and

we obtain the shape of the plasma jet by 2D shadowgraphy

at two different times for each shot.

We use two Gate Optical Imagers (GOI) with 120ps time

resolution (Fig. 2(b)). We have a common shape that is more

or less a plasma jet with a higher aspect ratio (jet radius/jet

length) for cones without washer than for cones with washer.

It confirms the guiding role of the washer. This result has to

Fig. 1 Experimental set-up for visible transverse diagnostics

Fig. 2 Transverse diagnostic results for cone target at 50 mg/cc withwasher: (a) VISAR: the line slope provides the front jet edge velocitywhich is around 176 km/s. (b) GOI: Shadowgraphy of the jet at 5.5 ns,aspect ratio ∼0,72

be compared with the emissive radius evolution where it is

smaller.

2.2 Rear side diagnostics

Thanks to the self-emission diagnostic, we measure the emis-

sive radius evolution (Fig. 3(a)). We notice that both for

50 mg/cc and 100 mg/cc foam density, two phases on the

emission radius variation arise. For the target without washer,

we have a second phase where the radial expansion speeds

up (Fig. 3(b): upper curve) whereas with washer the radial

expansion slows down (lower curve).

Finally, we also performed a 2D X-ray monochromatic

shadowgraphy of the jet with the He-like Vanadium line at

5.4 keV using a spherical bent crystal (Experimental set-up

in Fig. 4(a)). Figure 4(b) shows a result of high-Z material

foam target. We can observe a dense jet core, radius around

40µm, smaller than the 50µm cone rear side radius hole.

This result lets us thinking of a jet structure where a dense jet

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Fig. 3 Self-emission results: (a) Streaked image for 100 mg/cc withwasher, (b) Radius evolution for cone target at 50 mg/cc foam den-sity without washer (upper curve) and 100 mg/cc with washer (lowercurve)

core (shape from x-ray shadowgraphy) is surrounded with an

envelope by comparing GOI and self-emission radius.

Furthermore, with an absolute calibration of the self-

emission diagnostic we measure an equivalent black body

temperature of the jet equal to 4 eV for 50 mg/cc foam den-

sity and 2.5 eV for 100 mg/cc.

3 Analytical model

We consider the jet as a cylinder with an infinite length.

The cylinder has an initial radius R0 and a central density

ρc and we try to derive the time-dependent radius, R(t), of

the jet. We consider that the plasma can be described by the

hydrodynamics equation and the jet evolution is then given

by the Euler’s equations:

∂ρ

∂t+ 1

r

∂

∂r(rρv) = 0,

∂v

∂t+ v

∂v

∂r= − 1

ρ

∂P

∂r,

P = κργ , (1)

Fig. 4 X-ray monochromatic shadowgraphy: (a) Experimental set-up,(b) X-ray monochromatic shadowgraphy of the jet

where ρ(r, t), P(r, t), v(r, t), γ and κ are respectively the

density, the pressure, the velocity, the polytropic coefficient

and the polytropic constant in the flow. In order to solve

these equations, we use the rescaling theory, a method based

on group transformations (Munier and Feix, 1983; Bouquet

et al., 1985; Ribeyre et al., 2005). Following Ribeyre et al.

(2005), we introduce the four scaling functions A(t), B(t),

C(t) and D(t), such as:

r = C(t)r ; dt = A2(t)dt ; P(r, t) = B(t)P(r , t);

ρ(r, t) = D(t)ρ(r , t); v = C

A2v + Cr ; (2)

where ‘∧’ denotes the quantities in the new space(

r , t)

.

We impose the following initial conditions: C(0) = A(0) =B(0) = D(0) = 1. At t = t = 0, the two spaces (r, t) and(

r , t)

coincide and in the new space, the system (1) becomes:

∂ρ

∂ t+ 1

r

∂

∂ r(r ρv) + A2

(

D

D+ 2

C

C

)

ρ = 0, (3)

∂v

∂ t+ v

∂v

∂ r+ 2A2

(

C

C− A

A

)

v

+ C A4

Cr = − 1

ρ

B A4

C2 D

∂ P

∂ r, (4)

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P = Dγ

Bκργ , (5)

where the upper dot stands for d/dt in (3) and (4). We now

search a static solution, i.e., v ≡ dr/dt = 0, in the new space.

In order to remove the explicit time-dependence in Equations

(3)–(5) we must impose different constraints on the rescaling

functions, namely:

D = C−2, B A4 = C2 D, B = Dγ ,C A4

C= 2, (6)

where the dimension of the free parameter is the inverse

of a time.

After some simple manipulations, we have to solve the

ordinary differential equation CC2γ−1 = 2 and we obtain

its implicit solution t(C), which is written:

(t − t0) ∝ C(t)

× 2 F1

(

1

2,

1

2 − 2γ;

3 − 2γ

2 − 2γ; [C(t)]2(1−γ )

)

, (7)

where t0 is an arbitrary constant and where 2F1 is the hyper-

geometric function. The scaling function C(t) provides the

time evolution of the radius of the jet since from Equations (2)

we have, in particular, R(t) = R0C(t). This law is a wide ex-

tension of the usual self-similar solutions (SSS) for which the

radius (and the various physical quantities) evolve as a power

law of t. Once the scaling function C(t) is obtained, the addi-

tional functions A(t), B(t), and D(t) can be computed from

relations (2) and the density, pressure and velocity profiles

ρ(r, t), P(r, t) and v(r, t) respectively can be obtained from

(2) also.

It should be noticed that this approach is not the classical

one to derive SSS’s since in our case (Munier and Feix, 1983;

Bouquet et al., 1985; Ribeyre et al., 2005), we have a new

space including a new time t . This transformation is called

“the zooming coordinates method” in astrophysics (Blottiau

et al., 1988; Hanawa and Matsumoto, 2000; Hennebelle,

2001; Shadmehri and Ghanbari, 2001). It is clear that, pro-

vided we look for stationary or static solutions (∂/∂ t = 0) in

the new space(

r , t)

, this technique reduce to SSS’s; however,

they are much more general than the tn-classical SSS’s.

Introducing conditions (6) into Equation (4), we get the

density profile in the (real) physical space (r, t):

ρ(r, t)

ρc

= 1

C2

(

1 − 1

C2

(

r

R0

)2) 12−γ

. (8)

4 Discussion

The analytical evolution of R(t) presented previously is

compared with the experiments (Fig. 5) and it is in good

Fig. 5 Analytical (full line) and experimental radius comparison for50 mg/cc target with washer

agreement. To plot the analytical radius evolution, a sound

velocity in the jet core around 19 km/s has been assumed. We

estimate from the experimental result a sound velocity about

15 km/s. This difference is due to the measurement of the tem-

perature and the estimation of the ionization rate. Thanks to

the self-emission diagnostic we measure an equivalent black

body temperature of the jet around 4.5 eV. The temperature

measurement does not take into account possible absorption

in the visible band and it corresponds merely to the jet sur-

face temperature. In order that the analytical model matches

the experimental values of the radius, we have to consider

the core jet temperature. Therefore, when we take 4.5 eV,

we under-evaluate the temperature. Furthermore, up to now,

we could only evaluate an ionization rate from 1D MULTI

simulations (Ramis et al., 1988).

For 100 mg/cc foam density (experimental radius evolu-

tion Fig. 3(a)), the polytropic model fits the experimental data

over 5 ns roughly. This can be understood as the apparition

of a different dynamic regime.

To conclude, we have measured:

– the jet shape by visible shadowgraphy,

– its propagation velocity,

– the jet radius evolution from the self-emission diagnos-

tic. We have a new analytical work which is in quite

good agreement (for 50 mg/cc foam density) with the

experiment. We also tested the faisability of x-ray 2D

radiography and high-Z material doped foam (work in

progress).

References

Blue, B.E., et al.: Phys. Rev. Lett. 94, 095005 (2005)Blottiau, P., et al.: Astron. Astrophys. 207, 24–36 (1988)Bouquet, S., et al.: Astrophys. J. 293, 494–503 (1985)

Springer


Celliers, P.M., et al.: Rev. Sci. Instrum. 75, 4916–4929 (2004)de Gouveia Dal Pino, E.M.: astro-ph/0505521Farley, D.R., et al.: Phys. Rev. Lett. 83, 1982–1985 (1999)Hanawa, T., Matsumoto, T.: Astrophys. J. 540, 962–968 (2000)Hennebelle, P.: Astron. Astrophys. 378, 214–227 (2001)Koenig, M., et al.: Phys. Plasmas 6, 3296–3301 (1999)Lebedev, S.V., et al.: Plasma Phys. Control. Fusion 47, B465–B479

(2005)

Munier, A., Feix, M.R.: Astrophys. J. 267, 344–357 (1983)Raga, A.C., et al.: Astrophys. J. 364, 601–610 (1990)Ramis, R., et al.: Comput. Phys. Comm. 49, 475 (1988)Reipurth, B., et al.: Astron. J. 123, 362–381 (2002)Ribeyre, X., et al.: Astrophys. Sp. Sc. 298, 75–80 (2005)Shadmehri, M., Ghanbari, J.: Astrophys. Sp. Sc. 278, 347–355

(2001)Shigemori, K., et al.: Phys. Rev. E 63, 8838–8841 (2000)

Springer


DOI 10.1007/s10509-006-9275-9


Magnetic Fields for the Laboratory Simulationof Astrophysical Objects

D. Martinez · C. Plechaty · R. Presura


Abstract Strong magnetic fields were generated using a

fast pulsed power generator, to investigate the interaction

of plasma flows with magnetic fields and magnetized back-

ground plasmas. The inductive loads used in these experi-

ments were designed using a filament and a finite element

modeling approaches. Magnetic fields up to 2 MG (200 T)

were measured by using the Faraday rotation technique.

Keywords Faraday rotation . Laboratory astrophysics .

Finite element method . Filament model . Magnetic field

Introduction

The interaction of intense laser produced plasmas with

strong, externally generated magnetic fields is a robust so-

lution for creating in the laboratory matter with parameters

relevant to astrophysical phenomena. Depending on the fact

studied, the magnetic field can determine the plasma dy-

namics or can provide a small-scale dissipation mechanism.

Experiments to study these effects require a wide range of

plasma and field parameters. Such experiments are being

developed at the Nevada Terawatt Facility (NTF) taking ad-

vantage of the 10 TW laser Tomcat coupled with the 2 TW

pulsed power generator Zebra. With this set-up, the parame-

ters of the plasma and its environment can be varied indepen-

dently. Conceptual designs of experiments for the laboratory

simulation of the interaction of astrophysical jets with the

magnetized interstellar medium are presented in a compan-

ion paper (Presura et al., this volume). These experiments

focus on the stability of magnetized plasma jets and on their

D. Martinez · C. Plechaty () · R. PresuraNevada Terawatt Facility, University of Nevada, Renoe-mail: [email protected]

interactions with magnetized ambient medium. In another

type of experiment, the isochoric heating of a solid target is

pursued by attempting to control the heat transport by the hot

electrons, produced by a high intensity laser, with a strong

external magnetic field (Sentoku, 2006). The expected result

is solid density matter uniformly heated to several hundred

eV in volumes large enough (∼105 µm3) and for durations

long enough (several ps) to investigate radiation transport.

This would be directly relevant to the energy transport in

stellar interiors.

Experiments to generate the required magnetic fields were

performed at the NTF at the University of Nevada, Reno us-

ing Zebra, a pulsed power generator which can produce a

0.6 MA current with a fast rise time (200 ns). Two types

of coil designs were used, namely the two-turn helical coil

(TTHC) and the horseshoe coil (HSC). Two different meth-

ods of modeling were used for designing the coil loads. The

first method, used to determine the magnetic field spatial dis-

tribution, was based on solving the Biot-Savart equation and

implemented in MatLab. The second method utilized Com-

sol Multiphysics, a commercial solver that uses the Finite

Element Method (FEM), to solve for the magnetic and elec-

tric potentials. To illustrate the results obtained, two TTHC

shots and one HSC shot are presented. The magnetic fields

generated by these loads were measured using the Faraday

effect. Fields up to 2 MG were measured.

Coil designs

The filament model solves for the magnetic field of current

carrying filaments using the Biot-Savart law. Equations de-

scribing the load geometry were derived for the TTHC and

the HSC and a MATLAB program was written to solve and

display the magnetic fields generated by various coil designs.

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Fig. 1 The smooth curve is themagnetic flux density calculatedalong the axis of the HSC withthe filament model. The secondcurve is the result fromCOMSOL. The resistivity oftantalum, a loop radius of 3.2mm and a coil height of 8.1 mmwere considered

The FEM models were solved in COMSOL Multiphysics

and setup to solve for the magnetic and electrical potentials

using a time harmonic quasi-static approximation. In this

way, the skin depth and inductive currents are taken into ac-

count. Using the current waveform information from Zebra

and the geometry of the coil, COMSOL, taking into consid-

eration the resistivity of the coil material, was able to solve

the potentials.

The spatial magnetic field distributions calculated far from

the wire with the two different models were similar. For

example, the HSC models show very close matches to the

magnetic field with 90 T in the center of the coil with a cur-

rent of 0.7 MA (Fig. 1). The TTHC models, on the other

hand, predict different magnetic fields. Figure 2 shows that

the distributions are similar until the very center of the coil.

The FEM solution has a magnetic field strength of 220 T at

the center while the filament solution predicts 255 T, both

with a current of 0.7 MA through the coil. To determine the

origin of this difference, we solved in COMSOL a filament

model similar to that treated in MATLAB. The magnetic field

was practically the same with that predicted by the filament

model (250 T) as shown in Fig. 2. We have determined that

the difference observed in the FEM and filament model have

a physical origin, namely the preferential distribution of the

current on the lateral surfaces of the coil. This is supported

by the accuracy of the results for the HSC.

To assess the importance of the magnetic pressure on the

coil, the Maxwell stress was calculated. For the examples

shown here we calculated that the Maxwell stress for the

TTHC at its highest point is 3.3E10 Pa (0.33 Mbar) and the

HSC has 1.7E10 Pa (0.17 Mbar). The stress distribution will

be used to calculate the motion and deformation of the coil

on the experiment time scale in the future. Present estimates,

that ignore the heating of the coil, indicate that this effect is

negligible on the time scale of 100 ns.

Faraday diagnostics

The Faraday effect was used to measure the magnetic fields.

To obtain localized, non-perturbative, and time resolved mea-

surements of the magnitude of the magnetic field in vacuum,

the Faraday-active probes used were F2 (flint glass) disks,

3 mm in diameter and 1.75 mm thick. The glass probes

were calibrated with a known magnetic field that was pro-

duced by an electromagnet. The measured Verdet constant

was VF2 = 16.2 rad/(T·m), which is comparable with the

value obtained by others, 14.2 rad/(T·m) (Lide, 2003).

The Faraday diagnostic setup used on Zebra is shown in

Fig. 3. A 532 nm, 250 mW diode pumped solid state laser

beam is first collimated and then sampled by a photodiode

that monitors the laser stability. After sampling, the beam

is polarized by a Glan-Thompson prism and then is passed

through a λ/2 plate, used to control the polarization plane.

The beam is then focused into the Faraday probe located

in the vicinity of the coil. After passing through the probe,

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Fig. 2 This figure compares theaxial distribution of themagnetic flux density along theaxis of a TTHC, with 2.7 mmradius and 2.2 mm pitch,calculated with the filamentmodel (smooth curve), theCOMSOL filament model(dashed curve), and thecomplete COMSOL model(stepwise curve)

Fig. 3 Faraday diagnostic setupon Zebra

the beam is sampled again, by the “emission” photodiode,

which monitors for light generated from electrical breakdown

– pinpointing the time in which the Faraday measurement is

no longer valid.

During a shot, the polarization plane of the beam under-

goes a rotation, and its new polarization state is analyzed with

a Glan-Taylor prism. This allows the rotation to be deter-

mined from an intensity measurement, according to Malus’

Law. For the shots presented later in this paper, the Fara-

day diagnostic set up was used to measure rays with polar-

ization perpendicular and parallel with that of the original

beam. With this differential method, the signal-to-noise ratio

is increased.

The plasma formation and evolution in the vicinity of

the Faraday probe were monitored with laser shadow and

Schlieren imaging.

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Fig. 4 Faraday diagnostic data for a TTHC shot showing the measureddifferential Faraday rotation signal, the calculated Faraday rotation sig-nal, the current waveform, the self-emission signal and shadow images

taken at the stated times. (A) Point in which the first maximum inMalus’ law is obtained. (B) Point at which the Faraday measurement iscompromised. (C) Parameters of the TTHC used in the Zebra shots

Fig. 5 Faraday diagnostic datafor a TTHC shot showing thesame quantities as Fig. 4 andnegative Schlieren images at thetimes stated. (A) Point at whichthe Faraday signal becomes nolonger valid

Experimental results

Two shots with TTHC loads made of 1 mm diameter 316 L

stainless steel wire (Fig. 4C) are presented here. The F2 glass

probe was placed in different positions in the two cases an-

alyzed. In one case, the back face of the Faraday probe was

placed about 0.6 mm in front of the coil. The results for

this shot are shown in Fig. 4. As one can see, the calculated

Faraday signal, based on measurements of the current, and

the measured Faraday signal closely follow each other until

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Fig. 6 The Faraday diagnosticdata for a HSC shot showing thecalculated Faraday signal, thecomponents of the differentialFaraday measurement, and thecurrent waveform. (A) The HSCsetup. (B) Point at which themeasured signal diverges fromthe calculated one. (C) Point inwhich the components of theFaraday signal diverge

t = 50 ns (point B) at which point they diverge, making the

Faraday measurement invalid. It is theorized that at t = 50

ns the Faraday beam path becomes compromised (the emis-

sion photodiode starts to record a signal). In addition, the

last shadow image taken during the shot shows plasma on

the surface of the glass probe. Since the current peaks at

t = 150 ns, the maximum possible magnetic field was not

reached before the Faraday signal was compromised. At this

time, the magnetic field through the volume of the glass was

estimated to be 65 T, which leads to an estimate of 160 T in

the center of the coil and 290 T near the conductor surface.

In another TTHC shot, the front face of the glass was

placed flush with the surface of the coil. Instead of shadow

images, Schlieren images were taken (Fig. 5). In this shot,

the Faraday beam path is compromised at t = 0 ns (point A),

and practically at the same time, the emission photodiode de-

tects electrical breakdown. The current peaks approximately

100 ns later. Near the current peak, as seen by the Schlieren

images, plasma had formed on both the glass and the coil. At

the latest time when the Faraday measurement is valid, the

magnetic field is estimated to be 75 T at the location of the

probe, which leads to an estimated 100 T at the center of the

coil and 160 T near the conductor surface.

The two TTHC shots show that if the probe is placed too

close to the coil, the probe may flashover or the load may

electrically break down, causing the Faraday measurement

to become compromised earlier in the current waveform, as

shown by changing the probe placement.

Attempting to produce strong magnetic fields without cre-

ating a measurable amount of plasma, a four-wire HSC was

shot (Fig. 6). In this case, two Faraday probes were used to

increase the rotation effect and were placed roughly in the

middle of the HSC with the front face of the first probe about

0.25 mm inside. For this shot, the differential Faraday signal

diverges from the calculated Faraday signal when t = 300 ns

(point B). However, the components of the differential setup

do not diverge from each other until t = 800 ns (point C). It

is theorized at t = 300 ns, the current stops following along

the horseshoe path and arcs through plasma that forms in

the gap between the leads of the horseshoe. The current in

the horseshoe structure decays exponentially due to resis-

tive effects until t = 800 ns when the Faraday beam path is

compromised. At the current peak, the estimated magnetic

field was 50 T at the coil center. In another experiment with

a smaller HSC, magnetic fields of the order of 100 T were

measured, without any plasma formation.

Since the Faraday probe has a finite volume, the measure-

ment taken is not a simple line integral. Each point in the

volume of the glass contributes to the overall measurement.

In order to estimate the effect of the probe volume on the field

measurement, a MatLab program was written, based on the

filament model. The magnitude of this effect was estimated

as a function of the probe location.

Conclusion

To date, magnetic fields up to 2 MG were produced. Fields

of the order of 1 MG were obtained without generating any

measurable plasma. The experiments show that the magnetic

field strength predicted in the simulations closely matches

the field strength actually produced. The simulations have

also been useful in interpreting the results from the Faraday

measurements.

In future models, a more comprehensive study will be

preformed to obtain a self-consistent treatment of magnetic

Springer


and electric fields, coil heating, and coil deformation. In

addition, in future Faraday measurements, a smaller probe

will be used to minimize the effect of electrical breakdown

as seen in the two TTHC shots and to better localize the

measurement.

Acknowledgements We would like to thank V. Ivanov and P. Laca forlaser diagnostics. This work was supported by DOE/NNSA under theUNR grant DE-FC52-01NV14050 and by UNR undergraduate awardsgranted to D. Martinez and C. Plechaty.

References

Presura, R., Neff, S., Wanex, L.: Experimental design for the laboratorysimulation of magnetized astrophysical jets. Astrophys. Space Sci.,DOI 10.1007/s10509-006-9286-6 (2006)

Sentoku, Y., Kemp, A., Bakeman, M., Presura, R., Cowan, T.E.: Iso-choric heating of hot dense matter by magnetization of fast elec-trons produced by ultra-intense short pulse irradiation. J. PhysicsIV 133, 521 (2006)

Lide, D.R. (ed.): CRC Handbook of Chemistry and Physics, 84thedition. CRC Press LLC (2003)

Springer


DOI 10.1007/s10509-006-9255-0


Assessing Mix Layer Amplitude in 3D Decelerating InterfaceExperiments

C. C. Kuranz · R. P. Drake · T. L. Donajkowski · K. K. Dannenberg · M. Grosskopf ·

D. J. Kremer · C. Krauland · D. C. Marion · H. F. Robey · B. A. Remington ·

J. F. Hansen · B. E. Blue · J. Knauer · T. Plewa · N. Hearn


Abstract We present data from recent high-energy-density

laboratory experiments designed to explore the Rayleigh–

Taylor instability under conditions relevant to supernovae.

The Omega laser is used to create a blast wave structure that

is similar to that of the explosion phase of a core-collapse

supernova. An unstable interface is shocked and then decel-

erated by the planar blast wave, producing Rayleigh–Taylor

growth. Recent experiments were performed using dual, side-

on, x-ray radiography to observe a 3D “egg crate” mode and

an imposed, longer-wavelength, sinusoidal mode as a seed

perturbation. This paper explores the method of data analysis

and accurately estimating the position of important features

in the data.

Keywords Rayleigh-Taylor instability . Supernova .

Laboratory astrophysics

Introduction

In 1987, a core-collapse supernova (SN) occurred ∼160000

light years away, making it the closest SN in modern times.

The proximity of SN1987A made it possible to use con-

temporary astronomical instruments to collect data from the

C. C. Kuranz () · R. P. Drake · T. L. Donajkowski ·K. K. Dannenberg · M. Grosskopf · D. J. Kremer · C. Krauland ·D. C. MarionUniversity of Michigan, Ann Arbor, MI, USA

H. F. Robey · B. A. Remington · J. F. Hansen · B. E. BlueLawrence Livermore National Laboratory, Livermore, CA, USA

J. KnauerUniversity of Rochester, Rochester, NY, USA

T. Plewa · N. HearnUniversity of Chicago, Chicago, IL, USA

SN. At the time, existing models did not agree with the data

collected, specifically, the high velocities and early x-ray

emission of dense core elements. These discrepancies mo-

tivated improvement of the understanding of core-collapse

SNe. Current models have started to explain the mysteries of

SN1987A, but many questions remain unanswered. Of par-

ticular interest is the effect of hydrodynamic instabilities on

the transport of the heavy core elements.

High Energy Density (HED) facilities make it possible to

study specific, well-scaled areas of astrophysical phenom-

ena, in our case, the blast-wave-driven interface of a core-

collapse supernova. Intense lasers can create the extremely

large energies in mm-scale targets previously seen only in

astrophysical systems. Experiments of this type have been

done or are planned at numerous laser facilities (Drake et al.,

2004; Kane et al., 2000; Robey et al., 2001; Remington et al.,

2000). It is possible to compare the SN and the experiment

because the targets can be well-scaled to the SN explosion

phase so that the two will have similar hydrodynamic evolu-

tion (Ryutov et al.,1999).

The Rayleigh–Taylor (Rayleigh, 1900; Taylor, 1950) in-

stability occurs when a system has a density gradient and

effective pressure gradient in opposing directions. This is the

case both the SN, where a blast wave propagates from the

dense core through less dense, outer layers of the star, and in

the laboratory experiment, where a planar blast wave moves

through a dense plastic layer into a less dense foam layer. The

resulting evolution is the flow of dense elements “sinking”

outward in the form of fingers or spikes. Also, the less dense

material “floats” inward and is referred to as bubbles.

Experiments

During the experiment ten Omega (Boehly et al., 1995) laser

beams irradiate a 150µm layer of polyimide of a density

Springer


1.41 g/cc. The total energy of the beams is ∼5 kJ and the

irradiance is ∼1015 W/cm2, producing an ablation pressure

of ∼50 Mbars, which creates a strong shock in the plastic

layer of the target. After 1 ns, the laser pulse ends, causing

the material to rarify. When the rarefaction wave overtakes

the shock wave, a planar blast wave is formed. After about

2 ns, the blast wave crosses an interface between the plastic

and carbonized resorcinol formaldehyde (CRF) foam. The

foam has a density of 50 mg/cc, making the density drop

between the plastic and foam similar to that expected in the

case of the H/He interface in SN1987A. The interface is

initially accelerated by the blast wave and then decelerated

over a long period of time by the foam layer. The interface

is unstable to both Richtmeyer-Meshkov (Richtmyer, 1960;

Meshkov, 1969) and Rayleigh–Taylor instabilities. However,

interface growth due to Rayleigh–Taylor dominates after the

first few nanoseconds.

Diagnostics

This experiment uses an ungated Static Pinhole Camera

Array (SPCA) loaded with Direct Exposure Film (DEF)

behind Be, plastic and Ti or Sc light shields. To protect

the ungated diagnostic from laser beams and hot plasma

created during the experiment, a large gold shield is part

of the target structure. The polyimide and foam compo-

nents are placed inside a polyimide tube and attached to

the gold shield. The target package and gold shield as well

as the placement of pinhole backlighters and the two SP-

CAs can be seen in Fig. 1. The target package is placed at

the center of the Omega chamber and each pinhole back-

lighter is perpendicular to the polyimide tube. The two

backlighters are orthogonal to each other. The diagnostics

are on the opposite side of the target from each pinhole

backlighter.

The main diagnostic is dual, orthogonal, x-ray radiogra-

phy. There are two pinhole backlighters each having a 5 mm

square Ta foil with a stepped pinhole in the center. The step

refers to a large hole on one side of the Ta and a smaller

hole on the other. The pinhole backlighters are very sensi-

tive to rotational alignment. Therefore, a stepped structure

increases the size of the source while maintaining high reso-

lution. The large opening is about 50µm stepped to 20µm.

About 500µm behind the pinhole is a 50µm thick plastic

square; attached to the rear of the plastic is a 500µm square

foil of either Ti or Sc. These foils are irradiated with 4 omega

laser beams that have 200–400 J/beam, 1000–1200µm spot

size and a 1 ns square pulse. These beams overfill the metal

foil, irradiating the plastic under the foil so that the expand-

ing plastic provides radial tamping of the expanding metal

plasma. The Sc and Ti create 4.09 and 4.51 keV x-rays, re-

spectively. These x-rays pass through the pinhole in the Ta

then pass through the target to the ungated DEF on the op-

posite side of the target.

On the rear surface of the polyimide piece, a 200µm wide,

50–75µm deep slot has been machined out of the plastic. A

“tracer” strip of 4.3 at.% bromine doped plastic, C500H457

Br43 (CHBr), is glued into that slot. The CHBr has a density

1.42 g/cc. Since the CHBr and the polyimide have similar

densities and are both predominately low Z materials, they

will have similar evolutions in response to extreme pressures.

The tracer strip is used because the bromine component of

the CHBr more readily absorbs x-rays than the CH or poly-

imide; therefore, it provides contrast on the x-ray radiographs

Fig. 1 Image of target withpositions of backlighters andungated detectors. Inside thepolyimide tube attached to thegold shield contains a 150µmplastic layer followed by a2–3 mm CRF foam layer

Springer


Fig. 2 (a) Single-mode perturbation, a = 2.5µm, k = 2π /(71µm). (b) 2-mode perturbation, a = 2.5µm, additional mode is k = 2π /(212µm)

obtained by the primary diagnostics of the experiment. Also,

since the strip is in the center of the target it allows the di-

agnostic to “look through” the polyimide since it is nearly

transparent to the He-alpha x-rays used to diagnose the ex-

periment. This allows the radiograph diagnose primarily the

center of the target where the experiment is the least af-

fected by target walls and sound waves created during the

experiment.

After the tracer strip is in place a seed perturbation is ma-

chined onto the rear surface of the plastic component. This

paper will discuss two types of perturbations. The basic pat-

tern for both perturbations is two orthogonal sine waves with

a0 = 2.5µm and k = 2π /(71µm). The result is an “egg

crate” pattern as seen in Fig. 2a, which will be referred to as

a single-mode perturbation. The second type of perturbation

has an additional mode whose wave vector is parallel to the

long edge of the tracer strip. In this case, the additional mode

has a0 = 2.5µm and a k = 2bπ /(424µm). This perturba-

tion is referred as a 2-mode perturbation and can be seen in

Fig. 2b. The reason to add additional modes is to explore

enhanced spike penetration that these modes may produce.

This has been seen in past experiments (Drake et al., 2004)

and in simulations (Miles et al., 2003). This experiment uses

dual, orthogonal radiography with one diagnostic line of sight

down the tracer strip and the other across the tracer strip. The

view across the strip allows one to see about 13 spikes on the

radiograph and view down the strip allows one to see 3 or 4

spikes.

Results and discussion

Radiographs from recent experiments taken at 17 ns after the

laser beams have fired can be seen in Fig. 3. Figure 3a is a

radiograph of a single-mode target with the view across the

tracer strip. Figure 3b is also a view across the tracer strip,

but of a 2-mode target. The shock and interface are moving to

the right in both images. Also, the tube walls are seen around

Y = ±470µm and a gold grid is seen in each figure for

calibration of magnification and position. In Fig. 3a there are

several very bright lines due to scratches on the film. Notice

that Fig. 3a has less contrast and more noise than Fig. 3b.

This is because the radiograph in Fig. 3a is from a second

layer DEF, where the first layer was overexposed and acted as

a filter in this case. The resulting lineouts of this radiograph

have been adjusted so that it is possible to compare relative

positions between the two radiographs for the purposes of

this paper.

The positions of notable features are more clearly seen

in the horizontal lineouts taken from each radiograph seen

in Fig. 4a and 4b for the single-mode and 2-mode cases,

respectively. Two lineouts were taken for each radiograph in

order to estimate the distance from the spike tip to bubble tip.

One lineout was taken across a Rayleigh–Taylor spike, shown

by the dark grey line, and the other across a bubble structure,

shown by the black line. The location of each lineout is shown

on the corresponding radiograph by a black rectangle. On

each lineout the position of the shock, spike tip and bubble

head are shown. The lineouts across the spike and bubble

have a sharp decrease in intensity across the shock. Notice the

sharp differences in the lineout across the spike as compared

to the one across the bubble. The lineout across the spike

then has a gradual decrease in intensity and then another

abrupt decrease at the tip of the spike. The lineout across the

bubble also has a gradual decrease in intensity after the shock

followed by an abrupt decrease and then a gradual increase

in intensity from the remaining plastic layer. However, the

bubble has a higher intensity than the spike since it appears

lighter in the radiograph. The sharp increased in intensity

on the left portion of the single-mode lineout are from the

scratches in the film mentioned earlier.

The positions of the shock, spike tip, and bubble head are

shown by abrupt transitions in intensity, although the lineout

shows them spread out over some horizontal distance. This

is due to the finite resolution in experiment, the curvature of

Springer


Fig. 3 Radiograph at 17 ns of (a) single-mode perturbation. (b) 2-modeperturbation

these features, and perhaps larger-scale variations in density

and spike shape. In order to accurately and consistently de-

fine the locations of these features, a systematic method of

analysis must be developed. The present, preliminary method

involves finding the midpoint of the sharp decreases in in-

tensity from the lineout in the portion determined to be the

shock, spike tip or bubble head. The position of the bubble

head is subtracted from the position of the spike tip to esti-

mate the amplitude of the interface. At 17 ns, the amplitude

of the single-mode perturbation is 143µm and that of the 2-

mode perturbation is 168µm. The larger interface amplitude

of the 2-mode perturbation is consistent with the results of

simulations indicating that additional modes may cause in-

creased growth (Miles et al., 2003). We are now working on

the analysis required to subtract out the effects of refraction,

which will be necessary to determine a meaningful growth

rate for comparison with theory.

Fig. 4 (a) Lineouts at 17 ns across a spike and bubble in radiographfrom single-mode perturbation and a (b) 2-mode perturbation

An additional radiograph can be seen in Fig. 5. This image

is of a single-mode perturbation, but it is the view is down

the tracer strip. Therefore, it is looking down 3 or 4 rows of

about 13 spikes each. Note that on either side of the high

contrast tracer strip are fainter spikes. These are spikes of

the polyimide material surrounding the tracer strip. While

polyimide is nearly transparent to the He-alpha x-rays used

in this experiment, these spikes are a result of seeing an en-

tire row of spikes aligned to be seen as one. For the same

reason it is very difficult to see the location of the bubble

head within the tracer strip. Looking through a row of ∼13

aligned spikes and bubbles causes a blurring of features in

the radiograph. In contrast, the view across the tracer strip is

only looking through ∼3 rows of aligned spikes and bubbles.

The inaccuracy of the bubble head position in the view down

the tracer strip can be seen from a lineout across the spike and

bubble from the radiograph in Fig. 6. There is only a slight

Springer


Fig. 5 Radiograph at 17 ns from single-mode perturbation down thestrip view

Fig. 6 (a) Lineout from radiograph at 17 ns of single-mode data acrossbubble and spike from a view down the strip as compared to (b) theacross the strip view

difference in the lineouts of the bubble and spike, making it

almost impossible to locate the bubble head. Compare these

lineouts to the lineouts from a view across the strip for the

same perturbation at the same time in Fig. 4a. Notice that

the bubble position in the across the strip view is much more

distinct than in the down the tracer strip view.

Conclusions

Recent experiments have been successful in obtaining data

from targets well scaled to the expected conditions at the

H/He interface in the explosion phase of SN1987A. Initial

findings have shown that adding additional modes over the

basic “egg crate” perturbation create larger amplitudes of

the mixed layer. Dual, orthogonal radiography has allowed

two views of the interface and the resulting growth of the

mix layer from the Rayleigh–Taylor instability. Estimating

this amplitude can be done by finding the positions of the

spike tip and bubble head. This paper has shown that it is

challenging to estimate the position of the bubble head in

the view down the strip because many spikes form a line

blurring its position. However, the bubble head position can

be found more easily and more accurately from the view

across the tracer strip. The view down the strip is still very

useful for confirming the spike tip and shock position as well

as diagnosing target abnormalities.

Acknowledgements The author would like to acknowledge KaiRavariere, Aaron Miles, Dave Arnett, and Casey Meakin for their usefultechnical discussions. Financial support for this work included fundingfrom the Stewardship Science Academic Alliances program throughDOE Research Grant DE-FG03-99DP00284, and through DE-FG03–00SF22021 and other grants and contracts. This work is also supportedin part by the U.S. Department of Energy under Grant No. B523820 tothe Center for Astrophysical Thermonuclear Flashes at the Universityof Chicago.

References

Boehly, T.R., Craxton, R.S., et al.: Rev. Sci. Instr. 66(1), 508 (1995)Drake, R.P., Leibrandt, D.R., et al.: Phys. Plasmas 11(5), 2829

(2004)Kane, J., Arnett, D., et al.: ApJ 528, 989 (2000)Meshkov, E.E.: Fluid Dyn. 4, 101 (1969)Miles, A.R., Edwards, M.J., et al.: The Effect of a Short-wavelength

Mode on the Nonlinear Evolution of a Long-wavelength Perturba-tion Driven by a Strong Blast Wave. Inertial Fusion and ScienceApplications, Monterey, CA (2003)

Rayleigh, L.: Scientific Papers II. Cambridge, England, Cambridge(1900)

Remington, B.A., Drake, R.P., et al.: Phys. Plasmas 7(May), 1641(2000)

Richtmyer, D.H.: Commun. Pure Appl. Math. 13, 297 (1960)Robey, H.F., Kane, J.O., et al.: Phys. Plasmas 8, 2446 (2001)Ryutov, D.D., Drake, R.P., et al.: ApJ 518(2), 821 (1999)Taylor, S.G.: Proc. R. Soc. A201, 192 (1950)

Springer


DOI 10.1007/s10509-006-9237-2


The Formation of a Cooling Layer in a Partially Optically ThickShock

A. B. Reighard · R. P. Drake


Abstract The mechanics of a radiative shock which has

“collapsed,” or been compressed to high density, via radia-

tive cooling is discussed. This process is relevant to an ex-

periment in xenon gas that produced a driven, radiatively

collapsed shock, and also to a simulation of the supernova

1987A shock wave passing through the outer layers of the

star and into the low-density circumstellar material.

Keywords Radiation . Radiation hydrodynamics .

Optically thick shock . Driven shock . Collapsed shock .

Laboratory astrophysics

Introduction

To reach a regime where the transport of radiation through a

system can affect the hydrodynamic properties of the mate-

rial, two conditions must be met. Firstly, one must create a

system with sufficient “optical depth,” a measure of the atten-

uation of radiation. Said another way, there must be enough

material that the radiation is affected by its presence. This

system must also be relatively hot, on the order of at least

tens of eV. This can be difficult in laboratory systems, requir-

ing energy sources capable of delivering high laser power or

X-ray flux to a small target, driving shock waves into ma-

terials of relatively low density. In astrophysics, energetic

explosions into diffuse material, like a supernova explosion,

produce similar effects.

We have performed experiments at the Omega laser that

have created a planar, driven, radiatively collapsed shock

wave in xenon gas. This system is optically thick (highly

absorptive of radiation) downstream in the shocked gas, and

A. B. Reighard () · R. P. DrakeDepartment of Atmospheric, Oceanic, and Space Sciences,University of Michigan, Ann Arbor, MI 48109, USA

after a brief transition period where the upstream gas is ion-

ized by precursor radiation, optically thin upstream in the un-

shocked gas. This experiment (described further in Reighard

et al. (2006)) has produced a shock wave that radiated away

enough energy to cool both the electrons and the ions in the

shocked material, causing the gas to become highly com-

pressed in response. A metrology image of a target used in

this experiment is shown in Fig. 1a. The beryllium drive sur-

face is in the lower right-hand corner of the image, mounted

on a polyimide tube 5 mm in length and 0.6 mm OD. Shown

in Fig. 1b is a radiograph of collapsed xenon gas in this ex-

periment, taken via side-on backlit pinhole radiography. The

shock is moving to the right in this figure, where the x co-

ordinate is the distance along the target axis from the initial

position of the drive disk, while the y coordinate is the radial

coordinate of the target. The dense xenon shows as a dark

absorption feature. The shock front is just to the right of the

dark absorption feature. The shape of the dense collapsed

layer is affected by line of sight of the diagnostic and drag

along the walls of the target, as well as any non-uniformity

in the production of the layer from instabilities.

It is then natural to seek astrophysical shocks with the

same optical depth structure for comparison to this system.

In an article by Ensman and Burrows, (1992), 1D hydrody-

namic simulations of shock breakout in SN 1987A show a

highly compressed, cooled shock as the disturbance passes

through the outer layers of the star and into the lower den-

sity circumstellar material. The temperature and density pro-

files show a remarkable similarity to temperature and den-

sity profiles calculated in the experiment described above,

as shown in Fig. 2. The density and temperature profiles for

the experimental system in Fig. 2b. were generated using

Hyades (Larsen and Lane, 1994), a 1D lagrangian code with

a three-fluid treatment of the material and a multigroup treat-

ment of the radiation with flux-limited diffusion.

Springer


Fig. 1 (a) Metrology image from experiments described in Reighardet al. Drive beams hit a Be disk (lower right corner of image), andaccelerate it into the xenon filled plastic tube. The experiment is ap-proximately 5 mm long. (b) Data image from Reighard et al. at 8.2 nsafter drive beams turn on. The thin layer of collapsed xenon shown byX-ray radiography is approximately 65 µm thick, and has an averagevelocity of 140 km/sec. The gold grid used as a spatial and magnifica-tion indicator is in the top right of the image, while the edges of thetarget tube are visible as horizontal lines near the bottom and top of theimage

These systems will produce this kind of collapsed shock

structure in fundamentally the same way. This begs a causal

explanation of the physics producing this structure in both

systems. Here, we offer a step-by-step explanation of the

physics behind the formation of a collapsed radiative shock.

For each step in the discussion, parameters from the ex-

periment described above (and in more complete detail in

Reighard et al. (2006)) are used as an example.

Radiatively collapsed shock formation

As in any shocked system, a shock wave heats ions as it passes

through cold material. The shock accelerates everything to a

higher velocity, but most of the energy of the shock goes into

heating the heavier ions. This happens quickly, over the span

Fig. 2 1D Lagrangian simulations of the two systems discussed. (a) 1DVISPHOT calculation adapted from Ensman and Burrows (1992). (b)1D Hyades simulation of discussed experimental system. Both systemsshow a layer of highly compressed material accompanied by quickcooling through a region that is thin compared to the distance the shockhas traveled

of just a few ion-ion mean free paths. Through conservation

equations, the matter is also compressed by a certain amount,

depending on the polytropic index of the material, and the

pressure increases.

In an ionized system that can exchange energy by colli-

sions, the ions begin to transfer energy to the cooler electrons.

The rate at which this happens is dictated by the ion-electron

collision rate, ν ie, given by

νie = 3.2 × 10−9 ni Z3 ln

AT 1.5e

, (1)

where ni is number density in cm−3, Z is the average ion-

ization state, ln is the coulomb logarithm, A is the atomic

weight of the material, and Te is electron temperature in eV

(Drake, 2006). The rate of ion cooling depends on the differ-

ence between the ion temperature and the electron tempera-

ture;

∂Tion

∂t= −νie(Tion − Te). (2)

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Tem

pera

ture

(eV

)

Ion temperature

Electron temperature

Ion temperature

Electron temperature

Tem

pera

ture

(eV

)

(a)

(b)

Fig. 3 Ion and electron temperatures behind the shock front for theparameters from the experimental system, assuming a shock moving at150 km/sec. The shock front is located at 0 cm. (a) Temperature profileswith only collisional heat exchange. (b) Temperature profiles with bothcollisional heat exchange and radiative cooling of the electrons. In thissystem, radiative cooling becomes important before collisional heatexchange can equilibrate the electron and ion temperatures

This process slows as the temperatures equilibrate. In xenon

gas, with ni = 1.5 × 1019, A = 131.3, with initially Te

negligible, and Ti approximately 500 eV, this equilibration

process takes on the order of a few hundred picoseconds.

The ion and electron temperature profiles from collisional

heat exchange in xenon gas are shown in Fig. 3a. The initial

ion temperature is dictated by the strong shock equations for

a shock wave moving at 150 km/sec, for which the equilibra-

tion length would be approximately 30 µm. This calculation

is relevant to the experimental system described above, where

a driven shock moves through xenon gas at velocities in ex-

cess of 100 km/sec. 1D Hyades simulations give a similar

result, showing a equilibration length of approximately 25

µm at an instantaneous velocity of 160 km/sec.

In some systems, the electron temperature may become

large enough to cause the free electrons in the system to

radiate. As the electrons in the system get warmer, they begin

to radiate more. Though electrons in the system may continue

to gain energy from collisions with ions, now they may also

cool through radiative losses, at a fractional energy radiation

rate given by the ratio of the radiative flux from two surfaces

of a planar slab, 2Frad to the energy content of the shocked

material of density ρ, slab thickness d , specific opacity κ ,

and specific heat (per unit Te) cv . This ratio is

νrad = 2Frad

ρdcvT= 2κdρσT 4

e

ρdcvT(3)

for an optically thin material, where σ is the Stefan-

Boltzmann constant. On the approximation that T = Z Te +Ti = (Z + 1)Te if Te ≈ Ti , where Z is the average ionization

state, this rate simplifies to

ν∗rad = 2.2

A

(Z + 1)κT 3

e (4)

in sec−1 for Te in eV and κ in cm2/g. For the model calcu-

lation presented, we evaluated κ to be 2500 cm2/g using a

SEASAME table value for xenon at 100 eV, averaged over

values for relevant densities (Leibrandt et al., 2005). If the

collisional heating happens quickly, the heating and cooling

of electrons may be treated as a two-step process. For the

experimental system, significant radiative cooling begins be-

fore the ion and electron temperatures equilibrate, so the two-

step approximation does not hold. Figure 3b. shows the tem-

perature profiles produced by the combination of electron-

ion heat exchange and radiative losses evaluated as just de-

scribed.

If the shocked gas can quickly radiate away a significant

fraction of its energy, the system will form a collapsed layer in

response to this loss. The cooling layer, where radiation cools

the system, must be optically thin for energy to escape. As

the electron temperature rises in the system not only will the

radiative rate become large, the opacity of the system κ will

decrease, making the hot material optically thin, allowing the

energy to escape via radiation.

As the shocked material loses energy, the system must

respond to keep certain parameters constant across the shock.

The conservation equations for mass and momentum must

still hold, given by

ρ1u1 = ρ2u2 (5a)

p1(ρ, T ) + ρ1u1 = p2(ρ, T ) + ρ2u2. (5b)

Pressure, density, and temperature are linked through these

equations, so as the temperature drops another quantity must

respond to keep the system in balance. If the pressure were

to drop, there would be a void in the system, and the material

would compress in response to energy loss. Pressure can be

expressed as a function of temperature and density using the

ideal gas law,

p = ρRTi , (6a)

where R is the gas constant,

R = (Z + 1)kB

Am p

, (6b)

and kB is Boltzmann’s constant.

Springer


Fig. 4 Density profile when affected by radiation losses. In this simpleanalytic model, the radiative transfer equation was not fully employed,nor was the influence of increasing opacity as electron temperaturedecreased. Both these factors would affect the final density of the coolinglayer. Here, the final density is dictated by the loss of all electron energyvia radiation

Estimating the density profile from this kind of relation-

ship requires some understanding of how the “gas constant”

R changes with ionization. Using an estimate of how the

average ionization Z changes with temperature allows an es-

timate of R. Using the Saha equation to calculate the average

ionization state (only strictly valid in a equilibrium distribu-

tion, where ionization balances recombination exactly), Zbal

is

Zbal = 19.7

√

Te(1 + 0.19) ln

(

T3/2

e

n24

)

− 1

2, (7)

where Te is in eV and n24 is number density in units of 1024

cm−3. Given these substitutions, pressure can be expressed

solely as a function of temperature and density. The density

profile can then be calculated from the temperature profile

and the momentum conservation equation. While the profile

generated in this way should be qualitatively correct, one

would need to solve the radiative transfer equation to actually

predict the final density value, taking into account the amount

of radiation emitted and absorbed in each differential slab. In

Fig. 4 we show the qualitative profile of radiative collapse,

here without a full treatment of the radiative transfer. The

layer continues to collapse until the system has radiated away

all of the thermal energy of the system, where the electron

temperature approaches zero, and the ion temperature is very

low. Therefore in this incomplete model, the final value of

the compression is not indicative of the value produced in

the experimental systems.

As the system becomes more compressed, the optical

depth of the dense, collapsed layer increases, and the opac-

ity can change as the system cools. As the optical depth of

the material increases, it becomes more difficult to remove

energy from within the dense material, in response to which

the rate of density increase eventually reaches zero. If the

system relaxes to a quasi-steady state, where the shocked

system maintains the same general shape, the ultimate thick-

ness of the layer will depend on the balance of energy flowing

through the layer. The optically thick downstream material

will radiate at its blackbody temperature. This radiation will

pass through the optically thin cooling layer but must be bal-

anced by the radiation from the cooling layer to have steady

state. The cooling layer will radiate equally both upstream

and downstream. Because it is optically thin, the upstream,

unshocked gas will radiate negligibly, and energy will escape

from the system through it. Flux at the boundaries of the layer

coupled with the hydrodynamic equations then give the final,

post-shock temperature and the spatial extent of the cooling

layer, as discussed in more detail in Drake (2006).

In a driven system like the experiment described, the

amount of momentum in the system is fixed. As the pis-

ton driving the shock amasses collapsed xenon, the system

will decelerate. It is possible that the system will eventu-

ally slow to the point where the driving forces no longer

heat the system to the point where radiation cooling can be

effective. At this point, the evolution of the shock will be-

come hydrodynamic in nature, and radiative collapse will

cease.

Conclusion

Radiative cooling in a system that is optically thick down-

stream (behind the shock) and optically thin upstream (in the

unshocked material) can lead to dramatic effects in the over-

all structure of the shocked material. Collapse via radiative

cooling of the shocked gas can lead to compression of ma-

terial to much higher densities than those in a strong shock

with no radiative cooling.

Future work includes analysis of radiative effects at dif-

ferent initial driving velocities. This is achieved by vary-

ing the thickness of the Be layer illuminated by the laser.

Computational efforts include work to better understand

the effects of opacity on xenon at temperatures between

50 and 200 eV. In addition, by watching the long-term

evolution of the shocked layer, such experiments might

observe the onset of hydrodynamic instabilities like those

discussed by Vishniac and Ryu (1989). Beyond such work,

this type of system could be developed as a radiation source

for experiments to examine other issues such as radiation

transport.

Acknowledgements The authors acknowledge the vital contribu-tions of the Omega technical staff and the target fabrication groupat the University of Michigan. This work is supported by theNational Nuclear Security Agency under DOE grants DE-FG03-99DP00284 and DE-FG03-00SF22021, and by other grants andcontracts.

Springer


References

Drake, R.P.: High Energy Density Physics: Foundations of Inertial Fu-sion and Experimental Astrophysics. (Springer, New York, 2006)

Ensman, L., Burrows, A.: ApJ 393, 742 (1992)

Larsen, J.T., Lane, S.M.: J. Quant. Spectrosc. Radiat. Transfer 51(1),179 (1994)

Leibrandt, D.R., Drake, R.P., Reighard, A.B., et al.: ApJ 626, 616 (2005)Reighard, A.B., Drake, R.P., Dannenberg, K.K., et al.: Phys. Plasmas

13, 082901 (2006)Vishniac, E.T., Ryu, D.: ApJ 337, 917 (1989)

Springer


DOI 10.1007/s10509-006-9278-6


Measurement of the Growth of Perturbations on Blast Wavesin a Mixed Gas

A. D. Edens · R. G. Adams · P. K. Rambo · I. C. Smith ·

J. L. Porter · T. Ditmire


Abstract We have performed a series of experiments exam-

ining the properties of high Mach number blast waves. Ex-

periments were conducted on the Z-Beamlet laser at Sandia

National Laboratories. We created blast waves in the labora-

tory by using ∼1000 J laser pulses to illuminate millimeter

scale solid targets immersed in gas. Our experiments stud-

ied the validity of theories forwarded by Ryu and Vishniac

(1987, 1991) and Vishniac (1983) to explain the dynamics

of perturbations on astrophysical blast waves. These exper-

iments consisted of a systematic scan of the decay rates of

perturbations of known primary mode number induced on

the surface of blast waves by means of a regularly spaced

wire array. The amplitude of the induced perturbations rela-

tive to the radius of the blast wave was tracked and fit to a

power law in time. Measurements were taken for a number of

different mode numbers in a mixed gas consisting of 7.5 Torr

xenon and 2.5 Torr nitrogen and the results are compared to

theoretical predictions. It is found that two of the three mode

numbers imply one polytropic index while the third case,

which is the most complicated for several reasons, implies a

higher polytropic index.

Keywords Vishniac overstability . Z-beamlet . KiloJoule

laser . Laboratory astrophysics . Blast wave . Radiating

shock

Supernovae are some of the most energetic and impres-

sive phenomena in the universe. There are a number of

A. D. Edens () · R. G. Adams · P. K. Rambo · I. C. Smith ·J. L. PorterSandia National Laboratories, Albuquerque, NM 87123

T. DitmireUniversity of Texas at Austin, Austin, TX 78712

instabilities associated with different times in the supernova

process, from the collapse of the star to the merging of the

supernova remnant (SNR) into the background medium. The

particular instability we are interested in was theorized by

Vishniac (1983), and is therefore known as the Vishniac over-

stability. In this overstability, illustrated in Fig. 1, there is a

mismatch between the ram and thermal pressures at the blast

wave surface. This mismatch can create an oscillating rip-

ple in the blast wave surface. Depending on the thickness

of the blast wave and the wavelength of the perturbation in

its surface, this oscillating ripple can grow. After the initial

paper Vishniac and Ryu wrote several papers generalizing

their theory for more realistic blast waves.

Vishniac and Ryu determined that the amplitude of a per-

turbation on a blast wave varies as a power law in time,

A ∝ Ct s . For a given wavelength, C is a constant that de-

pends on the spherical harmonic for the mode number in

question, A is the amplitude of the perturbation, and s is the

growth rate. The growth rate should depend on two variables:

the mode number of the perturbation and the thickness of the

blast wave, measured by its polytropic index. The polytropic

index is a measure of the number of degrees of freedom for

a gas, and decreases (along with the blast wave thickness)

with an increasing number of degrees of freedom (Grun et al.,

1991). Radiation provides an effective degree of freedom for

a gas, lowering its polytropic index.

There have been a number of experiments looking at laser

produced blast waves. The first we could find in the litera-

ture was performed in 1972 by Basov et al. (1972), who used

multi-sided illumination of a spherical target to create a blast

wave in 15 Torr of residual air in order to gauge the amount of

laser energy absorbed by that target. Later, Grun et al. (1991)

looked at blast waves produced by single-sided illumination

of plastic foils in 5 Torr of nitrogen or xenon gas in order

to look at the Vishniac overstability. They showed that blast

Springer


Fig. 1 Illustration of the physics behind the Vishniac overstability. Amismatch in the direction of the two main forces on a blast wave surface,the thermal pressure directed normal to the surface and the ram pressuredirected opposite the direction of motion, creates an oscillating ripplethat can grow if conditions are right

waves in xenon were unstable to the Vishniac overstability,

while those in nitrogen remained stable. Unfortunately, there

was interference on the growth rate measurement due to the

effects of the drive laser (Edens et al., 2004). More recently

there have been a number of experiments looking at blast

waves in planar (Keiter et al., 2002), cylindrical (Ditmire

et al., 2000; Edwards et al., 2001; Shigemori et al., 2000),

and spherical (Edens et al., 2004, 2005a, b; Hansen et al.,

2006) geometries at a number of facilities. The present au-

thors have been involved in experiments (Edens et al., 2005a,

b) measuring the evolution of perturbations induced on blast

wave and comparing those evolution rates to the theoreti-

cal predictions of Vishniac and Ryu. In the present work

we present the results of those experiments for blast waves

traveling in a mixture of nitrogen and xenon gas.

We performed our experiments on the Z-Beamlet laser at

Sandia National Laboratories (Rambo et al., 2005), and the

general experimental setup is described in a previous publi-

cation (Edens et al., 2005b). The only changes from the setup

described in that publication are the design of the wire array

and the choice of gas. In order to reduce the modal noise

caused by the use of a planar wire array, we used a half-

cylinder array for this experiment. The array was designed to

be one half of a 1 cm diameter cylinder with extended sides,

and an illustration of the design can be seen in Fig. 2. With

this setup it is possible to change both of the primary vari-

ables involved in the growth rate of perturbations on blast

waves. The polytropic index of the gas will depend on the

amount of radiation emitted by the gas, which will vary with

the gas type. The primary mode number of the induced per-

turbation will vary with the spacing of wires in the array. The

experiments reported here were performed in one gas (our

mixed gas) and with three different wire array spacings.

Fig. 2 Illustration of half-cylinder array used in blast wave experi-ments. There are 1 mm spaced grooves surrounding the open area

The gas used was 7.5 Torr xenon and 2.5 Torr nitrogen and

will be referred to as the mixed gas from this point on. This

mixture was chosen to maximize the radiative properties of

the gas while still providing useable data. When experiments

were attempted in pure xenon, the radiation from the main

blast wave irradiated the wires in our array, creating small

blast waves off the surface of the blast wave. These addi-

tional blast waves were large enough to obscure the main

blast wave and make obtaining useable data impossible. To

combat this, we decided to add in some nitrogen gas, which

is less radiative, to reduce the overall radiation level. We first

tried a mixture of 5 Torr xenon and 5 Torr nitrogen, and the

additional blast waves with this mixture were small enough

to see the main blast wave clearly. We then moved on to

7.5 Torr xenon and 2.5 Torr nitrogen and found the data were

marginally useable depending on the choice of wires in the

array. As can be seen in Fig. 3 when plastic coated copper

wires were used in the array, the additional blast waves ob-

scured the main blast wave. However, when uncoated copper

wires were used, the main blast wave could be seen and mea-

sured. One can barely discern in the coated wire image a main

blast wave similar to that seen in the uncoated wire image,

but it is very faint. This dependence of the image quality on

the wire type indicated to us we were at a radiation level that

was at the edge of our tolerance and therefore we used this

gas mixture for the experiments.

Images of blast waves traveling in our mixed gas are seen

in Fig. 4. The edge of the main blast wave was traced out

in each image and the resulting plot transformed into polar

coordinates. This transformed data was then interpolated to

give constant spacing between the points and the mean radius

of the blast wave was subtracted out. The angular coordinate

was plotted in fractions of a circle and the graph was Fourier

transformed so that we could isolate the frequency of interest.

The amplitude of this frequency was then examined as a

fraction of the mean radius and plotted versus time. We then

fit a power law in time to the normalized amplitude versus

time plot in order to compare the fit exponent to the theoretical

predictions and an example of this can be seen in Fig. 5.

The comparison of our experimentally determined evo-

lution rates for perturbations on blast waves in mixed gas

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Fig. 3 Two Schlieren images of blast waves traveling in mixtures of7.5 Torr xenon gas and 2.5 Torr nitrogen gas past a 3 mm spaced wirearray. The wires in the array in the left image are coated in plastic, whilethose in the right hand image are bare copper wires. You can clearlymake out a blast wave in the right hand image, while the blast wave in

the left hand image is obscured by additional blast waves off the wires inthe array. This difference based on the wires in the array indicated thatwe had chosen a mixture of gas that provided the maximum radiationwhile maintaining discernable data

Fig. 4 Images of blast waves in 10 Torr of our mixed gas traveling pasta 3 mm spaced wire array

to theoretical predictions for several different values of the

polytropic index are shown in Fig. 6. We can see that the

growth rates for the two lower mode number perturbations

imply a polytropic index between 1.1 and 1.2, but that the

decay rate for the highest mode number perturbation is more

consistent with a polytropic index of 1.3.

There are several possible explanations as to why the high-

est mode number data point does not agree with the other two.

The first is that the oscillating nature of the overstability is

Fig. 5 Plot of the normalized amplitude versus time for perturbationswith a mode number corresponding to that induced by traveling past a4 mm spaced wire array. The results are fit to a power law in time sothat they can be compared to theoretical predictions

not taken into account when analyzing the data. The theory

says the oscillation rate should be highest at the higher mode

numbers. It may be that the effect of the oscillation during the

time period studied was to reduce the amplitude of the per-

turbation and increase the apparent decay rate, thus raising

the implied polytropic index. The next explanation for the

behavior of the high mode number perturbation is that the

smaller wavelength perturbation was simply more difficult

to resolve from the data. The smaller length scale of the per-

turbations may increase the error in the raw data. Finally, the

wavelength of the perturbation may be similar to the thick-

ness of the blast wave during the time period studied and

if so, the physics becomes more complicated and this may

account for the behavior of the array. Unfortunately, simu-

lations like those performed in support of our earlier work

(Edens et al., 2005b) are not possible, due to the fact that

there is no available equation of state for such a gas mixture.

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Fig. 6 Comparison of experimentally determined growth rates for per-turbations on blast waves traveling in mixed gas to the theoretical predic-tions of Vishniac and Ryu. Note that the growth rates for the two lowermode number perturbations correspond to a polytropic index (gamma)of 1.1–1.2 while the highest mode number perturbation has a growthrate that implies a polytropic index of 1.3

In conclusion, we have looked at the evolution of pertur-

bations on blast waves traveling in a gas consisting of 7.5 Torr

xenon and 2.5 Torr nitrogen. For low mode number pertur-

bations, growth is observed and the implied polytropic index

is between 1.1 and 1.2, but at a higher mode number the ob-

served decay rate is more consistent with a higher polytropic

index of 1.3. There areseveral possible explanations for this

discrepancy, including oscillatory behavior of the perturba-

tions being more pronounced in the higher mode number

data. Future experiments should look at a gas that is more ra-

diative that nitrogen, but less than xenon (perhaps krypton).

Perturbations on a blast wave in such a gas may show similar

behavior while being possible to simulate.

References

Basov, N.G., Shikanov, A.S., Sklizkov, G.V., et al.: Sov. Phys. Jetp-Ussr35, 109 (1972)

Ditmire, T., Shigemori, K., Remington, B.A., et al.: Astrophys. J. Suppl.Ser. 127, 299 (2000)

Edens, A.D., Ditmire, T., Hansen, J.F., et al.: Phys. Plasmas 11, 4968(2004)

Edens, A.D., Ditmire, T., Hansen, J.F., et al.: Astrophys. Space Sci. 298,39 (2005a)

Edens, A.D., Ditmire, T., Hansen, J.F., et al.: Phys. Rev. Lett. 95, 244503(2005b)

Hansen, J.F., Edwards, M.J., Froula, D.H., et al.: Phys. Plasmas 13,022105 (2006)

Edwards, M.J., MacKinnon, A.J., Zweiback, J., et al.: Phys. Rev. Lett.8708, 085004 (2001)

Grun, J., Stamper, J., Manka, C., et al.: Phys. Rev. Lett. 66, 2738 (1991)Keiter, P.A., Drake, R.P., Perry, T.S., et al.: Phys. Rev. Lett. 89, 165003

(2002)Rambo, P.K., Smith, I.C., Porter, J.L., et al.: Appl. Opt. 44, 2421 (2005)Ryu, D., Vishniac, E.T.: Astrophys. J. 313, 820 (1987)Ryu, D., Vishniac, E.T.: Astrophys. J. 368, 411 (1991)Shigemori, K., Ditmire, T., Remington, B.A., et al.: Astrophys. J. 533,

L159 (2000)Vishniac, E.T.: Astrophys. J. 274, 152 (1983)

Springer


DOI 10.1007/s10509-006-9260-3


Colliding Blast Waves Driven by the Interaction of a Short-PulseLaser with a Gas of Atomic Clusters

Roland A. Smith · James Lazarus · Matthias Hohenberger · Alastair S. Moore ·

Joseph S. Robinson · Edward T. Gumbrell · Mike Dunne


Abstract Collisions between shocks are commonly found in

many astrophysical objects, however robust numerical mod-

els or laboratory analogues of these complex systems remain

challenging to implement. We report on the development

of scaled laboratory experiments which employ new tech-

niques for launching and diagnosing colliding shocks and

high Mach number blast waves, scalable to a limited sub-

set of astrophysically-relevant regimes. Use of an extended

medium of atomic clusters enables efficient (>80%) cou-

pling of 700 fs, 1 J, 1054 nm laser pulses to a “cluster” gas

with an average density of ≈1019 particles cm−3, producing

an initial energy density>105 J cm−3, equivalent to ≈5 × 109

J/g. Multiple laser foci are used to tailor the spatial profile of

energy deposition, or to launch pairs of counter-propagating

cylindrical shocks which then collide. By probing the colli-

sion interferometrically at multiple view angles in 5 incre-

ments and applying an inverse Radon transform to the result-

ing phase projections we have been able to tomographically

reconstruct the full three-dimensional, time-framed electron

density profile of the system.

Keywords Hydrodynamics . Instabilities . Plasmas .

Radiative transfer . Shock waves . Supernova remnants .

Atomic clusters . Lasers . Laboratory astrophysics . Blast

wave collision . Tomography . Radon transform

R. A. Smith () · J. Lazarus · M. Hohenberger · J. S. RobinsonThe Blackett Laboratory, Imperial College, London, UKe-mail: [email protected]

A. S. Moore · E. T. GumbrellAWE Aldermaston, Reading, Berkshire, UK

M. DunneThe Rutherford Appleton Laboratory, Chilton, Didcot,Oxfordshire, UK

1 Introduction

The dynamics of shocks and blast waves and the behavior of

spatial instabilities are of key importance in many areas of

plasma physics, including fields as diverse as laser-matter in-

teraction experiments and the evolution of structures within

supernova remnants (SNR’s). Many systems are further com-

plicated by interactions between shocks, and shock collisions

can commonly be observed in astrophysical objects such

as nebulae and supernova remnants over an extraordinary

range of spatial and temporal scales. They can also occur in

laboratory-scale experiments, including high-energy-density

plasmas driven by Z-pinches (Lebedev et al., 2002) and en-

ergetic laser systems (Woolsey et al., 2001). Suitably-scaled

laboratory experiments may thus provide an important tool

for both improving physical insight and for code benchmark-

ing.

The understanding of the complex dynamics of many

high-energy-density astrophysical systems is underpinned by

numerical simulation (Stone and Norman, 1992; Casanova

et al., 1991). However, modeling of shock propagation, and in

particular shock collisions, remains challenging despite sev-

eral decades of effort. Both fluid and particle interactions

need to be tracked simultaneously over a large computa-

tional grid and the situation can be further complicated by

the interplay between plasma radiation processes and macro-

scopic hydrodynamics. For example, radiation produced by

the shock may pre-ionize surrounding material, thus modify-

ing the properties of the matter it propagates into. Address-

ing this interplay numerically may necessitate the compu-

tationally expensive use of linked atomic physics and hydro

routines. Woodward and Colella (1984) have also highlighted

shock collisions as an area that most challenges the abilities

of numerical models designed for the simulation of astro-

physical objects. There consequently exists a clear demand

Springer


for well-characterized laboratory experiments in this area to

help explain the complexities of the physics involved (Report

for the US NRC, 2003).

The absolute length, time and density scales in laboratory

experiments are clearly very different from those character-

istic of objects such as supernova remnants. However, the

underlying dynamics can be invariant under an Euler trans-

formation linking time, space and velocity (Ryutov et al.,

1991). This invariance can be maintained by matching the di-

mensionless parameters which describe dissipative processes

such as heat conduction, convection and viscous hydrody-

namic flow, and by careful control of initial conditions in

an experiment (Moore et al., 2006). This enables the use of

well-characterized laboratory experiments to test both so-

phisticated numerical codes, and the theories describing the

development of rich spatial structures seen for example in

SNR’s (Weiler et al., 1998). Recent developments of both

high-power laser systems and atomic cluster target sources

have enabled small-scale laboratory experiments to be con-

ducted in regimes of relevance to a limited range of astro-

physical systems. Here we describe new techniques for both

creating and diagnosing collisions between shocks which al-

low us to acquire a full three-dimensional, time-framed map

of the electron density profile of the shock collision.

2 Atomic clusters as a target medium

Atomic clusters are cold, weakly bound aggregates of a few

to many millions of cold atoms. These fragile entities can be

produced in low-temperature, expanding gas streams (Smith

et al., 1998) and, rather counter intuitively, a mm scale ex-

tended medium of such clusters can exhibit extremely ef-

ficient absorption of high intensity laser light. In contrast

to the <1% absorption typically seen in monatomic gases,

sub-picosecond laser pulses with energies of order 1 J can

deposit >80% of their energy in such a medium (Ditmire

et al., 1997a). This has resulted in their use to drive a diverse

range of processes including x-ray generation (Ditmire et al.,

1998b), the production of highly-charged ions (Smith et al.,

1999) and “tabletop” thermonuclear fusion (Zweiback et al.,

2000a,b). More recently it has been shown that laser ener-

gies up to ≈450 J (Moore et al., 2005a) can be deposited in

a Xe cluster medium, further increasing the available energy

density to ≈108 J cm−3, or equivalently ≈5×1010 J/g. The

resulting increase in blast wave velocity may enable access

to new regimes where strong growth of plasma instabilities

such as the Vishniac overstability are likely. The coupling of

single atomic clusters with sub-picosecond lasers has been

studied extensively and can be well explained in terms of a

transient Mie-like resonance (Ditmire, 1998) which drives

an energetic Coulomb or hydrodynamic explosion (Ditmire

et al., 1997b).

Heating of an extended cluster medium by a single high-

intensity laser beam focused into the plume produced by

a pulsed gas jet (Smith et al., 1998) creates a plasma fila-

ment with a diameter comparable to that of the laser focus

– typically a few 10 s of microns. This hot plasma cylinder

is created on a timescale comparable to the laser pulse dura-

tion (typically sub-picosecond) and is initially surrounded by

cold gas at an average atomic density of ≈1019 cm−3. On the

spatio-temporal scales of the subsequent energy transport,

this unique type of interaction can be accurately viewed as

delta function heat deposition. After this abrupt energy input

the system initially exhibits non-local effects and, over time,

thermalises, forming a hydrodynamic blast wave (Ditmire

et al., 2000). Energy flow in the system can be driven by elec-

tron transport (both non-local and diffusive), radiation trans-

port or hydrodynamic flow. Typically several of these mecha-

nisms will dominate the dynamics at various times during the

evolution. A more complete discussion of these processes is

given by Moore et al. (2006) in this issue. By selecting high Z

target species it is also possible to form radiative blast waves

(Moore et al., 2006; Ditmire et al., 1998a). This can lead to the

pre-heating of un-shocked material, significantly altering the

propagation dynamics. The fragility of atomic clusters also

provides a unique route for creating “tailored” shock and

blast waves in a range of interesting geometries. Clusters can

be broken apart by a lower intensity (<1014 W cm−2) laser

pulse in selected regions of a target, leaving a cold medium

with constant atomic density but strongly modulated absorp-

tion properties. A system for producing and optically probing

such a tailored cluster medium is shown in Fig. 1. Heating

of this pre-prepared medium with a second, high-intensity

(≈1017 W cm−2) laser pulse then launches a tailored ioniza-

tion wave (Fig. 2) (Symes et al., 2002; Moore et al., 2005b)

which could, for example, be used to study growth of insta-

bilities seeded by a range of spatial frequencies.

3 Creation of colliding blast waves

The creation of blast waves in the laboratory using high-

power lasers is an area of long standing interest (Grun et al.,

1991; Dunne et al., 1994; Drake et al., 2002). Sub-picosecond

lasers now allow energy deposition to be decoupled from

the subsequent 100 ps–100 ns hydrodynamic motion of a

plasma, greatly simplifying the understanding of these ex-

periments. As a result, laboratory studies investigating the

radial expansion of spherical and cylindrical blast waves in

gas media have made significant progress in the last few

years (Edens et al., 2004; Ditmire et al., 1998a). In contrast,

only a limited number of counter-propagating plasma exper-

iments to investigate shock collisions have been attempted

(Elton et al., 1994; Bosch et al., 1992). These (and older un-

published studies which utilized electromagnetically-driven

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Fig. 1 Experimental layout forproducing and imaging tailoredblast waves. A cluster mediumproduced in a gas jet expansionis “machined” by a low-power1054 nm beam at≈1014 W cm−2. A strong shockis then launched using ahigh-power 1054 nm picosecondpulse focused to ≈1017 W cm−2

and imaged using a time delayed527 nm interferometric probe

Fig. 2 Electron density map of a strongly modulated blast wave in anextended hydrogen cluster medium at 6 ns after the heating pulse. Athin-shelled blast wave is present but no evidence of instability growthis seen at this time

shock tubes) were not able to access the strong shock or blast

wave regime.

The creation of a well-characterized, high energy density

colliding shock experiment in the laboratory is rather chal-

lenging and very limited data on such systems has appeared

in the literature to date. Pairs of laser-driven foil targets have,

for example, been used to create counter-propagating plasma

flows with collisions diagnosed through X-ray spectroscopy

(Elton et al., 1994), while only a few attempts have been

made to recover the electron density profile of pairs of col-

liding shocks (Bosch et al., 1992; Gregory et al., 2005).

Our data contrasts significantly with these previous stud-

ies carried out using long pulse (ns) lasers, in which the

collisions occurred between steady state shocks, not blast

waves. Moreover, we have been able to provide significantly

improved spatial resolution for imaging of collisions. This is

pre-requisite to studying thin-shell shocks (blast waves).

The absorption characteristics of a cluster target medium

enables high-Mach-number shocks (Ditmire et al., 1998a) to

be created efficiently by delta function heating using a small

scale, high-repetition-rate laser system. This also enabled

us to quickly acquire large (>100 shot) data sets in order

to minimize statistical fluctuations arising from shot-to-shot

variations in laser energy, and to investigate shock collisions

over a broad range of initial particle and energy densities.

In order to launch pairs of colliding cylindrical blast waves

we employed a split focus system (Fig. 3). A 700 fs, 800 mJ,

1054 nm heating beam from an Nd:Glass CPA laser system

was spatially split into two halves using a shallow angle 15

Fresnel bi-prism. These two beamlets propagated at oppo-

site angles to the original laser axis and were recombined

using a second identical prism placed ∼0.25 m downstream.

By tilting the second prism about an axis normal to its tri-

angular cross-section, phase can be added to the two recom-

bined beamlets asymmetrically, resulting in a slight angular

deviation from parallel. Subsequent focusing with an F/10

plano-convex lens produced two focal spots of peak inten-

sity Imax ≈ 1017 W cm−2 with variable separation in space.

These foci were used to heat two near-parallel, cylindrical

regions of a 2 mm-scale cluster medium produced by a cryo-

genically cooled, pulsed gas jet (Smith et al., 1998). The

subsequent blast wave evolution was probed with a num-

ber of diagnostics including high-spatial resolution (≈3µm)

interferometric imaging perpendicular to the heating beam

axis. To create the probe a small amount of light split from

the main heating beam was frequency doubled to produce

a 527 nm, 500 fs pulse which was passed through a vari-

able time delay (0–150 ns) and used to backlight the plasma

channel (in a manner similar to Fig. 1). Transmitted light

was imaged through a Michelson interferometer onto a CCD

image capture system. Interferograms captured in this way

were unwrapped to retrieve a two-dimensional phase map

Springer


Fig. 3 A bi-prism arrangement for launching two parallel, cylindrical blast waves in a cluster gas medium. The system can be rotated around thelaser axis to allow optical probing at a range of angles. Here the viewing angle is defined as 90

Fig. 4 Experimentalline-integrated phase image of apair of colliding cylindrical blastwaves in an H2 cluster gasprobed at 15. The blast waveshave evolved 5 ns afterdeposition of ≈105 J cm−3

(5 × 109 J/g) into pre-shockedmaterial at an average atomicdensity of ≈1019 cm−3. There isan apparent enhancement ofphase shift in the collisionregion. However, the lack ofcylindrical symmetry precludesrecovery of the electron densityprofile via Abel inversion

and the three-dimensional electron density profile of a single

cylindrically-symmetric blast wave (as per Fig. 2) recovered

via an Abel inversion of the phase profile. However, the pres-

ence of two shocks or a shock collision region (Fig. 4) breaks

the cylindrical symmetry necessary for retrieval of the elec-

tron density by this technique.

4 Tomographic reconstruction of 3D electron

density profiles

Due to the strong departure from cylindrical symmetry in

the case of colliding blast waves, Abel inversion schemes

are no longer appropriate for retrieval of electron density.

To overcome this limitation we have implemented a tomo-

graphic technique, which acquires multiple two-dimensional

phase maps of colliding blast waves over a range of view an-

gles. By rotating the prism system used to generate pairs of

blast waves about the laser axis while keeping the optical

probe system fixed we are able to image the collision region

interferometrically in order to probe at different angles on

successive laser shots. A ±7% energy bin (together with the

modest E1/4 scaling of blast wave radius in cylindrical ge-

ometry with deposited energy) ensures that successive view

angles sample plasmas derived from very similar initial con-

ditions. We expect an E1/4 scaling of an adiabatic blast wave

radius here as a result of our cylindrical geometry, rather than

the more usual E1/5 case for spherical geometry. A projection

can then be built up from the two-dimensional phase maps

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Fig. 5 A 2D section takenperpendicularly through thetomographically reconstructed3D electron density profile oftwo colliding, cylindrical, thinshelled blast waves in an H2

cluster gas. 18 individualinterferograms taken at 5

intervals at t = 9.75 ns in a350 ± 25 mJ energy bin wereused in the reconstruction.Electron density spikes are seenat the two collision ‘cusps’where the thin shells intersect

obtained through standard unwrapping routines, and an elec-

tron density cross section can be calculated using the inverse

Radon transform (Kak and Slaney, 2001). Due to symmetry

considerations we were able to obtain a reconstruction with

5 resolution using 18 individual energy binned laser shots.

In this work we have assumed a single plane of symmetry

perpendicular to the shock collision to reduce the number

of angles required by a factor of two. However, we empha-

size that more generally this technique can recover the full

three-dimensional electron density profile of the system for

a single time interval, without recourse to any assumptions

about the symmetry of the system, provided a sufficiently

large number of view angles can be sampled.

Figure 5 shows a two-dimensional slice through the re-

constructed three-dimensional electron density profile of two

colliding cylindrical blast waves driven in a hydrogen cluster

gas at 9.75 ns after initiation and 1.75 ns after the blast waves

first begin to collide. Successive blast waves were launched

using 350 ± 25 mJ of laser energy approximately equally

split between the two individual blast waves and images were

recorded for a series of view angles taken at 5 intervals. The

faint striations visible in the image are the result of recon-

struction artifacts caused by the finite number of view angles

used in the tomographic process, however the structure of

the two thin-shelled blast waves and the collision boundary

between them is clearly well resolved. A slight curvature of

the boundary between the two blast waves is seen, which we

attribute to a small asymmetry in the energy split between

the heating beamlets resulting in a difference in internal en-

ergy density of the two waves. The thin-shell structure of the

regions of the individual blast waves expanding away from

the collision zone is also clearly visible, as is an enhance-

ment of the electron density by a factor of ≈1.5× where the

blast waves have collided. At the apex of the collision region

and the uncollided edges of the two cylinders two localized

spikes in electron density ≈2.5× the peak shell density can

also be seen. Figure 6 shows plots of electron density for

sections taken perpendicular to and parallel to the collision

region for clarity. The mechanism by which this additional

increase in density occurs is currently unclear and will be

investigated in future work. One possibility is that this is the

result of the early stages of a Mach stem being formed.

5 Discussion

While extremely powerful, the tomographic technique we

report here has a number of important limitations – and pos-

sible enhancements – which are worth noting. The method-

ology we have described above requires multiple laser shots

and is thus limited to situations where good shot-to-shot re-

producibility of the plasma dynamics can be assured. This

necessitates energy binning or a high degree of laser pulse

reproducibility, together with a sufficiently large data set.

More importantly, it also demands that the underlying physi-

cal processes should be robust under small-scale fluctuations

in deposited energy and gas density etc. For the gross dynam-

ics of the thin-shelled hydrogen blast wave that we have used

as an example case, these conditions are well met (Moore

et al., 2006). However, there are important situations in which

such a tomographic technique would be invaluable, which are

interesting precisely because shot-to-shot reproducibility is

likely to be poor. One key example would be the study of

instability growth where strongly non-linear processes can

result in significant variations in both small and large scale

spatial structures on near identical shots, for example the

Vishniac overstability (Vishniac, 1983; Grun et al., 1991).

In such a case a multi-shot technique is likely to blur out

important details. However, we believe that the tomographic

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Fig. 6 Electron density profiles of (a) a section perpendicular to thecollision region and (b) along the collision region for the pair of blastwaves described in Fig. 5. Enhancements in electron density comparedto the thin shell can be seen in both the collision region itself, and at theapex of the two thin shells and the collision region. The quantities x ∼0.1 mm and y ∼ 0.5 mm are the width and length of the collisionregion respectively

technique reported here could be implemented on a single

shot basis for cases where the increased complexity of the

optical system required is warranted by the payoff in terms

of the physical insight gained.

6 Conclusion

We have developed new techniques based on an extended

medium of atomic clusters which allow us to study the prop-

agation dynamics and collisions of high Mach number cylin-

drical shocks with unprecedented levels of detail. The com-

bination of high absorption efficiency of a cluster medium

and the fragility of individual clusters allows tailored blast

waves to be produced for instability growth studies. Pairs

of colliding blast waves can also be created using a mod-

est table-top-scale laser system. By combining a split prism

system able to vary the orientation of two colliding cylin-

drical blast waves with high resolution optical interferome-

try we have been able to apply tomographic reconstruction

techniques to a blast wave collision for the first time. This

has allowed us to reconstruct the fully three-dimensional,

time-framed electron density profile of a blast wave collision

for the first time. Studies are now underway to investigate

collision dynamics in a range of high and low Z gases with

the aims of improving physical understanding, and to pro-

vide high quality experimental data which can be used for

benchmarking astrophysical codes.

Acknowledgements This work was supported by grants from theEPSRC and AWE Plc. We are pleased to acknowledge the technicalsupport of P Ruthven, B Ratnasekara and M. Dowman.

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DOI 10.1007/s10509-006-9266-x


Investigating the Astrophysical Applicability of Radiativeand Non-Radiative Blast wave Structure in Cluster Media

Alastair S. Moore · James Lazarus ·

Matthias Hohenberger · Joseph S. Robinson ·

Edward T. Gumbrell · Mike Dunne · Roland A. Smith

Received: 15 May 2006 / Accepted: 17 October 2006C© Springer Science + Business Media B.V. 2006

Abstract We describe experiments that investigate the ca-

pability of an experimental platform, based on laser-driven

blast waves created in a medium of atomic clusters, to

produce results that can be scaled to astrophysical situa-

tions. Quantitative electron density profiles were obtained

for blast waves produced in hydrogen, argon, krypton and

xenon through the interaction of a high intensity (I ≈ 1017

Wcm−2), sub-ps laser pulse. From this we estimate the lo-

cal post-shock temperature, compressibility, shock strength

and adiabatic index for each gas. Direct comparisons be-

tween blast wave structures for consistent relative gas densi-

ties were achieved through careful gas jet parameter control.

From these we investigate the applicability of different ra-

diative and Sedov-Taylor self-similar solutions, and therefore

the (ρ, T ) phase space that we can currently access.

Keywords Laser-cluster interactions . Laser-driven shocks

and discontinuities . Hydrodynamic and radiative plasma

instabilities . Laboratory astrophysics . Radiative blast

waves . Blast waves

1 Introduction

Shocks are ubiquitous throughout the observed universe and

are thought to play a crucial role in the transport of energy

A. S. Moore () · E. T. GumbrellAWE Aldermaston, Reading, Berkshire, RG 74 PR

J. Lazarus · M. Hohenberger · J. S. Robinson ·R. A. SmithLaser Consortium, The Blackett Laboratory, Imperial College,London, SW7 2BZ, United Kingdom

M. DunneRutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire,OX11 0QX, United Kingdom

into the interstellar medium, setting in motion processes ob-

served in nebulae that eventually could lead to the creation of

new stars. Radiation is key to this since it plays a significant

role in energy transport over the vast distances encountered

between stellar objects, and can significantly modify the dy-

namics of a shock or blast wave. Consequently the study of

radiative shocks has been an active area of numerical and

theoretical research over many years, but recently has been

highlighted as an area requiring further experimental inves-

tigation (US National Research Council, 2003). A radiative

shock can broadly be defined as a shock in which radiative

cooling occurs over a shorter time than the hydrodynamic

timescale of the shock, τh ≈ shock thickness (h)/shock speed.

Astrophysical shocks, structures commonly found in super-

nova remnants (SNRs), often lie within this classification

and have been seen to exhibit many different physical forms

and modes of instability and overstability (e.g. Cassiopeia A

SNR (Hwang et al., 2000)). Very much related to this is the

study of radiative blast waves. A blast wave, in which the

rarefaction has caught up with the decelerating shock front,

is affected in a similar way by the presence of radiation. Such

radiation can transport energy ahead of the shock front and

preheat the upstream unshocked material, under certain con-

ditions causing small ripples or wrinkles in the blast wave

surface to grow non-linearly.

Extensive astrophysical research has focused on these

problems (Kimoto and Chernoff, 1997; Ostriker and McKee,

1988; Vishniac, 1983; Ryu and Vishniac, 1991, 1987), and

there is clear evidence from the structure seen in astrophys-

ical observations that instabilities must be driven in certain

phases of the expansion of supernova remnants. However,

current code capabilities do not enable adequate understand-

ing of the many areas of physics that must be included in

a full description due to the differing length-scales involved

(Drake et al., 2000). To aid the understanding of the physics of

Springer


some of these observations laboratory experiments have been

conducted with the aim to reproduce the unstable blast wave

structure which has been analytically predicted (Vishniac,

1983; Ryu and Vishniac, 1991, 1987; Grun et al., 1991;

Schappert et al., 1994).

As part of an ongoing campaign to experimentally inves-

tigate these instabilities, we report on new results that fo-

cus on the blast wave parameters – namely the strength and

structure – that can be created in the laboratory. In this in-

vestigation we use a well-established platform that exploits

the efficient absorption of short pulse (sub-ps) laser energy

in an atomic cluster medium to produce strong blast waves

(Ditmire et al., 2000; Shigemori et al., 2000; Edwards et al.,

2001). An intense laser pulse (∼1017 Wcm−2) heats a plasma

filament a few mm long, which expands to form a shock and

later a blast wave. We study the interaction of a 1 J, 700 fs

laser system with large H2, Ar, Kr, and Xe clusters. Lim-

iting our investigation to studying the structure of the blast

wave front, in this paper we aim to fully identify the cur-

rent range of hydrodynamic parameters that we are able to

access and the most appropriate physical regime to scale to

astrophysical blast waves. Through high-resolution interfer-

ometry using a sub-ps, second-harmonic optical probe beam,

we were able to extract time-framed, two-dimensional elec-

tron density information, and measure the shock structure

with ±3µm accuracy.

2 Creating hydrodynamically-similar laboratory and

astrophysical systems

Astrophysical phenomena and laboratory experiments dif-

fer vastly in both scale and temperature. However, this scale

difference can be overcome by fulfilling four criteria for hy-

drodynamic similarity (Ryutov et al., 1999): (i) localisation,

(ii) convective heat flow, (iii) negligible radiation flux, (iv)

viscosity. Quantitatively these must be satisfied to approxi-

mately the same degree in both the laboratory and astrophys-

ical case for the hydrodynamic results to be comparable.

In order to study radiative effects, criterion (iii) is relaxed

and the opacity of the upstream and shocked material and

the radiative cooling rate then become critical to the flow

characteristics. The differing peak temperatures of the plas-

mas in each case also results in different temporal charac-

teristics in a laboratory experiment from those seen in an

astrophysical object. For the case of a cooling SNR, which

has been extensively studied in the astrophysical community,

the temperature is initially much higher than in the labora-

tory (Tastro∼1–100 keV) and the system must evolve for a

long time (∼104 yrs) before radiative cooling becomes im-

portant (Blondin et al., 1998). Conversely, in the laboratory,

once local thermodynamic equilibrium (LTE) is achieved,

the typical plasma temperature is 0.1–100 eV and so causes

radiative effects to become apparent almost immediately that

the system becomes hydrodynamic.

3 Blast wave self-similarity

The commonly-used self-similar solution for blast wave

propagation, first formulated by Sedov (1959), is found

through consideration of the total energy in the system and

results in the reduction in the number of non-dimensional

parameters to just two. For a simple spherical or cylindrical

system this analysis yields five dimensional variables – den-

sity (ρ), pressure (p), energy (E), radius (r) and time (t) –

and one dimensionless variable – the adiabatic index, γ . In

the case of a strong shock (Mach number, M > 1.3) the am-

bient pressure can be neglected and the list of dimensional

parameters is further reduced to four, all of which are de-

pendent on three dimensions: mass, length and time. Since

the system is thus reduced to four equations and three di-

mensions, it is straightforward to re-write the equations for a

dimensionless parameter, ξ , confirming that the system can

be described as self-similar, reproducing the common pa-

rameters in spherical (sph) and cylindrical (cyl) coordinates

(Sedov, 1959; Zel’dovich and Raizer, 1966, 1967).

ξ = r

(

ρ0

Et2

)β

. . .

⎧

⎪

⎪

⎨

⎪

⎪

⎩

βsph = 1

5

βcyl = 1

4

(1)

Although this Sedov-Taylor solution has proven invaluable in

describing blast wave evolution, it is reliant on some critical

assumptions: namely that for robust self-similarity no other

length-scales are involved in the system. Consequently, this

derivation can only be applied during the energy-conserving

phase of blast wave evolution when all dissipative processes

via radiation or conduction are negligible. This limits the va-

lidity of the Sedov-Taylor solution to a stage in the evolution

of the system when the mass set in motion by the shock is far

greater than that initially heated and the energy can be con-

sidered as deposited in an instantaneous, point-source. Con-

sequently, the Sedov-Taylor description is appropriate during

the adiabatic phase and holds until the blast wave shell den-

sity becomes comparable to that of the ambient background

or until radiation losses become large. Importantly, the con-

cept of self-similarity can be further extended provided that

the fraction of energy radiated is constant throughout the

duration of the ‘problem’. As discussed by Barenblatt, this

extends the use of solutions of the form r ∼ tα beyond the

adiabatic case and Liang and Keilty extended this to show

that an analytic description exists for any constant energy

loss fraction, ε, where 0 > ε > 1 (Barenblatt, 1979; Liang

and Keilty, 2000). Therefore, while either the Sedov-Taylor

or this modified self-similarity holds, for a blast wave radius

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Table 1 Experimental gas jet parameters for each of the four gases, calculated to ensure that an approximately equalambient density is produced. (a) denotes the ideal case and (b) the experimentally-realised case

P (bar) T (K) ×10−4 (g cm−3) ∗ Nc Rc (nm)

(a) Ideal

H2 282.5 100 3.40 – 130000 3000000 30

Ar 42.5 300 3.40 – 35000 130000 11

Kr 20.2 300 3.40 – 42000 220000 13

Xe 12.9 300 3.40 – 64000 580000 20

(b) Real

H2 49.9 100 0.60 85% 48000 300000 14

Ar 50.5 300 4.04 0% 35000 140000 11

Kr 21.0 300 3.52 13% 26000 66000 9

Xe 20.3 300 5.35 32% 48000 300000 16

at a particular time tcomp, the structure in different gases can

be compared to determine the effect of increased radiation

flux. Then, provided that the deposited energy, background

density and radius are approximately equal at tcomp, the struc-

ture of the front can be directly compared to learn about the

energy loss through radiation.

Astrophysical blast waves from SNRs are rarely observed

to evolve through different phases owing to the short ‘hu-

man’ timescale over which observations can take place.

Conversely, in laboratory experiments all such phases are

readily observed, and so it becomes crucial to study the for-

mation, structure and evolution of laboratory blast waves if

they are to be comparable and applicable to astrophysical

problems. It is therefore important to acknowledge that, de-

pending on the conditions in the experiment, there is only

a specific length of time during which the blast wave ex-

ists, and possibly a reduced duration for which a self-similar

description (and hence any hydrodynamic comparison with

astrophysics) is appropriate.

When constructing experiments to study the change in

blast wave structure due to different clustered gas media,

it is crucial that resulting plasmas, of the same deposited

energy, are compared at a time when the blast waves have

unambiguously formed and have swept up equal amounts of

mass. The second point is important in order to compare the

structure, since if self-similarity does not hold (for example

due to substantial radiative flux in higher Z gases) then the

flow variables are no longer ‘frozen’ and a comparison can

only be made if the blast waves are expected to be in a similar

phase of evolution.

Consequently, by comparing the blast wave structure

at the same radius with the density held constant, these

factors are satisfied. Interestingly, from the simple analytical

theory leading to Equation (1), the only variables in ξ are

deposited energy (E), density (ρ0), radius (r), and time (t),

so if the first three of these factors are kept constant, any

deviations from the expected power-law in time are an

indication of the extent to which the self-similarity does

or does not hold. Finally it is important to note that the

analytical blast wave theory we have described here is

purely fluid-based, and assumes a single fluid remaining

instantaneously in thermodynamic equilibrium. In reality,

the timescales of different aspects of the flow such as

ionization, recombination or electron conduction may be

shorter than thermodynamic equilibrization processes.

3.1 Calculating experimental parameters

We present high spatial-resolution data comparing blast

wave structure in hydrogen, argon, krypton and xenon, and

so it is necessary to produce equivalent mass densities in all

four gases.

m = P0a

√

√

√

√

γ A

RT0

(

2

γ + 1

)γ+1γ−1

(2)

The rate of mass flow, m, through a nozzle is related to the

pressure and temperature of the gas pre-expansion (P0 and T0

respectively), nozzle diameter, a, and ratio of specific heats,

γ = C p/Cv , of the gas (atomic mass A) according to Equa-

tion (2) (Miller, 1988). The laser-interaction and subsequent

dynamics are instantaneous on the mass-flow timescale, and

the gas jet forms a steady flow prior to the laser interaction.

Consequently, since the volume profile of the gas jet is in-

dependent of gas species (Smith et al., 1998), provided the

mass flow is the same then the neutral gas density profile in

the jet will be equivalent, and thus the ambient density, ρ0,

from Equation (1) will be the same for each gas, provided P0

and T0 are scaled appropriately. The ratios of specific heats

in Equation (2) are estimated using the temperature depen-

dence of C p in ref Forsythe (1954, 2003). Table 1 shows the

equivalent pressures and temperatures required to achieve

this comparison together with the resulting individual clus-

ter parameters: Hagena parameter, Ŵ∗ (Hagena and Obert,

1972), which governs cluster formation, and average cluster

size, Nc, and radius, Rc.

Due to the large difference in mass between, for example,

hydrogen and xenon, it is technically difficult to achieve the

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Fig. 1 A schematic, based on experimental data, of the timescales in-volved in the formation of a blast wave from a laser-heated clustermedium. Initially non-local effects occur due to the fast, free-streamingelectron component (green). The electrons thermalise after several pi-coseconds, and form an electron thermal wave (red) that follows a t0.2

trajectory according to Kimoto and Chernoff (1997) Finally, after afew nanoseconds, the plasma becomes LTE, and a shock forms. Asthe rarefaction wave catches up with the shock and sufficient mass isshock-heated, a blast wave forms, the deceleration of which identifiesit as radiative (purple) or adiabatic (blue)

necessary pressures in the laboratory to directly compare

blast waves produced in hydrogen with heavier cluster gases –

an issue we will address in future work. While it is clear that

the individual cluster parameters differ significantly between

the gases, which will have an effect on the ps-timescale laser

interaction with each cluster, the large Hagena parameter

implies that clustering has saturated, and energy absorption

studies (Ditmire et al., 1997; Moore et al., 2004, 2005) show

that when clusters are present to this degree a high fraction

of the laser energy (>80%) is coupled into the plasma, so

on the timescales we investigate the detailed physics of the

individual laser-cluster interaction can be neglected.

3.2 Blast wave formation

As we have highlighted, to be confident of the blast wave

formation it is important to have an insight into the pro-

cesses that occur prior to this. Figure 1 shows a schematic

representation of data obtained at the different stages of ex-

pansion occurring after the initial nanoplasma explosions of

individual clusters have merged into a uniform plasma. Ini-

tially from timescales ∼10 ps, non-local electron transport, as

observed in Ditmire et al. (1998), is expected to occur, result-

ing in a precursor or ‘foot’ to the electron density profile due

to a small fraction of free-streaming electrons transporting

energy ahead of the diffusively-driven thermalized electron

wave. As more matter is ionised by the propagating electrons,

Fig. 2 Electron density profiles of blast waves formed in H2, Ar, Kr, andXe, at equivalent times in their self-similar trajectories. Shell-thinning(increased compression), should occur in higher atomic number gasesdue to the lower ‘effective’ adiabatic index caused by the increasedradiation drive. However, we observe that the shell thickness increaseswith increasing atomic number (Z), probably due to increased electronconduction in Kr and Xe

the ionization front slows. The electrons quickly thermalize

on a timescale of a few ps, but since ti i ∼ (mi/me)1/2tee,

the ions thermalize considerably more slowly on the 100 ps

timescale. Equilibration between electrons and ions will take

longer still: tei ∼ (mi/me)tee and so thermal equilibrium in

the plasma is only reached on a timescale of many nanosec-

onds. By this time the shock has already begun to form and

after sufficient time (∼10 ns), the swept-up mass is greater

than that initially heated and the rarefaction wave will have

caught up with the shock front and a blast wave forms. From

this point on it is possible to make the self-similar approxi-

mations described since we assume the plasma components

are in local thermodynamic equilibrium (LTE).

4 Achievable blast wave parameters

In Fig. 2 we show comparative electron density profiles for

the blast waves formed in each of the four gases studied.

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The gas jet backing pressures were as indicated in Table

1(b). The profile in hydrogen was taken at a backing pres-

sure of 50 bar, somewhat lower than that required by the mass

flow calculations since this was not achievable with current

equipment. Although this difference between hydrogen den-

sity and those of the other gases are quite large, this should

only marginally affect the blast wave radius according to the

all self-similar derivation in Equation (1).

The comparative blast wave structures in Fig. 2 show sev-

eral key differences. By assuming that the initially-heated

plasma filament extends over an ≈80µm focal spot, the mass

swept up by the blast wave at r ≈ 400 µm that we measure

here is approximately 100 times greater, allowing complete

confidence that a blast wave has formed. A clear distinction

can be seen between the blast wave structure in each case. In

all four gases a fairly clean, sharp shock is observed, however

krypton and xenon are subject to considerably more preheat-

ing (and thus ionization) of the upstream gas and the shock is

less steep. In both Kr and Xe there is significant ionization up

to 200µm ahead of the shock, providing confirmation that the

shock is radiating energy upstream, since the electron mean

free path is estimated to be ≈50 µm at most at this point in

time. Such radiative blast waves are typically classified into

three regions: pre-shock or pre-cursor, transition, and cavity

regions (Keilty et al., 2000; Mihalas and Weibel-Mihalas,

1999). For a monotonic gas, in the absence of radiation and

ionization, it is appropriate to use an ideal gas description

with a constant value for γ . However, radiative blast waves

of the type we produce can not be described in this simpli-

fied way and a more accurate description utilises a separate

γ in each region. In the regime of our experiments, the de-

gree of ionization on either side of the shock will alter the

equation of state. For simplicity we use two approximations

both utilising an ‘effective polytropic index’. First we simply

calculate a single-valued γid−eff equivalent to the analysis for

a monatomic ideal gas. However we also compare this to the

calculated γeff that takes account of different levels of ion-

ization either side of the shock, by redefining the equation

of state: γeff − 1 = p/ρǫ(Z∗ + 1) (Drake, 2006), so that γ eff

reflects more accurately the affect of just radiation. Here, ε

is the specific internal energy, Z∗ is the level of ionisation,

and ρ is the molar gas density. In this slightly more complex

approximation, while Z∗

is estimated on either side of the

shock, a single γeff is assumed, having the same value in the

pre-shock and post-shock gas (Liang and Keilty, 2000).

Blast waves that strongly radiate should undergo shell-

thinning, since the polytropic index of the post-shock gas is

reduced through radiation and the material becomes more

compressible. By measuring the shell thickness, h, as a func-

tion of the blast wave radius, R, in all four gases we calcu-

late the compressibility, C = ρshell/ρ0, through the relation

C = R2/(2Rh − h2). Provided that the blast wave is self-

similar and that all the mass can be assumed to be in the thin

shell, then the compression can be estimated to better than

10%, based on the resolution of the interferometric data.

The comparison in Fig. 2 shows much thicker shells in Kr

and Xe than in Ar and H2 for the same blast wave radius.

This implies, counter intuitively, that H2 and Ar are more

compressible. We measure a compression factor C ≈ 4 in

both H2 and Ar, demonstrating that we access the strong

shock limit for an ideal gas. Surprisingly, the compression we

measure for Kr and Xe is C ≈ 2–3, whereas a radiating gas

with a lowered γeff should lead to a compression exceeding

4. This contradiction indicates that either the blast waves we

generate are not in the strong shock limit or that another of

the assumptions implicit in the calculations is invalid.

A consequence of preheating gas ahead of the shock is the

considerable increase in the upstream sound speed, causing

a reduction in the Mach number. In Krypton, the shock speed

is ≈17 ± 2 km s−1. Without preheat this would correspond

to a Mach number in excess of 60. In fact it is more likely

that in the precursor region material is heated to several eV

ahead of the main shock, so that in the pre-shocked medium

the sound speed could be as high as 3 km/s. This drastically

reduces the Mach number to ∼6. However, we find that even

with precursor temperatures as high as 5 eV, as shown in

Table 2, the Mach numbers for both Kr and Xe do not drop

sufficiently to preclude being in the strong shock limit.

For an ideal gas, ignoring ionisation, compression is re-

lated to the adiabatic index and Mach number: C = (γid-eff +1)M2/[(γid-eff − 1)M2 + 2] (Shigemori et al., 2000). Using

this ideal case, the range of predicted adiabatic indices for

each gas are shown in Table 2. In H2 and Ar these are not

unreasonable agreeing, within experimental limitations, with

that of an ideal gas. However, the large error prevents any real

insight into whether or not the value has fallen due to any in-

crease in the effective number of degrees of freedom, f, in the

plasma. In the higher-Z, more-radiative cases of Kr and Xe

reduced compression leads to an increase in the polytropic

index. The cylindrical blast waves that we create are quasi-

2D, which can be argued to decrease f, resulting in a max-

imum γid-eff = 2. However, the larger-still calculated γid-eff

for Xe demonstrates the limitations of the ideal gas approx-

imation of a constant polytropic index across the shock. Re-

calculating C to take account of differing levels of ionization

ahead of and behind the shock produces much more realistic

values of γeff. In H2 γeff = 1.48 ± 0.13 and Ar, Kr and Xe

γeff is between 1 and 1.3. Interestingly, we achieve the low-

est effective index in Krypton, where radiation is significant

but not so strong that the shock strength is reduced like in

Xenon.

The large uncertainty in the precursor temperature leads to

error in the Mach number. The shock velocity, which can be

measured directly over a number of different laser shots or es-

timated from a single shot assuming a self-similar blast wave

trajectory and associated deceleration parameter, typically

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Table 2 Table of calculated plasma parameters that we access in H2, Ar, Kr and Xe, for the experimentally-realisedblast wave profiles in Fig. 2

Parameters H2 Ar Kr Xe

Experimental

Time (ns) 6.8 11.5 12.0 15.4

Shell thickness h (cm) 54 ± 4 58 ± 5 70 ± 5 100 ± 5

Shock velocity Vs × 106 (cm s−1) 3.0 ± 0.1 1.7 ± 0.1 1.7 ± 0.2 1.3 ± 0.1

Upstream Mach number M 56.2 ± 2.3 33.1 ± 29.3 6.7 ± 4.3 5.7 ± 4.3

Compression 4.0 ± 0.3 3.7 ± 0.3 3.1 ± 0.2 2.3 ± 0.1

Density ρ (g cm−3) 5 × 10−5† 4 × 10−4† 6 × 10−4† 3 × 10−4†

Pressure P0 (Pa) 1.5 × 108† 9 × 109† 2 × 1010† 9 × 1010†

Postshock temperature T2 (eV) 1.3 ± 0.2 7.3 ± 3.3 15.4 ± 3.3 16.2 ± 3.5

Ideal gas polytropic index γid-eff 1.5–1.7 1.4–1.7 1.5–1.9 2.0–2.4

Effective polytropic index γeff 1.48 ± 0.13 1.18 ± 0.17 1.07 ± 0.14 1.12 ± 0.14

Localised

Mean free path li i (cm) 2.7 × 10−5† 7.2 × 10−4† 1.5 × 10−3† 5.3 × 10−3†

Collisionality li i/h 4 × 10−4† 0.12† 0.2† 0.5†

Heat conduction/viscosity

Peclet number Pe 300† 1.2† 0.5† 0.5†

Reynolds number Re 1000† 10000† 30000† 40000†

Radiation

lR (cm) 0.3† 120† 860† 900†

lc (cm) 43000† 30000† 38000† 51000†

τthin (ns) 5200† 570† 310† 400†

Hydro

τhydro (ns) 2† 3† 4† 8†

e-i equilibration distance (µm) 3.2† 440† 2020† 5300†

Euler Number 0.8† 0.1† 0.1† 0.1†

Scaling parameters are estimated following the analysis of Hagena and Obert (1972) and indicate that in hydrogenthe system has become hydrodynamic and can be compared to the self-similar models. However, in the higheratomic number gases electron conduction remains significant, causing the low Peclet numbers we calculate† – order of magnitude estimate

introduces an error of ∼10%.

T2

T1

=(

M22γeff(γeff − 1) − (γeff − 1)2

(γeff + 1)2+ 2

M2(γeff + 1)

)

×(

Z∗1 + 1

Z∗2 + 1

)

(3)

Assuming an equation of state that includes ionization:

p = (Z∗ + 1)ρkB T /(Am p), the post-shock temperature can

be calculated using Equation (3), of which only the first term

remains in the absence of ionization and in the strong shock

limit. However, the dependence of the post-shock tempera-

ture on T1 is relatively small, since M2α1/T1. Consequently,

for large M, T2 is independent of T1. In the general case

this leads to the largest obstacle to accurately predicting T2

being the effective polytropic index, leading to the large er-

ror (∼20%) in the estimated post-shock temperature. In the

ideal gas derivations of Zel’dovich and Raizer (1966, 1967),

Mihalas and Weibel-Mihalas (1999), this is assumed constant

across the shock, which is clearly not the case here. With the

exception of hydrogen, where γeff = 5/3 can be used with

relative confidence, the preheating ahead of the shock will

undoubtedly change the adiabatic index of the upstream gas,

preventing an accurate estimation of the post-shock temper-

ature and Mach number.

The order-of-magnitude estimates of the scaling param-

eters provide an indication of the hydrodynamic parameter

space we are able to access. The limiting parameter is the

Peclet number, Pe, which must be ≫1 in order for heat con-

duction to be negligible. In hydrogen this is not a problem,

since the thermal diffusivity is relatively low. However, due

to the higher temperature and lower number density in Ar, Kr

and Xe, the electron transport is sufficient to limit the hydro-

dynamic scalability of the plasmas we create. In agreement

with the conclusions of Edwards et al. (2001), the relatively

thick shocks that we currently observe in the high Z gases

are due to electron conduction that transports energy over

10’s µm in the case of Kr and Xe. Finally, the e-i equilibra-

tion distances further demonstrate the need to account for

ionization. While there is probably LTE within the electron

and ion fluids, only in hydrogen is it likely that a single fluid

description is accurate, since for Ar, Kr and Xe the equili-

bration distances are on the order of or larger than the scale

of the hydrodynamic system.

Springer


5 Conclusions

These results provide convincing evidence that the blast

waves we produce in both Kr and Xe are strongly radiative

and that, because the gas medium we use is optically thin,

radiation transports energy ∼200 µm ahead of the blast wave

front, preheating the upstream material. We are able to pro-

duce high-Mach number (M > 20) blast waves in less radia-

tive gases, Ar and H2, and find that measurements of the shock

compression match well with the expected values for an ideal

gas. However, in the more radiative Kr and Xe, the assump-

tions of a single-valued effective polytropic index throughout

the shock are less appropriate due to the degree of ionization.

This is indicative that a simple monatomic ideal-gas theory

invoking a single-value for γid−eff is not sufficient to predict

our post-shock conditions. When Z∗ is included in the equa-

tion of state more accurate estimates of the post-shock tem-

perature and γeff can be made, however numerical modeling

that can manage generalized jump conditions in an ionized,

radiating gas is required to fully explain the behavior we

observe.

The radiative self-similar solutions in part address this

problem, but without accurate knowledge of the pre-shock

conditions, owing to the very strong preheating, it is not pos-

sible to calculate meaningful parameters. The scaling param-

eters necessary to link this work to an astrophysical situation

are closest in hydrogen, but the additional electron conduc-

tion in Ar, Kr and Xe prevent realistic scaling being made

without further modeling.

Acknowledgements This work was supported by grants from the EP-SRC and AWE. We are pleased to acknowledge useful conversationswith D. R. Symes and the technical support of P. Ruthven and B.Ratnasekara.

References

Barenblatt, G.I.: Similarity, Self-Similarity and Intermediate Asymp-totics. Consultants Bureau, New York (1979)

Blondin, J.M., et al.: Astrophys. J. 500, 342 (1998)Ditmire, T., et al.: Phys. Rev. Lett. 78, 3121 (1997)Ditmire, T., et al.: Phys. Rev. Lett. 80, 720 (1998)Ditmire, T., et al.: Astrophys. J. Supp. Series 127, 299 (2000)Drake, R.P.: High-Energy-Density Physics. Springer Publications

(2006)Drake, R.P., et al.: Phys. Plasmas 7, 2142 (2000)Edwards, M.J., MacKinnon, A.J., Zweiback, J., et al.: Phys. Rev. Lett.

87, 8708 (2001)Forsythe, W.E.: Smithsonian Physical Tables, 9th edition. Table 157,

Knovel (1954, 2003)Grun, J., et al.: Phys. Rev. Lett. 66, 2738 (1991)Hagena, O.F., Obert, W.: J. Chem. Phys. 56, 1793 (1972)Hwang, U., et al.: Astrophys. J. Lett. 537, 119 (2000)Keilty, K.A., et al.: Astrophys. J. 538, 645 (2000)Kimoto, P., Chernoff, D.: Astrophys. J. 485, 274 (1997)Liang, E.P., Keilty, K.A.: Astrophys. J. 533, 890 (2000)Mihalas, D., Weibel-Mihalas, B.: Foundations of Radiation Hydrody-

namics. Dover Publications Inc. (1999)Miller, D.R.: in Scholes, G. (ed.) Free Jet Sources Vol. 1 of Atomic and

Molecular Beams. Oxford Univ. Press (1988)Moore, A.S., et al.: CLF Annual Report 2004–2005, p. 34Ostriker, J., McKee, C.: Rev. Mod. Phys. 60, 1 (1988)Ryu, D., Vishniac, E.T.: Astrophys. J. 313, 820 (1987)Ryu, D., Vishniac, E.T.: Astrophys. J. 368, 411 (1991)Ryutov, D., et al.: Astrophys. J. 518, 821 (1999)Schappert, G.T., et al.: AGEX II, LANL Quarterly Report (1994)Sedov, L.I.: Similarity and Dimensional Methods. Academic Press, New

York (1959)Shigemori, K., et al.: Astrophys. J. Lett. 533, L159 (2000)Smith, R.A., et al.: Rev. Sci. Inst. 69, 3798 (1998)US National Research Council: Frontiers in High Energy Density

Physics. National Acadamies Press, Washington, D.C. (2003)Vishniac, E.: Astrophys. J. 274, 152 (1983)Zel’dovich, Raizer: Physics of Shock Waves and High Temperature

Phenomena. Dover Publications Inc. (1966, 1967)

Springer


DOI 10.1007/s10509-006-9269-7


Mass-Stripping Analysis of an Interstellar Cloud by a SupernovaShock

J. F. Hansen · H. F. Robey · R. I. Klein · A. R. Miles


Abstract The interaction of supernova shocks and interstel-

lar clouds is an important astrophysical phenomenon since

it can result in stellar and planetary formation. Our experi-

ments attempt to simulate this mass-loading as it occurs when

a shock passes through interstellar clouds. We drive a strong

shock using the Omega laser (∼5 kJ) into a foam-filled cylin-

der with an embedded Al sphere (diameter D = 120µm)

simulating an interstellar cloud. The density ratio between Al

and foam is ∼9. We have previously reported on the interac-

tion between shock and cloud, the ensuing Kelvin-Helmholtz

and Widnall instabilities, and the rapid stripping of all mass

from the cloud. We now present a theory that explains the

rapid mass-stripping. The theory combines (1) the integral

momentum equations for a viscous boundary layer, (2) the

equations for a potential flow past a sphere, (3) Spalding’s

law of the wall for turbulent boundary layers, and (4) the

skin friction coefficient for a turbulent boundary layer on a

flat plate. The theory gives as its final result the mass stripped

from a sphere in a turbulent high Reynolds number flow, and

it agrees very well with our experimental observations.

Keywords Shock . Interstellar . Cloud . Turbulent .

Turbulence . Mass-loading . Star formation

J. F. Hansen () · H. F. Robey · A. R. MilesLawrence Livermore National Laboratory, Livermore, CA 94550,USAe-mail: [email protected]

R. I. KleinLawrence Livermore National Laboratory, Livermore, CA 94550,USA; Department of Astronomy, University of California,Berkeley, CA 94720, USA

1 Introduction

We report here the results from experiments designed to study

the interaction between a supernova shock and interstellar

accumulations of matter (ISM) or “molecular clouds.” The

experiments were carried out at the Omega laser at the Lab-

oratory of Laser Energetics in Rochester, New York (Boehly

et al., 1997). We observe rapid stripping of all mass from a

cloud, and we present a model based on turbulent boundary

layers where the time scale of the mass-stripping agrees with

the experiment. This may be important for understanding

mass-loading in the astrophysical case, and for comparison to

numerical simulations of the interaction between shocks and

interstellar clouds which have shown that turbulence can gen-

erate structure which acts as seeds for star formation (Clark

and Bonnell, 2005). We note that aided by the forerunner

of this experiment (Klein et al., 2000; Robey et al., 2002),

shock-cloud interaction evolved to a late time has recently

been observed in supernova remnant Puppis A (Hwang et al.,

2005).

2 Experimental setup and diagnostics

The strong shock of a supernova explosion is simulated at

experiments at Omega in the following manner: a small Be

shock tube (2.25 mm long; 0.8 mm inner diameter; 1.1 mm

outer diameter) is filled with a low density (300 mg/cm3) car-

bonized resorcinol formaldehyde foam (CRF). The CRF at one

end of the shock tube is then ablated by laser beams, causing

the ejection of ablated material in one direction to launch

a planar shock in the opposite direction. Good planarity of

the shock is ensured by using multiple (ten), superimposed

beams, each with a super-gaussian beam profile created by

a phase plate in the focusing optics; the super-gaussian is of

Springer


Fig. 1 HYADES calculation of the free-stream velocity U∞ = U∞ (t) ,the Mach number M , the compression ρ1 (t) /ρ10 (where the initialdensity ρ10 = 300 mg/cm3), and the temperature T = T (t)

order eight and has a flat top matching the diameter of the

shock tube. Each laser beam has an energy of ∼500 J with a

pulse duration of 1.0 ns.

The scaled ISM cloud is simulated by an Al sphere (ra-

dius R0 = 60µm) embedded in the CRF a short distance into

the shock tube (on the shock tube axis 500µm from the ab-

lated CRF surface). The density ratio between the Al (density

2.7 g/cm3) and the surrounding CRF is chosen to match the

density ratio for an actual ISM cloud and other experimental

parameters are also scaled to preserve the physics regime of

the astrophysical case by following the guidelines set out by

Ryutov et al. (1999). Physical quantities in the CRF (with-

out an embedded Al sphere) can be accurately calculated

using the 1D radiative hydrodynamic code HYADES (Larsen

and Lane, 1994). We have used this code to calculate the

free-stream velocity U∞ (t) , the temperature T (t) , and the

density ρ1 (t) for the CRF and these quantities are plotted in

Fig. 1. (These values were used in our Euler scaling estimates

above.) The simulation used an in-line quotidian equation of

state (QEOS) model with a bulk modulus of 3 × 109 Pa and

112 zones to represent the 2.25 mm long CRF, with the first

46 zones feathered for ablation with a zone-to-zone scaling

ratio of 1.15, the final 46 zones feathered for shock release

with a scaling ratio of 0.87.

The cloud is imaged using a gated x-ray framing camera

(Budil et al., 1996). X rays for the image are generated by a

second set of time-delayed laser beams (backlighter beams)

pointed at a metal foil, typically Ti, located on the opposite

side of the shock tube from the camera. He-α radiation from

the Ti (at 4.7 keV) moves through the shock tube and is im-

aged by either a 10µm pinhole located at the front end of the

camera (“area radiography”) or by a 20µm pinhole next to

the Ti foil (“point projection radiography”). The point pro-

jection radiography technique results in a higher photon flux

through the pinhole, and thus a better signal-to-noise ratio.

The imaging element of the camera is either a microchannel

plate (MCP) + film or MCP + charge coupled device (CCD),

and in both cases has a size of ∼35 mm. The exposure of

the MCP was kept in the linear regime (Landen et al., 1994).

The distance from shock tube to Ti foil is 4.0 mm for area

radiography, 6.5 mm for point projection radiography. The

time-delay for the backlighter beams is chosen to obtain an

image at a desired time t after the initial, ablative laser pulse

has started the shock in the shock tube. The camera MCP is

triggered to coincide with the backlighter beams. The MCP

pulse length was set to 500 ps in a trade-off between maxi-

mizing x-ray exposure on the MCP while minimizing motion

blurring (e.g., when the plasma moves 20 km/s the motion

blurring is 10µm, comparable to the pinhole diameter). The

experiment is repeated with different time-delays to generate

an image sequence.

3 Results

Results from the experiment can be seen in Fig. 2. As the

shock runs over the cloud (t = 5 ns), its speed inside the cloud

is greatly reduced, leading to a Kelvin-Helmholtz instability

and its characteristic roll-up (t = 12 ns). Soon thereafter, a

Widnall-type instability (Widnall et al., 1974) occurs, creat-

ing a low mode number azimuthal perturbation of order five

when viewed from a point on the extended shock tube axis

(Robey et al., 2002). Here we see the Widnall instability as

four “fingers” at the trailing edge of the cloud at t = 30 ns,

indicating a mode number of four to eight (depending on if

each finger is or is not overlapping another finger along the

line of sight). Material is constantly being stripped away from

the Al plasma cloud and is visible in the images as a cone of

diffuse material behind the cloud (t ≥ 19 ns). By t = 40 ns

this cone extends outside our diagnostic field of view. By

t = 60 ns so much material has been stripped away that the

remaining cloud is quite diffuse (we are showing the 60 ns

image at a higher contrast than the earlier point projection

radiography images in Fig. 2). By t = 100 ns the cloud has

been completely stripped away and can no longer be iden-

tified in the point projection radiography images. We also

obtained an image at 80 ns in which the cloud is completely

gone, but this target unfortunately had its Au spatial reference

grid mounted outside the view of the x-ray framing camera

and was therefore not included in Fig. 2.

Because the point projection radiography technique illu-

minates the MCP in a very uniform fashion, we can use the

point projection radiography images to estimate the cloud

mass using the formula I = I0 exp(−m/µA) + Ib where I

is the measured pixel intensity, I0 is the x-ray source intensity

Springer


Fig. 2 A time sequence of images showing how the cloud evolves afterthe passage of a shock. In each image, the direction of motion of theshock is approximately from left to right, and is perpedicular to the im-aged shock at time t = 19 ns, and the imaged Au grids at times t = 30 nsand 40 ns . In the first image, at t = 5 ns, the shock is intersecting thecloud and the left hand side of the cloud is compressed by a factor of4, the strong shock limit for a polytropic gas with an adiabatic indexγ = 5/3. The cloud undergoes a classical Kelvin-Helmholtz roll-up, asseen at t = 12 ns and later. Cloud material is stripped away from the

cloud. Stripped material is clearly evident trailing the cloud at t ≥ 19 nsand is shaped as a cone that extends all the way to the shock (or ex-tends outside the field of view at t = 40 ns). A rarefaction changes thedirection of the surrounding flow at approximately t = 40 ns, and byt = 60 ns the reverse flow has caused the right hand side of the cloudto become fairly round. By t = 100 ns the cloud has disappeared. Thefirst four images were obtained with area backlighters, the last four withpoint projection radiography

(i.e., the intensity we would have expected to measure had

there not been mass attenuation), m is the integrated line

of sight mass, µ is the x-ray attenuation coefficient in units

of mass per area, A is the pixel area in the image, and Ib

is background intensity from all sources of non-directional

exposure, such as film fogging, non-directional x rays, en-

ergetic particles, etc. We estimate the cloud mass to be

0.67 ± 0.11µg at t = 30 ns and 0.54 ± 0.11µg at t = 40 ns

(it is coincidental that the error is 0.11µg in both of these

images – the error is normally different from image to im-

age). This can be compared to the original sphere mass of

2.44µg.

4 Analysis

We present in this section a new mathematical model that

describes mass stripping from a cloud under turbulent, high

Reynolds number conditions. We compare this model to

our experimental data and to an existing model (Taylor,

1963; Ranger and Nicholls, 1969) for laminar mass strip-

ping. Our model combines four separate concepts of fluid

mechanics: (1) the integral momentum equations for a vis-

cous boundary layer, (2) the equations for a potential flow

past a sphere, (3) Spalding’s law of the wall for turbulent

boundary layers (Spalding, 1961), and (4) the skin fric-

tion coefficient for a turbulent boundary layer on a flat

plate.

We begin with the integral momentum equations for a

stationary, viscous boundary layer:

∂

∂x

∫ δ1

0

u1 (U − u1) dy1 + dU

dx

∫ δ1

0

(U − u1) dy1

+ 1

r

dr

dx

∫ δ1

0

u1 (U − u1) dy1 = ν1

∂u1

∂y1

∣

∣

∣

∣

y1=0

(1)

∂

∂x

∫ δ2

0

u22dy2 + 1

r

dr

dx

∫ δ2

0

u22dy2 + 1

ρ2

dp

dxδ2

= −ν2

∂u2

∂y2

∣

∣

∣

∣

y2=0

(2)

ρ1ν1

∂u1

∂y1

∣

∣

∣

∣

y1=0

= −ρ2ν2

∂u2

∂y2

∣

∣

∣

∣

y2=0

(3)

where x is a coordinate along the surface of the cloud (we will

approximate the cloud with a sphere at all times so that x = 0

at the flow stagnation point and x = π2

R at the equator), y

is a coordinate perpendicular to the cloud surface, r is the

distance from the cloud surface to the cloud axis of symmetry,

U = U (x) is the free stream flow velocity behind the shock,

u = u (x, y) is the flow velocity inside the boundary layer,

ν is the kinematic viscosity, ρ is the density, p = p (x) is

the pressure, and δ = δ (x) is the thickness of the boundary

layer. The flow properties U, u, ν, ρ, p, and δ are also

functions of time t. Subscript 2 denotes plasma in the cloud

and subscript 1 the surrounding flow, e.g., δ2 is the boundary

Springer


Fig. 3 The flow geometry around the cloud is modelled with the poten-tial flow around a sphere with the boundary layer flow calculated usinga local cartesian coordinate system where the coordinate x is along theflow (i.e., along the surface of the sphere) and y is the distance intothe boundary layer from the sphere surface. The velocity in the bound-ary layer is u1 outside the sphere radius R and u2 inside the sphereradius. At the outside edge of the boundary layer y1 = δ1 the velocityu1 = U, i.e., matches the potential flow velocity. At the sphere radiusu1 = u2 = AU, where A is a constant. At the inside edge of the bound-ary layer y2 = δ2 the velocity u2 = 0. The cylindrical coordinate r isthe distance from the axis of symmetry to the sphere surface

layer thickness inside the cloud. The geometry is sketched

out in Fig. 3.

We will assume that the free stream velocity around the

cloud follows the potential flow of a sphere,

U (x) = 3

2U∞ sin

( x

R

)

, (4)

whereU∞ is the flow velocity far from the sphere. This has the

advantage that dU/dx = 0 at the equator x = π R/2, which

will simplify the algebra substantially. Also, at the equator

dp/dx = −ρgUdU/dx = 0 and trivially dr/dx = 0.

To express the boundary layer velocity u as a function of

the distance y from the sphere surface, we will use Spald-

ing’s law of the wall for turbulent boundary layers (Spalding,

1961):

y+ = u+

+ e−κB

(

eκu+ − 1 − κu+ − (κu+)2

2− (κu+)3

6

)

(5)

where the dimensionless coordinate y+ ≡ yv∗/ν and the

dimensionless velocity u+ ≡ u/v∗, or in the appropriate

coordinate frame of reference u1 = v∗1u+

1 + AU and u2 =AU − v∗

2u+2 , where A is a constant such that AU is the flow

velocity on the sphere surface. The wall-friction velocity v∗ is

defined through v∗2 = ν (du/dy)|y=0 and Coles (1995) gives

the coefficients κ = 0.41 and B = 5.0.

This is as far as we can go without saying something about

the wall-friction velocity v∗, or equivalently the skin-friction

coefficient C f , as the two are related through

v∗2 = 1

2C f (x) U 2 (x) . (6)

For calculations of skin-friction drag, many renowned re-

searchers, beginning with Dryden and Kuethe (1930) and

Millikan (1932), have used velocity distributions for a flat

plate in non-flat geometries and found that the results do not

differ seriously from measured values (Goldstein, 1965). We

will do the same and use the skin friction coefficient for a

turbulent boundary layer on a flat plate (White, 1974):

C f (x) ≈ 0.0592 Re−1/5x (7)

where the Reynold’s number Rex = U x/ν, but with a mod-

ification; if we use Equation (7) as is, the problem is overde-

termined. We replace the coefficient 0.0592 with a coefficient

that will be determined by our system of equations. We set:

C f (x) = 2

α2(U x/ν)−1/5 . (8)

We can now rewrite Equation (1) as

1200κ2eκB K 91 (1 − A)α−1

1 = 120eξ [(1 + 2A) ξ 2

− (3 + 2A) ξ + 4] − 3 (3 + 10A) ξ 5 +−20 (1 + A) ξ 4 + 20(κ−1eκB − 1) (1 + 2A) ξ 3

−120 (1 − 2A) ξ − 480 (9)

where

ξ = κK1(1 − A)α1. (10)

K1 =(

3πU∞ R

4ν1

)1/10

. (11)

We expect A to be a fairly small quantity (it will certainly

be smaller than unity) so one might be tempted to linearize

Equation (9) w.r.t. A, but this only simplifies terms where A

(or ξ ) does not appear in the exponents and does not lead to

an analytical solution for A (as it does in the laminar model).

Consequently some form of simple numerical scheme must

be employed to calculate A, and we have therefore chosen

to not linearize Equation (9) w.r.t. A, but to keep the exact

form.

Springer


Similarly in the Al cloud defining

K2 =(

3πU∞ R

4ν2

)1/10

(12)

allows us to rewrite Equation (2) as

600κ2eκB K 92α

−12 = 120eη (η − 2) − 3η5 − 10η4

+ 20

(

1

κeκB − 1

)

η3 + 120η

+ 240 (13)

where

η = κK2 Aα2. (14)

Next relate α1 to α2 by rewriting Equation (3) as

α1 = K3α2 (15)

where

K3 =(

ρ1

ρ2

)1/2 (ν1

ν2

)1/10

. (16)

Eliminate α1 by substituting Equation (15) in Equations (9)–

(10), leaving us with two equations, Equations (9) and (13),

for two unknown coefficients A and α2. This equation pair

can easily be solved numerically, e.g., Equation (9) can be

solved for A by simple iteration as it converges nicely, and

a simple regula falsi (secant) method can be used for Equa-

tion (13), but other numerical schemes will work, too, and

we used a globally convergent Newton’s method. With A and

α2 at hand, one easily calculates the mass stripped from the

cloud by integrating the cloud material flowing through the

boundary layer at the equator (Ranger and Nicholls, 1969):

dm

dt= 2π Rρ2

∫ δ2

0

u2dy2 = 2π Rρ2ν2ψ (η) (17)

where we have defined a mass-strip coefficient

ψ (η) = 1

2κ2η2 + 1

κ

× e−κB

(

1 − η − 1

2η2 − 1

6η3 − 1

24η4 + eη

)

. (18)

It should be noted that dm/dt is not proportional to R, ρ2, or

ν2 because η = η(R,U∞, ρ1, ρ2, ν1, ν2) from the numerical

solution above.

Using the specific physical quantities for our experi-

ment, we can now calculate the mass stripped as a function

Fig. 4 Cloud mass remaining as a function of time calculated usinga laminar model (Taylor, 1963; Ranger and Nicholls, 1969) (dashedline) and the turbulent model presented in this manuscript (solid line),compared to experimentally measured values of the cloud mass (twosquares). The turbulent model agrees with the measured values andalso predicts that the cloud is completely stripped by ∼90 ns, whichcompares well with the experimental observation of the cloud beingstripped by 80 ns–100 ns . In the laminar model (assuming unchangedcondition from 80 ns) the cloud is not stripped until ∼1µs

of time and see how the calculation compares to our ex-

perimental data. For the cloud radius R (t) we use mea-

sured values from the experiment images, and interpolate

to other times. For values of the free stream flow veloc-

ity U∞ (t), the density ρ1 (t), and the temperature T (t) ,

we use values from HYADES. The density ρ2 (t) is obtained

by applying the same compression as for ρ1 (t). All of

these values are in full agreement with values from CALE.

Additionally, the peak compressions are independently veri-

fied from the experiment at t = 5 ns where the left side of the

sphere is compressed to an ellipsoid shape with minor radius

∼30µm, corresponding to a compression of ∼4 (which is the

strong shock limit for a polytropic gas with adiabatic index

γ = 5/3).

With our given physical quantities, the coefficients

K1 ≈ 5, K2 ≈ 3, and K3 ≈ 14

at all times. From solving

Equations (9), (13) and (15) we calculate the coefficients

A ≈ 1/5, α1 ≈ 6, and α2 ≈ 27 at all times, and we find that

the compound quantity η varies between 4 η 8 (except

very briefly when the rarefaction changes the direction of

the flow) so that the mass-strip coefficient ψ is in the range

4 × 102 ψ 4 × 103. The mass of the cloud as a func-

tion of time is plotted in Fig. 4 and reaches m = 0 (fully

stripped) by t ≈ 90 ns . This agrees well with the experi-

ment where the cloud can no longer be observed by 80 ns–

100 ns . By comparison, the equivalent mass-strip coefficient

ψ =(

2π Rρlνl

)−1dm/dt in the laminar model is 4 × 102

for all times of interest in the experiment, which is too low to

Springer


achieve the cloud being completely stripped by t ≈ 80 ns; if

the mass-stripping was done by laminar flow and continued

under the same conditions past t = 80 ns (ignoring experi-

mental limitations) the laminar mass-stripping time would be

∼1µs. As a final note, to illustrate the non-linearity between

dm/dt and the various physical quantities one can arbitrarily

double, say, the value of the viscosity ν2 and see that this

leads to only a 12% increase in dm/dt .

5 Summary

We observe the rapid stripping of all mass from a simulated

interstellar cloud in a laser experiment. We present a model

that agrees very well with our experimental observations.

The model combines (1) the integral momentum equations

for a viscous boundary layer, (2) the equations for a potential

flow past a sphere, (3) Spalding’s law of the wall for turbu-

lent boundary layers, and (4) the skin friction coefficient for

a turbulent boundary layer on a flat plate. By comparison, a

laminar model overestimates the stripping time by an order

of magnitude. This suggests that mass-stripping in the exper-

iment must be of a turbulent nature, and with its even higher

Reynolds numbers, this must hold also in the astrophysical

case.

Acknowledgements We would like to thank C. F. McKee, Departmentsof physics and Astronomy, University of California, Berkeley for hissupport of this project. This work was performed under the auspicesof the U. S. Department of Energy by the University of California,Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

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UCRL-JC-112384 Gain Uniformity, Linearity, Saturation andDepletion in Gated Microchannel-Plate X-Ray Framing Cam-eras, Lawrence Livermore National Laboratory, Livermore (1994).Copies may be obtained from the National Technical InformationService, Springfield, VA 22161

Larsen, J.T., Lane, S.M.: J. Quant. Spectrosc. Radiat. Transfer 51, 179(1994)

Millikan, R.A.: Trans. Amer. Soc. Mech. Eng. 54, 29 (1932)Ranger, A.A., Nicholls, J.A.: AIAA J. 7, 285 (1969)Robey, H.F., Perry, T.S., Klein, R.I., Kane, J.O., Greenough, J.A.,

Boehly, T.R.: Phys. Rev. Lett. 89, 085001 (2002)Ryutov, D., Drake, R.P., Kane, J., Liang, E., Remington, B.A. Wood-

Vasey, W. M.: Astrophys. J. 518, 821 (1999)Spalding, D.B.: J. Appl. Mech. 28, 455 (1961)Taylor, G.I.: The shape and acceleration of a drop in a high speed

air stream. In: Batchelor, G.K. (ed.) The Scientific Papers of G.I.Taylor. University Press, Cambridge (1963)

White, F.M.: Viscous Fluid Flow. McGraw-Hill, New York (1974)Widnall, S.E., Bliss, D.B., Tsai, C.: J. Fluid Mech. 66, 35 (1974)

Springer


DOI 10.1007/s10509-006-9227-4


Hydrodynamics of Supernova Evolution in the Windsof Massive Stars

Vikram V. Dwarkadas


Abstract Core-Collapse supernovae arise from stars greater

than 8 M⊙. These stars lose a considerable amount of mass

during their lifetime, which accumulates around the star

forming wind-blown bubbles. Upon the death of the star in

a spectacular explosion, the resulting SN shock wave will

interact with this modified medium. We study the evolution

of the shock wave, and investigate the properties of this in-

teraction. We concentrate on the evolution of the SN shock

wave in the medium around a 35 solar mass star. We discuss

the hydrodynamics of the resulting interaction, the formation

and growth of instabilities, and deviations from sphericity.

Keywords Supernova remnants . Hydrodynamics .

Instabilities . Stellar winds . Massive stars . Wind-blown

bubbles . Shock waves

1 Introduction

Mass-loss from stars is a ubiquitous process. Massive stars

(>8M⊙) lose a considerable amount of mass before they ex-

plode. This material collects around the star, forming a cir-

cumstellar (CS) wind-blown bubble. At the end of its life, the

star will explode in a cataclysmic supernova (SN) explosion,

and the resulting shock wave will interact with this medium.

The further evolution of the resulting supernova remnant will

depend on the properties of this medium.

In this paper we discuss the evolution of the surrounding

medium around massive stars, and the subsequent interac-

tion of the SN shock wave with this medium following the

V. V. DwarkadasAstronomy and Astrophysics, University of Chicago,5640 S Ellis Ave. AAC 010c, Chicago, IL 60637e-mail: [email protected]

star’s death. The rich and complex dynamics of the various

interactions leads to the formation and growth of a variety

of hydrodynamic instabilities, which we will focus on in this

paper. It may be possible to simulate some of these hydro-

dynamic situations with available laboratory apparatus, and

we hope that this work will further stimulate laboratory ex-

periments of realistic astrophysical phenomena, particularly

those involving radiative shocks.

2 SN-Circumstellar interaction

It has been realized over the years that the medium around

a core-collapse SN is continually being sculpted during the

progenitor star’s lifetime, by the action of winds and out-

bursts. Chevalier and Liang (1989) discussed the interaction

between a SN shock wave and the surrounding wind-blown

bubble formed by the pre-SN star. However analytic argu-

ments can only be extended so far, and numerical simulations

are required to study the subsequent non-linear behavior. A

series of papers in the early 90’s (Tenorio-Tagle et al., 1990,

1991; Rozyczka et al., 1993) explored some aspects of this.

Since then our observational knowledge of this phenomena

has multiplied exponentially, thanks to the availability of

space based data in the optical, X-ray and infrared bands,

and the stream of data pouring in from observations of SN

1987A. The latter has become the poster-child for SN evo-

lution in wind-blown bubbles, having shaped and confirmed

many of our views.

The basic details of SN interaction with wind-blown bub-

bles were outlined in the papers listed above, and further

elaborated on by Dwarkadas (2005). The mass loss results in

the formation of a circumstellar (CS) wind blown cavity sur-

rounding the star, bordered by a thin, dense, cold shell. The

typical structure of this wind-blown bubble for constant wind

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Fig. 1 Density and pressureprofiles for a circumstellarwind-blown bubble

properties (Weaver et al., 1977) is shown in Fig. 1, and con-

sists of an outwards expanding shock wave (Ro), and a wind

termination shock (Rt ) that expands inwards in a Lagrangian

sense, separated by a contact discontinuity (Rcd ). In general

most of the volume between Rt and Rcd is occupied by a

low-density, high pressure shocked wind bubble, surrounded

by the extremely dense shell. Most of the mass is contained

in the dense shell. When the SN shock wave interacts with

this bubble, it quickly finds itself in a medium with density

much lower than that of the ISM. Consequently, the emission

from the remnant, which arises mainly from CS interaction

(Chevalier and Fransson, 1994), will be considerably reduced

compared to evolution within the ISM.

It comes as no surprise then that the subsequent evolution

depends primarily on a single parameter , the ratio of the

mass of the dense shell to that of the ejected material. For

very small values ≪ 1 the effect of the shell is negligible

as expected. For values of 1 interaction with the shell

results in considerable deceleration of the SN shock wave.

The X-ray luminosity can increase by orders of magnitude

upon shock-shell collision. A transmitted shock wave enters

the shell, while a reflected shock wave moves back into the

ejecta. If X-ray images were taken just after the interaction,

they would show the presence of a double-shelled structure as

the reflected shock begins to move inwards. In about 10 dou-

bling times of the radius the SN begins to ‘forget’ about the

existence of the shell. The remnant density profile changes

to reflect this, and consequently the X-ray emission from the

remnant, which depends on the density structure, will also

change. The reflected shock will move to the center and pre-

sumably be reflected back, while the transmitted shock will

slowly exit the shell and eventually separate from it.

As the ratio increases, more of the kinetic energy from

the remnant is converted to thermal energy of the shell. The

transmitted shock is considerably slowed down, and in ex-

treme cases ( ≫ 1) may even be trapped in the shell. The

high pressure behind the reflected shock will impart a large

velocity to the shock, and therefore thermalization of the

ejecta is achieved in a much shorter time as compared to

thermalization by the SN reverse shock. Upon reaching the

center the reflected shock bounces back, sending a weaker

shock wave that will collide with the shell. In time a series

of shock waves and rarefaction waves are seen traversing the

ejecta. Each time a shock wave collides with the dense shell

a corresponding (but successively weaker) rise in the X-ray

emission from the remnant is seen.

We have outlined the basics of SN interaction with

CS wind-blown shells. One-dimensional models are fully

described in Dwarkadas (2005). We wish to present herein

results from multi-dimensional models.

3 CS medium around a 35 M⊙ star

The above description considered an idealized wind-blown

bubble formed by the interaction of a fast wind with the sur-

rounding medium, where the properties of both are constant

in time. In reality, as a massive star evolves, the wind proper-

ties change with time. In particular after a star leaves the main

sequence, its mass-loss properties change considerably. This

will give rise to a much more complicated bubble structure

than is shown in Fig. 1.

In order to explore more realistically the medium sur-

rounding a core-collapse SN, we have taken stellar evolution

calculations from several groups, and investigated the evo-

lution of the surrounding medium as the star evolves. In this

paper we discuss the evolution of the medium around a 35

M⊙ star, from an evolutionary model provided to us by Nor-

bert Langer. The star begins its life on the main sequence as

an O star, then expands to become a Red Supergiant (RSG),

and finally ends its life as a Wolf-Rayet (WR) star. The mass-

loss rate and wind velocity over the evolution are shown in

Fig. 2. During the main-sequence stage, the mass-loss rate is

a few times 10−7 M⊙/yr, and the wind velocity is about 3000–

4000 km/s. Once the star swells to become a red supergiant,

the wind velocity reduces by more than 2 orders of magni-

tude, and the mass-loss rate increases to almost 10−4 M⊙/yr.

The WR wind shows a slight drop in mass-loss rates by a

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Fig. 2 Evolution of the mass-loss rate (left) and wind velocity (right) for the 35 M⊙ star during its lifetime

factor of a few from the RSG phase, but a steep increase in

wind velocity by two orders of magnitude. We use these val-

ues as input boundary values at each timestep to our code,

which then computes the structure of the nebula over time.

Unlike a previous computation (Garcia-Segura et al., 1996)

our computation is fully two-dimensional right from the start.

We use a grid consisting of 600 zones in both the radial and

azimuthal directions. The code used is the VH-1 numerical

hydrodynamics code, a multi-dimensional code that solves

the equations of fluid dynamics on a Lagrangian grid, and

then remaps them onto an Eulerian grid. Radiative cooling

is included via a cooling function, but we have not included

the effects of ionization. A grid that expands outwards with

the outgoing shock wave is used, although no new zones are

added, i.e. the grid is not adaptive. Initially the wind occupies

about 20 zones on the grid, depending on the grid resolution.

The mass-loss rate and velocity of the wind are used to com-

pute the density and velocity of the inflow, which are used

as the input boundary conditions at each timestep. The ini-

tial setup is uniform and no perturbations are applied to the

system. Perturbations that arise are due to effects such as

non-spherical shocks on the spherical grid.

The evolution of the medium is shown in Fig. 3. In the

main sequence stage (3a,b), although the wind properties

are changing continuously, the nebular structure is not very

different from that expected from the idealized, two-wind

case of a fast wind interacting with a slower wind, both

of which have constant wind properties. A thin shell of

swept-up material is formed, and the volume of the nebula is

mostly occupied by a hot, low density bubble. The shell is on

the whole mostly stable, although it shows some wrinkles.

These arise mainly from shearing, due to flow of gas along

the contact discontinuity. However these instabilities are not

highly pronounced, and do not appear to grow to any signifi-

cant extent. In lower-resolution simulations presented earlier

(Dwarkadas, 2001, 2004) we had suggested that the shell

is unstable to some type of thin-shell instability (Vishniac

1983). We have seen the same instability in simulations of

the medium around a 40 M⊙ star. The higher resolution cal-

culations presented here do not show a strong presence of

such an instability however. This is a topic still under inves-

tigation.

The interior of the nebula shows significant fluctuations in

density and pressure, and vortices are visible in the velocity

flow. Since the mass-loss rate and wind velocity are changing

at every timestep, the position of the reverse shock is not fixed

on the grid, but moves slightly every timestep with respect to

the outer shock. The changing position of the reverse shock

from one timestep to another results in the deposition of

vorticity into the shocked wind, which is then carried out

with the shocked flow. This results in an interior that is quite

inhomogeneous, with significant density fluctuations.

When the star leaves the main sequence and becomes a

RSG star, its radius increases considerably, the wind velocity

(Vwind ) drops by two orders of magnitude, while the mass-

loss rate (M) increases appropriately. Thus the wind density,

proportional to M/Vwind , goes up by several orders of magni-

tude. A new pressure equilibrium is established, and a shock

front is formed in between the RSG wind and main sequence

bubble as the RSG wind is decelerated by the bubble pres-

sure. The RSG wind piles up against this shock, forming a

thin dense shell of RSG material. No hot, low-density cav-

ity is present. The shell decelerates as it expands outwards,

satisfying the classic case of Rayleigh-Taylor (R-T) insta-

bility, and Rayleigh-Taylor fingers are seen expanding out-

wards from the high-density shell into the low-density ambi-

ent medium (3c). Some of the filaments show the presence of

sub-filaments growing from the main one, and the expanded

heads of many of the filaments are a sign of Kelvin-Helmholtz

instabilities resulting from the shear flow in between the

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Fig. 3 Density evolution of the medium a round a 35 M⊙ star with time.The wind properties at each stage are given in the top right hand cornerof each panel. The velocity is in km/s, the mass loss rate in M⊙/yr, and

the time in years. The color bar shows the logarithm of the gas densityin units of g cc−1

filaments and the surrounding medium. Unfortunately, since

we need to resolve the entire bubble, the resolution is not

large enough to study the growth of the fingers in detail.

The star leaves the RSG phase and loses its outer hydrogen

envelope, becoming a WR star in the process. The compact

star now gives off a very fast wind, not unlike in the O star

stage, but with a mass-loss rate that is much higher than in the

main sequence, and just a few times lower than the RSG stage.

The supersonic WR wind creates a wind-blown bubble in the

RSG wind. The dense W-R shell is accelerated by the high

pressure, low density interior as it expands outwards, leading

to the triggering of the Rayleigh-Taylor instability (3d). In

this case R-T fingers are seen expanding inwards from the

dense shell into the low-density cavity. The large momen-

tum of the WR shell causes the RSG shell to fragment, and

carries the material outwards (3e), speeding up as it enters

the low-density bubble. Due to the fact that the WR wind

is carrying fragments of the RSG material, and that it trav-

els through a medium with considerable fluctuations in den-

sity and pressure, its expansion is not completely spherical.

The collision of this slightly aspherical wind with the main

sequence shell gives rise to a reflected shock that moves back

into the bubble. The asphericity is accentuated in the reflected

shock, which moves inwards, before finally coming to rest

in a wind-termination shock where the ram pressure of the

freely expanding wind is equal to the thermal pressure within

the bubble. The wind-termination shock when it forms is con-

sequently also not spherical but slightly elongated along the

equator (3f). As we shall show later this has important con-

sequences for the expansion of the SN shock wave.

4 SN-CSM interaction in the case of the 35 M⊙ star

At the end of the WR stage, the stellar mass remaining is

9.1 M⊙. We assume that the star then explodes in a SN explo-

sion. A remnant of 1.4 M⊙ is left behind, and the remaining

mass is ejected in the explosion. We use the prescription of

Chevalier and Fransson (1994) to describe the ejecta struc-

ture as a power-law with density, with power-law index of

7. We compute the evolution of the SN described by this

density profile expanding into the unshocked wind, and then

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Fig. 4 Pressure evolution of the SN shock wave within the WR bubble

map it onto the grid containing the bubble simulation. This

calculation was also carried out using 600 × 600 zones.

The evolution of the SN shock wave is shown in Fig. 4.

It starts out as expected, with the formation of a forward

and reverse shock structure (Fig. 4a). The interaction of the

spherical forward shock with the aspherical wind termina-

tion shock, susceptible to the Richtmeyer-Meshkov instabil-

ity, reveals quite interesting dynamics. Since the SN shock

is spherical while the wind termination shock is slightly

more elongated towards the equator, the interaction first takes

place close to the symmetry axis. A transmitted shock moves

out into the shocked bubble, while a reflected shock moves

back. Different parts of the SN shock collide with the wind-

termination shock at different times, leading to transmit-

ted shocks with a small but non-negligible velocity spread

(Fig. 4b). The composite transmitted shock then expands in

the inhomogeneous medium, interacting with several large

density fluctuations on the way. The net result is a very cor-

rugated shock wave that expands outwards towards the main

sequence shell (Fig. 4c). The wrinkles are similarly preva-

lent in the reverse shock also. The wrinkled shock collides

in a piecemeal fashion with the main sequence shell, with

some parts of the shock colliding before others (Fig. 4d,e).

Each collision with the shell will give rise to an increase in

the optical and X-ray emission at that point. Therefore some

parts of the shell will brighten before others. It is interest-

ing to note that a similar phenomenon has been observed in

SN 1987A, where bright spots appear successively around

different parts of the equatorial ring (Sugerman et al., 2002)

As each portion of the forward shock wave collides with

the shell, a reflected and transmitted shock pair is formed.

The shell is dense enough that the transmitted shock does not

emerge from the shell for a long period. The reflected shock

meanwhile travels back towards the origin. However, as seen

in Fig. 4, the velocity of each piece differs considerably from

the next, both in magnitude as well as direction. The shape

of the reflected shock therefore deviates significantly from

spherical, and some parts of the reflected shock reach the

symmetry axis before the rest has traveled far into the interior

(Fig. 4f). This gives the remnant a very asymmetric shape,

and results in some portions of the ejecta being much hotter

than others.

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5 Conclusions

The surroundings of massive stars are shaped by the mass-

loss from the progenitor star. This can lead to a complicated

density structure for the surrounding medium, the formation

and growth of various hydrodynamical instabilities, deposi-

tion of vorticity and onset of turbulence. When the star ex-

plodes as a SN remnant, the SN shock wave will interact with

this ambient medium. The inhomogeneous structure of the

ambient medium can cause distortions in the SN shock wave

as it expands outwards, which are magnified by the turbulence

in the wind-blown structure. In this paper we have shown that

the end result can be a wrinkled shock wave whose impact

with the surrounding shell occurs in a piecemeal fashion. As

each part of the shock wave hits the shell, it will brighten up

in the optical and X-ray regime, a phenomenon that is ob-

servable in SN 1987A. We do caution that this comparison

is illustrative only. Our numerical models are not meant to

simulate SN 1987A, whose progenitor star was a much lower

mass B3Ia star.

Our simulations show that the complicated structure of

the medium may result in deviations from spherical symme-

try for the SN shock wave. Even though the expansion starts

out as spherical, the final shape of the remnant may devi-

ate considerably from sphericity. Most of the emission from

the remnant arises from the high pressure region in between

the forward and reverse shocks. The distorted shape of this

emitting region is clearly visible in Fig. 4, and this will be

reflected in observations of the remnant.

Herein we have summarized the features of multi-

dimensional models of SN evolution in the environments

shaped by massive stars. Further details are available from

Dwarkadas (2006).

Acknowledgements Vikram Dwarkadas is supported by award # AST-0319261 from the National Science Foundation, and by NASA throughgrant # HST-AR-10649 from STScI. We thank the anonymous refereefor suggestions that helped to improve this paper. We acknowledge use-ful discussions with Roger Chevalier which were particularly helpful inidentifying the various instabilities that were observed. We are gratefulfor comments from John Blondin and Thierry Foglizzo. This researchwas supported in part by the National Science Foundation under GrantNo. PHY99-07949 to the KITP.

References

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mic explosions in three dimensions : asymmetries in supernovaeand gamma-ray bursts. Cambridge: CUP, pp. 74 (2004)

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RAS 251, 318 (1991)Tenorio-Tagle, G., Bodenheimer, P., Franco, J., Rozyczka, M.: MN-

RAS 244, 563 (1990)Vishniac, E.T.: ApJ 428, 186 (1994)Vishniac, E.T.: ApJ 274, 152 (1983)Weaver, R., McCray, R., Castor, J., Shapiro, P., Moore, R.: ApJ 218,

377 (1977)

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DOI 10.1007/s10509-006-9203-z


Theoretical and Experimental Studies of Radiative Shocks

C. Michaut · T. Vinci · L. Boireau · M. Koenig ·

S. Bouquet · A. Benuzzi-Mounaix · N. Osaki ·

G. Herpe · E. Falize · B. Loupias · S. Atzeni


Abstract This paper deals with the radiative shock from

both theoretical and numerical points of view. It is based

on the whole experimental results obtained at Laboratoire

d’Utilisation des Lasers Intenses (LULI, Ecole Polytech-

nique). Radiative shocks are high-Mach number shocks with

a strong coupling between radiation and hydrodynamics

which leads to a structure governed by a radiative precursor.

These shocks are involved in various astrophysical systems:

stellar accretion shocks, pulsating stars, interaction between

supernovae and the interstellar medium.

In laboratory, these radiative shocks are generated using

high power lasers. New diagnostics have been implemented

to study the geometrical shape of the shock and the front

shock density. Data were obtained varying initial conditions

for different laser intensities and temperature. The modeling

of these phenomena is mainly performed through numerical

C. Michaut ()· G. HerpeLUTH UMR 8102, Observatoire de Paris, 92195 Meudon cedex,Francee-mail: [email protected]

L. BoireauLUTH UMR 8102, Observatoire de Paris, 92195 Meudon cedex,France, CEA/DIF, Departement de Physique Theorique etAppliquee, 91128 Bruyeres-le-Chatel, France

T. Vinci · M. Koenig · A. Benuzzi-Mounaix · N. Osaki ·B. LoupiasLULI, Ecole Polytechnique, 91128 Palaiseau cedex, France

S. Bouquet · E. FalizeCEA/DIF, Departement de Physique Theorique et Appliquee,91128 Bruyeres-le-Chatel, France

S. AtzeniDipartemento di energetica, Universita di Roma La Sapienza andINFM, Italy

simulations (1D and 2D) and analytical studies. We exhibit

results obtained from several radiative hydrodynamics codes.

As a result, it is possible to discuss about the influence of the

geometry and physical parameters introduced in the 1D and

2D models.

Keywords Radiative shock . High-power laser .

Hydrodynamics . Radiation transfer . Plasma

1. Introduction

In astrophysics, radiative shocks are very common phenom-

ena. A radiative shock is a shock sufficiently strong that radia-

tion transport alters the hydrodynamic structure. The ionized

gas emits a radiation flux, and a part of this radiation goes

ahead the shock and heats up the gas before the compression

wave arrives. This heating is called the radiative precursor. In

this case, the temperature is high during the compression like

for a classical shock. But the temperature profile presents a

slow decreasing curve ahead the shock. As an example, in the

supernova remnant Puppis-A, one can shows details of the

strong shock wave disrupting an interstellar cloud (Hwang

et al., 2005), or in the Rotten Egg Nebula (also known as

the Calabash Nebula and OH231.8 + 4.2,), the central re-

gions are contracted into a white dwarf. The gas strikes the

surrounding medium, with a velocity around 300 km/s. A

supersonic gas front forms ionizing hydrogen and nitrogen

(Bujarrabal et al., 2002).

In order to better understand these phenomena, astro-

physicists need radiative hydrodynamics models which have

to be validated by confrontation with experimental results.

Since the hydrodynamics is structured by radiation effects,

the relevant models are sensitive to the treatment of radia-

tion transport and its coupling with hydrodynamics. Code

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benchmarking is one of the motivations for the development

of high-energy density studies. High-energy density labora-

tory astrophysics (HEDLA) experiments are mostly driven

on large-scale lasers. But the connection of experimental re-

sults to astrophysical situation is not straightforward.

In this work, our step is to check theoretical assumptions or

computational models through radiative shock experiments

performed with LULI 2000 laser.

2. Theoretical considerations on radiative shocks

In laboratory, the shock velocity is determined by the laser

energy and the pulse duration as a technology limitation. At

LULI, the laser intensity is typically about 1014 W/cm2, and

the pulse duration is around 1 ns. Koenig et al., have shown

(Koenig et al., 1999) that, under these conditions, the max-

imum achievable shock velocity at the interface pusher/gas

cannot exceed 60 km/s. In order to produce radiative regime,

Bouquet et al., have shown (Bouquet et al., 2000) from a sim-

ple analytical model in an ideal gas, that the shock velocity

Us should be larger than the following threshold:

Us ≥ Csteρ1/3

A2/3. (1)

This threshold value depends on the gas density ρ and the

atomic weight A. Since Us is determined from laser spec-

ifications and target design, this ratio should be as low as

possible. Consequently, we need a low gas density ρ and

a high atomic weight A. For these reasons, the heavy ele-

ment xenon has been chosen for experiments. Equation (1)

has been derived assuming that the magnitude of the radiative

pressure and the thermal pressure is equivalent. Other authors

(Keiter et al., 2002; Reighard et al., 2006) consider that the

radiative regime is reached provided a radiation flux exists.

They obtain a condition similar to Equation 1 but with less

constraining exponents. In more details, one can discuss the

kind of regime which is achieved according to the importance

of radiative flux and pressure.

Once the appropriate gas is chosen, we can define various

types of shocks from a simple analytical model considering

Table 1 Threshold value Us (km/s) for the radiative shock calculatedat three densities ρ0 (g/cm3) and required to reach two regimes respec-tively where the radiative flux equal the thermal flux and the radiativepressure equals the thermal pressure. The corresponding temperatureT (eV) is also evaluated for these two cases

Fr = Fth Pr = Pth

ρ0 Us T Us T

10−4 20 7.5 165 110

10−3 47 23 271 256

10−2 97 74 420 595

Fig. 1 Representation in the xenon (ρ, T ) diagram of the three shockregimes. The working LULI domain is also drawn in grey oval

the diffusion approximation (Mihalas et al., 1999; Zel’dovich

et al., 2001; Drake, 2005). In Table 1, the shock velocity Us is

estimated for two cases. The first one arises when the radia-

tive flux Fr = σT 4 (σ Stefan-Boltzman constant, T tempera-

ture) is equal to the thermal flux Fth = ρ0 Cv T Us (ρ0 initial

density, Cv heat capacity at constant volume). The second

case occurs when the radiative pressure Pr = (4σ/3c) × T 4

(c light velocity) is equal to the thermal pressure Pth =ρ0 k T/m (k Boltzman constant, m average particle mass).

As expected, on the one hand, the velocity Us increases

for increasing densities and, on the other hand, the condi-

tion Pr = Pth requires a higher value for Us than for the

case Fr = Fth . Obviously, the same behavior is observed for

the shock temperature. The results are summarized in Fig. 1

where we plotted the two frontiers Fr = Fth and Pr = Pth

giving rise to three distinct regimes. Below the lower line

we have the pure hydrodynamical regime and above the up-

per line one find the fully radiative regime dominated by the

radiative pressure. In contrast to our approach developed ini-

tially (Bouquet et al., 2000) where the radiative shock was

expected only in the upper zone, we proved experimentally

that in between these two limits, the shock is also radiative

(its structure is governed by a radiative precursor (Bouquet

et al., 2004)). In the following, this intermediate region will

be called therefore “hybrid radiative domain”. Our experi-

ments at LULI are located in this area (see the grey patch

in Fig. 1). If we want to perform experiments in the fully

radiative regime, we need higher energy/power lasers.

3. Experimental set-up

In the laboratory the radiative shock is generated using the

LULI 2000 laser which interacts with the target. This target

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Fig. 2 Experimental set-up used to generate radiative shock driven byhigh-power laser and diagnostics arrangement

is composed by a three-layers pusher and a gas cell. The cell

filled with xenon is 7 mm long. When the laser energy is

deposited on the first layer, it produces an ablation of the CH

plastic. The laser energy is converted into mechanical energy,

in the direction of the laser interaction by rocket effect during

this ablation phase. As X-rays are emitted, a second layer in

Titanium screens these X-rays so they do not reach the xenon

cell. Titanium is heavier than plastic, therefore a third (CH)

plastic layer is added to re-accelerate the shock wave. Then

the shock emerges in the xenon cell. From then on the xenon

compression is strong, the gas is ionized and a radiation flux

goes faster than the shock. Therefore a matter flux is produced

combined with a faster radiation flux towards the gas cell end.

Figure 2 shows the experimental set-up which has been

previously described (Bouquet et al., 2004; Fleury et al.,

2002; Vinci et al., 2006) in details. The cell is placed in the

center of the vacuum chamber. Two laser beams, at 2ω, de-

liver 1 kJ maximum on the target with a square pulse ranging

from 1 to 5 ns.

Several diagnostics are implemented and they are shown

in Fig. 2 each one with a small picture of the type of data

recorded. On the transverse side of the target, a VISAR mea-

sures the precursor electron density and records both the

shock front and the precursor velocities. Two Gated Optical

Imager (GOI) take a picture of the shock front and precursor

by imaging the probe laser absorption, at two different times

per shot. We measure the shock curvature and follow its prop-

agation. As this diagnostic is the more recently implemented

we exhibit in this issue raw experimental data.

Figure 3 represents a series of GOI snapshots at 6 different

times from 5 to 10 ns after the high-power laser interaction.

For each image the exposure time is 100 ps, the shock takes

place in xenon with an initial pressure of 0.1 bar.

The sequence of images in Fig. 3 is obtained by performing

several experiments in the same conditions, and changing the

time delay for each picture. Thus indirectly we point out that

Table 2 Diameter and curvature depth of the radiative shock recordedby GOI from 5 to 10 ns

Time (ns) 5 6 7 8 9 10

Diam. (µm) 730 783 835 887 1043 1148

Depth (µm) 104 157 209 261 365 470

Fig. 3 Series of GOI snapshots at 6 different times imaging a transverseview of the shock front and its precursor

our way to generate radiative shocks and to measure its phys-

ical parameters are reproducible. This snapshot series allow

to follow the shock propagation and its curvature. One can

clearly see in Fig. 3 that the shape of the shock becomes more

and more curved with the time while its lateral extension in-

creases. At the beginning the shock front is around 400 µm

in diameter, equalling that of the focal spot laser. Table 2

summarizes the shock front diameter and its curvature depth

(penetration depth in the unshocked xenon) corresponding

to each image in Fig. 3. From values in Table 2, it is quite

obvious that the shock front is expanding and curving. This

phenomena seems relatively linear from 5 to 8 ns and it un-

dergoes an acceleration from 8 ns. In Section 4.4, we discuss

the radial shock expansion with numerical support.

On the rear side, two VISAR measure the third layer-

Xenon interface velocity. Therefore this CH rear side is used

as pusher and as witness to determine the shock velocity and

the temperature at the beginning. In addition since we have

also an emissivity diagnostic on the rear side, we know the

shock front temperature during all the experiment.

The typical results obtained have been widely discussed

elsewhere (Bouquet et al., 2004; Vinci et al., 2002, 2005a;

Koenig et al., 2005, 2006). We produce a shock with a ve-

locity typically in the range [50–100] km/s. The shock front

temperature is measured between 10 to 20 eV. The precursor

electron density is typically in the range [1017–1020] cm−3.

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And the precursor velocity is always around twice the shock

velocity. Of course the precursor velocity depends on the cho-

sen electron density. Even if the broad lines of experimental

data can be recovered from rough but simple analytical mod-

els, numerical simulations (1D and 2D) are quite necessary

to understand the fine structure and evolution of the radiative

shock.

4. Numerical simulations and experimental results

Numerical simulations require specific atomic data. But ac-

tually, xenon opacity calculations are very difficult especially

at low temperature. However we have the opportunity to use

three opacity tables from LULI (Eidmann, 1994), LUTH

(Michaut et al., 2004) and CEA (in FCI code (Dautray and

Watteau, 1993)).

4.1. A non-stationary code for astrophysics: HADES

For astrophysical purposes we developed a new numerical

code, namely, HADES. It deals with radiative hydrodynam-

ics in gas, without laser consideration. However, it can also

describe in details the physics of radiative shocks generated

in the gas-cell target during experiments. Therefore, it can

be also validated by comparing experimental data with nu-

merical results in the same way it is done with laser/matter

interaction codes. In addition, improved numerical schemes

and radiative transfer allow HADES to especially model var-

ious specific astrophysical situations.

HADES is based on the public package CLAWPACK

(Leveque, 2002) coupled with a moment method for radi-

ation transport (Dubroca and Feugeas, 1999; Gonzales and

Audit, 2005), for a 1D-geometry. Figure 4 shows the way a

nonstationary shock wave converges to a stationary structure

(provided by the code LASTAR (Michaut et al., 2004a,b))

as the time elapses. Here Hydrogen is chosen for astrophysi-

cal application. In Fig. 4 the precursor length increases with

time until reaching its final stable length is about 1.2 mm.

Based on this simulation, a laser radiative shock wave in H

gas might become steady in 1D at about 100 ns. This sug-

gests that a more direct connection between experiments and

astrophysics can be achieved in the near future with higher

energy/power lasers.

At the foot of the radiative precursor (z = 1.2 mm), the

discrepancy between HADES (t = 100 ns) and LASTAR

profiles comes from the fact that the stationary shock is cal-

culated in the diffusion approximation with a Local Ther-

modynamics Equilibrium (LTE). P. Drake suggested (Drake,

2005) we switch this approximation by a Non-LTE diffusion

model to recover the right shape of the precursor. As a matter

of fact, the photon mean free path is larger than the gradient

temperature length in the precursor foot. This property is in

contradiction with the LTE assumption.

Fig. 4 Non-stationary 1D calculation with the code HADES for a radia-tive shock in hydrogen (ρ0 = 1.2 × 10−3 g/cm3, T0 = 1 eV). The shockfront velocity is 150 km/s. All shock front positions are aligned to easilycompare precursor lengths. The code LASTAR gives the reference forthe stationary shock profile (grey curve)

At the moment, the coupling between CLAWPACK and

a radiation transport is in progress in cylindrical geometry to

obtain the first steps for the further HADES 2D.

4.2. Precursor length

1D simulations containing laser interaction are used to design

targets, especially the thickness of each pusher layer. Figure 5

compares the precursor length between a 1D simulation ob-

tained with MULTI (Ramis et al., 1998) and a 2D simulation

performed with DUED (Atzeni, 1987) with the same initial

conditions.The shock starts in xenon at 2 ns. One clearly see

in Fig. 5 that the 1D-precursor length is growing during at

least 10 ns, until around 350µm. In contrast, the 2D length

becomes constant from 6 ns and its length saturates about

190 µm. Thus the 1D or 2D behaviors of the precursor length

evolution are completely different. The same difference be-

tween 1D and 2D simulations was yet encountered in many

other confrontations between experimental data and simula-

tions (Koenig et al., 2005; Vinci et al., 2005a; Michaut et al.,

Fig. 5 Evolution of the precursor length for a 1D simulation (MULTI)and a 2D simulation (DUED) under same conditions

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Fig. 6 Experimental data of the evolution of the shock front temper-ature compared with a 1D simulation (MULTI) and a 2D simulation(DUED) under same conditions. The bump in the experimental data fort < 0 is due to emission from the shocked CH

2006), or theoretical approach (Leygnac et al., 2006), which

allows to affirm that 1D simulation always overestimates the

precursor. This fact is due to the intrinsic 1D-geometry which

does not take into account the lateral radiation losses.

In the same way, the evolution of the shock front temper-

ature can be studied with 1D and 2D simulations.

4.3. Temperature

The shock front temperature is measured by an emissivity

diagnostic which records the self light emitted by the shock

front during all the experiment (Vinci et al., 2006). Same

previous 1D and 2D simulations have been compared with

experimental results (Vinci et al., 2005b) and same previous

conclusions were drawn. Figure 6 shows experimental tem-

perature recorded during 7 ns in xenon and two numerical

profiles.

The experimental shock front temperature suddenly in-

creases to 16 eV during less than 2 ns and adopts a slow

decreasing to reach 13.5 eV at 7 ns. The 2D-DUED sim-

ulation gives a better agreement with experimental result,

excepted during the first ns. In opposition, the temperature

keeps growing up in 1D-MULTI simulation.

Another point which can be examined with 2D simulation

is the time-dependent radial expansion of the shock and its

precursor.

4.4. Radial expansion

In addition using DUED, one can reproduce the emissivity

diagnostic and like in experiments, the shock undergoes a

radial expansion with the time. Therefore in order to under-

stand the radial expansion of the shock, we perform a simula-

tion experiment with DUED. Figure 7 shows two simulations

reproducing the emissivity diagnostic: on the left side, the

Fig. 7 2D simulations with DUED reproducing the emissivity diagnos-tic comparing a full calculation (left side) and a pure hydrodynamicscalculation (without radiation, right side)

Fig. 8 2D simulation performed with FCI showing the shock frontcurvature and its precursor in xenon at 5 ns. Local velocities along theshock front and in the shock frame are indicated with black arrows

calculation is complete with hydrodynamics and radiation,

but on the right side radiation is turned off.

In Fig. 7 the laser spot is 400µm in diameter and the shock

is almost twice at 8 ns. One can notice that the lateral expan-

sion is almost the same in these two cases until 6 ns. Until at

least this time the radial heating is due only to hydrodynamic

effects. However, the radial temperature gradient is modified

because the shock, in the left case, penetrates material previ-

ously heated by the radiation flux. This very likely explains

the larger extent of the heating after 6 ns in the case with the

radiation flux.

Figure 8 is still a 2D simulation performed with the FCI

code, the radiative shock is in xenon at 5 ns. On the propaga-

tion axis, the shock velocity is 60 km/s. In black arrows, the

local velocity vectors are drawn, in the shock frame. There-

fore we have subtracted 60 km/s in the propagation direction

(z-axis).

The result is that along the shock front radius there are

large radial velocities. It explains why the shock is expanding

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radially. From different simulations in which we turned off

radiation or electronic conduction, we understand that this

expansion is due to the non uniformity of the laser spot.

5. Conclusions

In conclusion, we performed radiative shock experiments

with long pulse laser. From all analyzed experiments, one can

conclude that our experimental results are reproducible. Spe-

cial targets are designed according analytical model and 1D

simulation. These 1D-codes, taking into account laser/matter

interaction, are essential to prepare and to design experi-

ments. Because of their running times are compatible with

this type of work consisting to optimization instead of 2D-

codes. But 1D simulations always overestimate precursor

lentghs.

In our experiments, we measured the shock front and pre-

cursor velocities, the electron density in the precursor and

the shock front temperature. In addition, the time-dependant

shock curvature and radial expansion were recorded. Thus

2D behavior of the radiative shock was clearly identified.

With simulations we obtain good agreement for all pa-

rameters excepted the precursor length which depends on

the dimensionality of the code. We have shown that the 1D

evolution of both the precursor length and the temperature

is continuously increasing with time. And yet, same param-

eters reach almost their asymptotic value after few ns in 2D-

simulations like in experiments. The shock front and its ra-

diative precursor expand in the lateral direction as the time

elapses. This expansion is measured using GOI and is recov-

ered by 2D simulations. We have shown this effect is mainly

due to the non uniformity of the initial energy deposition

i.e. the laser spot is not spatially homogenous. The precursor

length remains difficult to predict, it probably depends on

opacities.

Acknowledgements Authors would like acknowledge Ravasio, A.,Rabec le Gloahec, M., Barroso, P., Bauduin, D., for their contributionto the experiments and for the target manufacturing.

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Springer


DOI 10.1007/s10509-006-9254-1


Electrostatic Potentials in Supernova Remnant Shocks

Matthew G. Baring · Errol J. Summerlin


Abstract Recent advances in the understanding of the prop-

erties of supernova remnant shocks have been precipitated by

the Chandra and XMM X-ray Observatories, and the HESS

Atmospheric Cerenkov Telescope in the TeV band. A critical

problem for this field is the understanding of the relative de-

gree of dissipative heating/energization of electrons and ions

in the shock layer. This impacts the interpretation of X-ray

observations, and moreover influences the efficiency of in-

jection into the acceleration process, which in turn feeds back

into the thermal shock layer energetics and dynamics. This

paper outlines the first stages of our exploration of the role of

charge separation potentials in non-relativistic electron-ion

shocks where the inertial gyro-scales are widely disparate,

using results from a Monte Carlo simulation. Charge density

spatial profiles were obtained in the linear regime, sampling

the inertial scales for both ions and electrons, for different

magnetic field obliquities. These were readily integrated to

acquire electric field profiles in the absence of self-consistent,

spatial readjustments between the electrons and the ions. It

was found that while diffusion plays little role in modulating

the linear field structure in highly oblique and perpendicular

shocks, in quasi-parallel shocks, where charge separations in-

duced by gyrations are small, and shock-layer electric fields

are predominantly generated on diffusive scales.

Keywords Shock acceleration . Cosmic rays . Supernova

remnants . Electrostatics

M. G. Baring () · E. J. SummerlinDepartment of Physics and Astronomy MS-108, Rice University,P.O. Box 1892, Houston, TX 77251, U.S.A.e-mail: [email protected]

1 Introduction

The understanding of the character of shells and interiors

of supernova remnants (SNRs) has been advanced consider-

ably by groundbreaking observations by the Chandra X-ray

Observatory. These have been enabled by its spectral res-

olution coupled with its impressive angular resolution. Of

particular interest to the shock acceleration and cosmic ray

physics communities is the observation of extremely nar-

row non-thermal X-ray spatial profiles in selected remnants

(see Long et al., 2003 for SN1006; Vink and Laming, 2003 for

Cas A; and Ellison and Cassam-Chenaı, 2005, for theoretical

modeling), which define strong brightness contrasts between

the shell, and the outer, upstream zones. If the synchrotron

mechanism is responsible for this non-thermal emission, the

flux ratios from shock to upstream indicate strong magnetic

field enhancement near the shock. These ratios considerably

exceed values expected for hydrodynamic compression at the

shocked shell, so proposals of magnetic field amplification

(e.g. Lucek and Bell, 2000) in the upstream shock precursor

have emerged.

Another striking determination by Chandra concerns elec-

tron heating by ions in the shock layer. Dynamical inferences

of proton temperatures in remnant shocks can be made us-

ing proper motion studies of changes in a remnant’s angular

size, or more direct spectroscopic methods (e.g. Ghavamian

et al., 2003). In the case of remnant 1E 0101.2-7129, Hughes

et al. (2000) used a combination of ROSAT and Chandra

data spanning a decade to deduce an expansion speed. Elec-

tron temperatures Te are determined using ion line diag-

nostics (assuming the equilibration Te = Tp), via both the

widths and relative strengths for different ionized species.

From these two ingredients, Hughes et al. (2000) observed

that deduced proton temperatures were considerably cooler,

i.e. 3kTp/2 ≪ m p(3u1x/4)2/2, than would correspond to

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standard heating for a strong hydrodynamic shock with an

upstream flow speed of u1x . The same inference was made

by Decourchelle et al. (2000) for Kepler’s remnant, and by

Hwang et al. (2002) for Tycho’s SNR. This property is nat-

urally expected in the nonlinear shock acceleration scenario

that is widely used in describing cosmic ray and relativistic

electron generation in SNRs: the highest energy particles tap

significant fractions of the total available energy, leading to

a reduction in the gas temperatures. This nonlinear hydrody-

namic modification has been widely discussed in the cosmic

ray acceleration literature (e.g. see Jones and Ellison, 1991;

Berezhko and Ellison, 1999, and references therein), and has

been extensively applied to multiwavelength SNR spectral

models (e.g. see Baring et al., 1999; Berezhko et al., 2002;

Ellison and Cassam-Chenaı, 2005; Baring et al., 2005).

The extent of equilibration between electrons and ions in

SNR shell shocks needs to be understood, and can potentially

be investigated by laboratory plasma experiments. A critical

ingredient is the electrostatic coupling between electrons and

protons in the shock layer, which offers the potential for con-

siderable heating of e−, coupled with cooling of protons, set-

ting m p(3u1x/4)2/2 ≫ 3kTe/2 ≫ me(3u1x/4)2/2 with Te =Tp. Probing this coupling is the subject of this paper. Here

we describe preliminary results from our program to explore

electrostatic energy exchange between these two species in

SNR shocks, using a Monte Carlo simulation of charged

particle transport, their spatial distribution and associated

electric field generation. The goal is to eventually obtain a

simulation with self-consistent feedback between the charge

separation potentials, and the Lorentz force they impose

on the charges. The research progress outlined here indi-

cates that the role of diffusion in quasi-parallel shocks is

very important, and can readily influence charge separation

potentials.

2 Shock layer electrostatics in supernova remnants

Cross-shock electrostatic potentials arise in the shock layer

because of the different masses of electrons and ions: up-

stream thermal ions gyrate on larger scales than do their

electron counterparts when they transit downstream of the

shock for the first time. In shocks where the field is oblique

to the shock normal by some angleBn1 upstream (and there-

fore a greater angle downstream), on average, protons will be

located further downstream of the shock than electrons. This

naturally establishes an electric field E, to which the plasma

responds by accelerating electrons and slowing down ions

to short out the induced E. A feedback loop ensues, medi-

ated by fields and currents that vary spatially on the order of,

or less than, the ion inertial scale, which is typically shorter

than the ion gyroradius in astrophysical shocks such as those

associated with supernova remnant shells.

Particle-in-cell (PIC) simulations are a natural technique

(e.g. see Forslund and Friedberg, 1971 for an early imple-

mentation) for exploring signatures of such electrostatics in

shock layers. These trace particle motion and field fluctua-

tions, obtained as self-consistent solutions of the Newton-

Lorentz and Maxwell’s equations, in structured zones or

cells in spatially-constrained boxes. Such simulations have

been used recently to probe the Weibel instability at weakly-

magnetized, perpendicular, relativistic pair plasma shocks

(see Silva et al., 2003; Hededal et al., 2004; Nishikawa et al.,

2005). They have also been used by Shimada and Hoshino

(2000) to treat electrostatic instabilities at non-relativistic,

quasi-parallel electron-ion shocks. Rich in their turbulence

information, due to their intensive CPU requirements, such

simulations have difficulty in modeling realistic m p/me mass

ratios, and fully exploring 3D shock physics such as diffu-

sive transport. Moreover, they cannot presently address the

wide range of particle momenta and spatial/temporal scales

encountered in the acceleration process; they often do not

obtain time-asymptotic states for the particle distributions.

Monte Carlo techniques provide an alternative method

that can easily resolve electron and proton inertial scales,

treat fully 3D transport and large dynamical ranges in spa-

tial and momentum scales, all at modest computation cost.

While they parameterize the effects of turbulence via dif-

fusive mean free paths (e.g. see Jones and Ellison, 1991),

they can accurately describe the microphysics of cross-shock

electrostatic potentials. This simulational approach has been

well-documented in the literature (e.g. Jones and Ellison,

1991; Ellison et al., 1996), with definitive contributions to

the study of heliospheric shock systems, cosmic ray produc-

tion, SNR applications and gamma-ray bursts. It models the

convection and diffusion of gyrating particles in spatially-

structured flows and fields, with transport back and forth

across the shock effecting diffusive Fermi-type acceleration

directly from the thermal population. The mean free path λ

is usually prescribed as some increasing function of particle

momentum p or gyroradius rg . Here we use this approach,

with λ ∝ p adopted as a broadly representative situation: see

Baring et al. (1997) for a discussion of evidence from obser-

vations and plasma simulations in support of such a special-

ization. Here, λ/rg = 5 is chosen for illustrative purposes,

to begin to investigate electrostatic influences on low-energy

particles in non-relativistic electron-ion shocks of arbitrary

Bn1. Clearly then, the diffusive scales for protons and elec-

trons are disparate by their mass ratio.

In the Monte Carlo simulation, the shock is defined mag-

netohydrodynamically, consisting of laminar, uniform flows

and fields upstream and downstream of a sharp, planar

discontinuity. The magnetic fields and flow velocities either

side of the shock are related uniquely through the standard

Rankine-Hugoniot solutions for energy and momentum flux

conservation (e.g. see Boyd and Sanderson, 1969). These

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Fig. 1 Electrostatic profiles for an electron-proton plasma shock ofspeed u1x = 9000km/sec and sonic Mach number MS = 10, with up-stream field obliquities Bn1 = 0 (green), Bn1 = 30 (blue) andBn1 = 60 (red), as labelled. Left panel: scaled charge density dis-tribution ρ(x); Right Panel: resulting “linear” electric field profileE ≡ Ex (x) computed by solving Gauss’ Law (the Bn1 = 0 case

displays Ex (x)/2). The profiles are exhibited on proton gyroscalesrg,p = m pu1x c/eB1, so that fluctuations on electron gyroscales arecollapsed into the shock layer (dashed vertical line). The panels dis-play both oscillations associated with proton gyrations in oblique cases,and diffusive upstream “precursors,” which are most prominent whenBn1 = 0 (a parallel shock).

solutions include essential elements of Maxwell’s equations,

such as the divergenceless nature of the B field. In the ref-

erence frames of the local upstream and downstream fluids,

the mean electric field is assumed to be zero, a consequence

of very effective Debye screening, so that the only electric

fields present in shock rest frames are u×B drift fields. The

charged electrons and protons (more massive ions are omit-

ted in this paper to simplify the identification of the princi-

pal effects) are treated as test particles, convecting into and

through the shock, initially with the prescribed upstream fluid

velocity u1. This neutral beam is entirely thermal, and more-

over is in equipartition, so that it has an input temperature

Te = Tp. The charges constantly diffuse in space to mimic

collisions with magnetic turbulence that is putatively present

in the shock environs, and in so doing, can be accelerated.

These non-thermal particles form a minority of the total pop-

ulation, and provide only a minor contribution to the fields

illustrated.

The charges transiting the shock distribute their down-

stream density in a manner that couples directly to their gy-

rational motion (e.g. see Baring, 2006), and the local den-

sities of electrons and protons can easily be tracked in the

Monte Carlo technique by accumulating “detection” data

at various distances from the shock. Monte Carlo simula-

tion runs clearly exhibit non-zero charge excursions within

a proton gyroradius of the shock, an effect similar to those

found in PIC codes. For example, a cold, neutral e − p up-

stream beam develops an electron concentration near the

shock in the downstream region, with protons distributed on

their larger inertial scales. The resulting charge distributions

ρ(x) depend on both the upstream field obliquity Bn1, and

also on the sonic Mach number MS ≈ u1x/√

5kTp/(3m p)

in situations where the upstream beam is warm. Due to the

steady-state, planar nature of the simulation, these distribu-

tions depend only on the coordinate x along the shock nor-

mal. It is straightforward using Gauss’ law for electrostatics,

∇ · E = 4πρ(x), to integrate the charge distribution profile

to obtain Ex (x) = −∂/∂x . Eventually, such “linear” fields

will then be used to compute the energy exchange between

electrons and ions as they cross the non-monotonic charge

separation potential (x).

Linear determinations of electrostatic spatial profiles are

shown in Fig. 1 to illustrate the key features; these did not

self-consistently include the acceleration of electrons and

protons in the produced E field. The left panel depicts large

charge density fluctuations that trace ion gyration in the

downstream magnetic field. Similar fluctuations of opposite

sign are created by electrons, but on much smaller scales

that are not resolved in the Figure. Accordingly, “striped”

zones of positive and negative charge density result, and

this electrostatic analog of a plasma oscillation integrates

to produce the E fields in the right panel that can accelerate

or decelerate electrons and protons. The outcome depends

on the shock obliquity Bn1 when Bn1<∼ 60, whereas

quasi-perpendicular shocks with Bn1>∼ 60 possess pro-

files fairly close to the Bn1 = 60 case depicted in Fig. 1,

since they all have field obliquities Bn2 ≈ 80–90 down-

stream. Note also, that while the gyrational contributions are

prominent, there is also a diffusive contribution, manifested

as an upstream precursor modification to ρ(x) and E. This is

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particularly marked in the parallel shock (Bn1 = 0) case,

where the diffusive scale along the field achieves a maxi-

mal component orthogonal to the shock plane. This diffu-

sive influence originates in accelerated particles returning

to the upstream side of the shock (x < 0), enhancing the

density there before convecting downstream again: protons

effect this on larger scales, and so control the precursors

seen in the figure (i.e. ρ(x) > 0 for x < 0). Since the fields

are established on the scale of a proton gyroradius, their

magnitude scales as Ex ∼ 4πρrg,p = 4πen p(m pu1x c/eB1),

so that |Ex/B1| ∼ 4πn pm pu1x c/B21 ≡ M2

A(c/u1x ) ≫ 1 for

Alfvenic Mach numbers MA > 1.

The competition between gyrational and diffusive influ-

ences on electrostatics is a principal conclusion of this paper,

defining a dichotomy delineating quasi-parallel and quasi-

perpendicular shocks. The Monte Carlo technique can ac-

curately trace both influences, while comfortably resolv-

ing the disparate scales for the e − p shock problem. Since

the “linear” results illustrated need to be upgraded to ac-

count for the E-field’s influence on e− and p motions, it is

presently unclear whether ions can energize electrons overall

(the right panel of the Figure suggests they may even decel-

erate them), and how the net work done depends on field

obliquity. A noticeable feature of the electric field profiles

in Fig. 1 is that for Bn1<∼ 60, these linear field calcula-

tions do not establish |E| → 0 asymptotically as |x | → ∞,

as required by net charge neutrality. The next step of this

program will be to solve the Newton-Lorentz equation of

motion dp/dt = q(E + v × B/c) to determine both drift and

accelerative contributions to the charges’ motions. These will

necessitate a recomputation of the E field profiles, and a

feedback loop will result, with shock layer currents gener-

ating magnetic field excursions via Ampere’s law, ∇ × B =4πJ/c. This iterative process will continue to convergence

(establishing |E| → 0 as |x | → ∞), with relaxation to equi-

librium occuring on the spatial response scale u1x/ωp, where

ωp =√

4πe2n p/m p is the proton plasma frequency. Since

u1x/(ωprg,p) ∼ u1x/(cMA) ≪ 1), this response scale is far

less than a proton gyroradius for typical SNR environmental

parameters, and indeed for any strong, non-relativistic astro-

physical shock.

The degree of electron energization in the cross shock

potential may offer significant insights into the well-known

electron injection problem at non-relativistic shocks. Elec-

trons do not resonantly interact with Alfven waves until

they become relativistic. Levinson (1992) suggested that e−

interaction with a presumably abundant supply of whistler

waves could effect pre-injection into diffusive acceleration

processes, if electrons could achieve energies in excess of

around 10 keV to access the whistler resonance branch. The

planned self-consistent extension of the developments out-

lined here will help determine whether this channel of access

to continued acceleration is opened up by shock layer elec-

trostatics. Moreover, crafted laboratory plasma experiments

may cast light on this aspect of shock physics.

3 Conclusion

In this paper, charge density and associated cross-shock elec-

tric field spatial profiles are presented for different mag-

netic field obliquities. It was found that in highly oblique

and perpendicular shocks diffusion plays little role in mod-

ulating the field structure, which is controlled by the mag-

netic kinking and compression on the downstream side of the

shock. In contrast, in quasi-parallel shocks, where the gyra-

tional charge separation is small, diffusion scales upstream

and downstream of the shock dominate the generation of

shock-layer electric fields. This is an interesting twist, sug-

gesting that observationally, thermal X-ray emission could

be distinctly different in portions of an SNR rim that es-

tablish quasi-parallel and quasi-perpendicular shocks. The

work discussed here paves the way for self-consistent de-

termination of the acceleration/deceleration of electrons and

protons, their spatial distributions, and the electric fields nor-

mal to non-relativistic shocks. This development will impact

the understanding of electron injection and acceleration in

shocks of all obliquities.

References

Baring, M.G.: On-line proceedings of the 2006 KITP/UCSB con-ference “Supernova and Gamma-Ray Burst Remnants” (2006)[http://online.kitp.ucsb.edu/online/grb c06/ baring/]

Baring, M.G., Ellison, D.C., Reynolds, S.P., Grenier, I.A., Goret, P.:ApJ 513, 311 (1999)

Baring, M.G., Ellison, D.C., Slane, P.O.: Adv. Space Res. 35, 1041(2005)

Baring, M.G., Ogilvie, K.W., Ellison, D.C., Forsyth, R.J.: ApJ 476, 889(1997)

Berezhko, E.G., Ellison, D.C.: ApJ 526, 385 (1999)Berezhko, E.G., Ksenofontov, L.T., Volk, H.J.: ApJ 395, 943 (2002)Boyd, T.J.M., Sanderson, J.J.: Plasma Dynamics. Nelson & Sons,

London (1969)Decourchelle, A., Ellison, D.C., Ballet, J.: ApJ 543, L57 (2000)Ellison, D.C., Baring, M.G., Jones, F.C.: ApJ 473, 1029 (1996)Ellison, D.C., Cassam-Chenaı, G.: ApJ 632, 920 (2005)Forslund, D.W., Freidberg, J.P.: Phys. Rev. Lett. 27, 1189 (1971)Ghavamian, P., Rakowski, C.E., Hughes, J.P., Williams, T.B.: ApJ 590,

833 (2003)Hededal, C.B., Haugbolle, T., Frederiksen, J.T., Nordlund, A.: ApJ 617,

L107 (2004)Hughes, J.P., Rakowski, C.E., Decourchelle, A.: ApJ 543, L61 (2000)Hwang, U., et. al.: ApJ 581, 110 (2002)Jones, F.C., Ellison, D.C.: Space Science Rev. 58, 259 (1991)Levinson, A.: ApJ 401, 73 (1992)Long, K.S., et al.: ApJ 586, 1162 (2003)Lucek, S.G., Bell, A.R.: MNRAS 314, 65 (2000)Nishikawa, K.-I., et al.: ApJ 622, 927 (2005)Shimada, N., Hoshino, M.: ApJ 543, L67 (2000)Silva, L.O., et al.: ApJ 596, L121 (2003)Vink, J., Laming, J.M.: ApJ 584, 758 (2003)

Springer


DOI 10.1007/s10509-006-9263-0


Non-Stationary Rayleigh-Taylor Instabilities in Pulsar WindInteraction with a Supernova Shell

X. Ribeyre · L. Hallo · V. T. Tikhonchuk · S. Bouquet ·

J. Sanz


Abstract The Rayleigh-Taylor instability (RTI) plays an im-

portant role in the dynamics of several astronomical objects,

in particular, in the supernovae (SN) evolution. In the present

paper we examine the dynamics of a shell (representing a

type II SN remnant) blown-up by a wind emitted by a central

pulsar. The shell is accelerated by the pulsar wind and its

inner surface experiences the RTI. We develop an analytical

approach by using a specific transformation into the coor-

dinate frame co-moving with the SN ejecta. We first derive

a non-stationary spherically symmetric solution describing

an expansion of a gas shell under the pressure of a central

source (pulsar). Then, we analyze its 3D stability with respect

to a small perturbation on the inner shell surface. The dis-

persion relation is derived in the co-moving reference frame.

The growth rate of the perturbation is found and its tempo-

ral evolution is discussed. We compare our result with the

previous published studies and apply it to the Crab nebula

evolution.

Keywords Rayleigh-Taylor instabilities . Pulsar wind

nebulae . Supernova remnants

X. Ribeyre () · L. Hallo · V. T. TikhonchukCentre Lasers Intenses et Applications, UMR 5107, CNRS,Universite Bordeaux 1, CEA, Universite Bordeaux 1, 351, Coursde la liberation, 33405 Talence, Francee-mail: [email protected]

S. BouquetCommissariat a l’Energie Atomique, DIF/Departement dePhysique Theorique et Appliquee, BP 12, 91680,Bruyeres-le-Chatel, France

J. SanzE.T.S.I., Aeronauticos, Universidad Politecnica de Madrid,Madrid 28040, Spain

1 Introduction

The Crab nebula observations lead us to conclude that the

expansion of the supernova remnant (SNR) is non station-

ary (Trimble, 1968). Indeed, the supernova (SN) ejecta are

accelerated by the central pulsar wind and a initially homo-

geneous shell is decomposed in a complex filamentary struc-

ture. For a typical filament position about r ≃ 1 pc from the

central pulsar, with an age about t ∼ 930 years (Davidson

and Fesen, 1985), the ratio r/t gives the average filament ve-

locity about 1050 km s−1; but the current velocity is 10%

greater (∼1150 km s−1). This result shows clearly that the

nebula was accelerated (Trimble, 1968; Davidson and Fesen,

1985). Moreover the Hubble Space Telescope high resolu-

tion observations of the Crab nebula show that the filaments

are arranged in a structure that morphologically is similar

to the non linear stage generated by the RTI. It develops at

the interface between the pulsar-driven synchrotron radiation

and a shell of swept-up ejecta (Hester et al., 1996). The Crab

nebula belongs to the so-called family of pulsar wind nebula

(PWN). Hereafter we use this more general expression.

A hydrodynamic model of PWN has been developed by

Blondin et al. (2001) and Chevalier (2005). Several analytical

and numerical studies are devoted to the interaction between

the pulsar wind and the SN ejecta (Jun, 1998). Reynolds et al.

(1984) studied a self-similar homologue spherical expansion

model where the radial velocity is proportional to the radius.

The fact that the synchrotron luminosity of the pulsar de-

creases in time plays an important role in the PWN evolution.

However, in his simulations (Jun, 1998) considers a constant

pulsar wind pressure in interaction with expanding ejecta.

He carried out 2D simulations of the RTI development in a

thin shell expanding during several thousand years, while the

pulsar luminosity decreases in a shorter time scale. Blondin

et al. (2001) studied the evolution phase beyond 104 years,

Springer


Fig. 1 Density profile for a PWN, the density is in g/cm3 and the radiusin cm. There are an expanding pulsar wind, a region of shock pulsarwind at radius r0 (pulsar bubble), a swept shell of ejecta is boundedby a shock at r1 (or Rp). We consider stability of the shell of ejecta ofthickness r0-r1. From Blondin et al. (2001)

when the reverse shock in the ejecta (denoted R2 in Fig. 1),

due to the interaction between the ejecta and the interstellar

medium interacts with the pulsar wind. More recently, Buc-

ciantini et al. (2004) carried out a MHD simulation to take

into account the magnetic field effect on the RTI development

over a time scale of 2000 years.

However, the relation between SN fragments characteris-

tics and the wind pressure remains unclear. In this paper, our

study is based on the analytical model developed by Ribeyre

et al. (2005), which is similar to that of Bernstein and Book

(1978). In this model, the pulsar wind composed by high en-

ergetic particles and photons, is modelled by a pressure law

that varies during the shell expansion. The evolution phase

that we consider in this paper is the same that Jun (1998)

studied. It concerns the period about 1000 years after the SN

explosion and we are taking into account the non-stationarity

of the shell evolution, and the acceleration phase is followed

by the ballistic movement.

The following assumptions have been made:

– The mass brought into the expanding shell is negligible

and consequently the mass of the shell is constant.

– The pulsar wind density is small compared to the density

at the inner shell interface.

– The outer shell interface is in contact with vacuum.

The first assumption is valid if one considers a period of

evolution of the PWN not too close to the explosion time,

i.e., when the shell was already formed and consequently, its

mass does not evolve much more (Chevalier, 2005).

The second assumption is valid because the pulsar wind

density is low and it brings little matter to the shell. In our

study, we do not consider the effect of the ablation process on

the instability development driven by the wind (Atzeni and

Meyer-Ter-Vehn, 2004; Sanz and Betti, 2005). Although ab-

lation is possible, it should not have a strong effect since the

radiation of the pulsar is trapped inside the shell volume and,

Fig. 2 Simplified model of the PWN. A shell of ejecta is blown bythe pulsar wind and the RTI develops at the inner shell interface. Weassume that the pulsar wind have a negligible density and that the outershell is in contact with vacuum

consequently, the ablation and the deposition of the wind mat-

ter on the inner shell surface balance. Moreover, it is known

that ablation stabilizes the small wavelengths, whereas we

are interested in large wavelengths compared to the shell

thickness.

The third assumption is valid for a shell expanding in an

interstellar medium of a very low density or for a young

supernova remnant. Moreover, the derivation below shows

that the external boundary condition of the shell has a little

influence on the RTI growth rate, because only the inner

interface of the shell is unstable.

In our paper we consider a perturbed amplitude smaller

than the shell thickness. This configuration is appropriate for

studies of the shell fragmentation. The case of thin shell,

where the perturbed shell amplitude is greater than the shell

thickness, was considered in Ref. (Kull, 1991).

2 Shell space-time evolution

Our analysis of expansion of a shell blown by a pulsar wind

(see Fig. 2) is based on the model developed by Ribeyre

et al. (2005). We consider a family of solutions of the Euler

equations, for a polytropic gas, such as p = Kργ . The effect

of the pulsar wind is treated by a pressure law acting on the

inner interface of the ejecta.

A non-stationary spherically-symmetric solution for the

unperturbed radial shell flow is described by [see in Ribeyre

et al., 2005]:

ρ(r, t) = ρ0

C3

(

1 − R20

)−1/(γ−1)(

1 − r2

C2 r21

)1/(γ−1)

, (1)

p(r, t) = Kρ

γ

0

C3γ

(

1 − R20

)−γ /(γ−1)(

1 − r2

C2r21

)γ /(γ−1)

, (2)

vr (r, t) = r

C2τ

[

β + (β2 + 1)t

τ

]

, (3)

Springer


where ρ(r, t), p(r, t), vr (r, t) are respectively the density,

the pressure and the radial velocity, r0 and r1 are the inner

and outer initial shell radii respectively. The parameter β

definines the magnitude of the velocity profile at t = 0 and

τ is the characteristic time for the dynamics of the shell.

Moreover, R0 ≡ r0/r1 and ρ0 ≡ ρ(r0, 0) is the initial density

at the inner border of the shell. A relation exists between the

inner density, the constant K and r1:

ρ0 =[

r21 (γ − 1)

(

1 − R20

)

/2Kγ τ 2]1/(γ−1)

. (4)

Finally, the scale function C(t) is given by the solution of the

following differential equation:

τ−2 = C C3γ−2, (5)

where the upper dot stands for the time derivative of C .

For the polytropic constant γ = 5/3, i.e., for a mono-

atomic ideal gas, an analytical expression for C(t) reads as:

C(t) =√

(

1 + βt

τ

)2

+ t2

τ 2. (6)

As explained earlier, it is clear that the parameter τ is

the characteristic time of shell expansion and the constant

β ≡ τ C(0) characterizes its initial velocity, v0(r ) = βr/τ .

Moreover, the temporal evolution of the internal face is given

by r0(t) = C(t)r0 and for the outer interface r1(t) = C(t)r1.

The scale function C(t) is a linear function of t for t ≫ τ ,

i.e., when the shell is in ballistic motion.

The function C(t) characterizes also the acceleration of

the shell. Indeed, the temporal derivative of v(t), is:

v = r

τ 2C3γ−1. (7)

3 Rayleigh-Taylor instability

Bernstein and Book (1978) obtained an exact expression

for the incompressible perturbation evolution for a given

radial flow. They consider a radially symmetric shell flow

described by Equations (1)–(3) and a 3D perturbation with

an angular dependence corresponding to the spherical har-

monic: Ylm(θ, φ). Applying their expression to the case of an

ideal gas, γ = 5/3, the time evolution of the perturbation δ,

is given by:

δ ∝ C(t) cosh [√

l + 1 arctan (t/τ )], (8)

for β = 0, i.e., without initial shell velocity.

A similar expression can be obtained by solving the per-

turbed Euler equations in the expanding co-moving frame

(Bouquet et al., 1985; Ribeyre et al., 2005). More precisely,

for γ = 5/3, one can express the angular perturbed displace-

ment evolution as:

δ ∼ eωt/τYlm(θ, φ), (9)

where the relation between the co-moving time t and the time

in the physical space t is given by:

t ≡ τg(t) = τ arctan[β + (β2 + 1) t/τ ] − arctanβ. (10)

Moreover, although Bernstein and Book (1978) do not suc-

ceed to exhibit a dispersion relation, the one we obtain is

very simple. The parameter ω is given by the roots of the

following equation :

ω4 − ω2 − l(l + 1) = 0. (11)

Therefore, there are four linearly independent eigenmodes:

ω1,2 = ±√

l + 1, ω3,4 = ±i√

l. (12)

One mode ω1 is unstable and three others are stable. One ob-

tains the same expression (8) that Bernstein and Book (1978)

by considering a linear superposition of the modes ω1 and

ω2 for β = 0. However this particular solution does not de-

scribe a general perturbation of the inner shell surface. By

taking account of all four modes, one can describe various

initial conditions for the shell perturbations and study their

stability as well as the interaction between the inner and outer

shell interfaces.

4 Qualitative RTI analysis

The dispersion relation defined by the Equation (11) and the

solutions (12) are surprisingly simple and they correspond to

the incompressible perturbations.

In particular, if one considers only the unstable mode

ω1 =√

l + 1, then, the perturbation evolves as exp [ω1 t/τ ]

[see Equations (9) and (10)]. This behavior can be com-

pared with a simple model, by supposing that at every mo-

ment the growth rate of the RTI is given by the formula

ω = τ√w0k0 (Rayleigh, 1883), describing the instability of

a plane surface in acceleration w0. In our case the accelera-

tion of the inner interface is defined by Equation (7). Con-

sequently, the temporal evolution of the perturbation δ(t) in

this model is given by:

δ(t) ∝ exp

[∫ t

0

√

k0(t ′)w0(t ′) dt ′]

, (13)

Springer


Table 1 Comparison of the growth rate and the amplification of theperturbation for t → ∞, given by Eq. (15) with β = 0 and the solutionof Bernstein and Book (1978) [see the Eq. (51) in Bernstein and Book(1978)], for γ = 4/3, 5/3, 2 and for l = 100. One gives for each valueof γ respectively, the growth rate and also the amplification whichcorresponds to exponential of the product growth rate multiplied bytime

Our approximate Bernstein and Book (1978)

γ model solution

4/3 21.63–2.48 × 109 20.06–5.20 × 108

5/3 15.09–3.60 × 106 15.09–3.60 × 106

2 12.19–1.98 × 105 12.84–3.78 × 105

where k0(t) is the wave number of the perturbation. In spher-

ical geometry, the wavelength of the perturbation λ evolves

in time as, 2π/λ(t) = (l + 1)/[r0 C(t)] (where l is the mode

number of the perturbation). Then using expression (7) in

Equation (13), one obtains:

δ(t) ∝ exp

[√l + 1

τ

∫ t

0

dt ′

C3γ−1

2

]

. (14)

In particular for γ = 5/3, one finds the exact solution (8) if

β = 0 or a more general solution if β = 0.

One can go further and calculate the growth rate for an

arbitrary γ . Indeed, the expression of k0(t) does not change

and it is enough to calculate the new acceleration w0(t, γ ).

Then, the perturbation growth (14) can be presented in the

following way:

δ(t) ∼ exp

⎡

⎣

√l + 1

∫ C(t)

0

C (1−3γ )/2dC√

β2 + 23(γ−1)

(

1 − 1C3γ−3

)

⎤

⎦.

(15)

Starting from this expression, one can find the asymptotic

growth rate [C(t) → t for t → +∞] withβ = 0 which could

be compared with the expression of the asymptotic growth

rate provided by Bernstein and Book (1978) for an arbitrary

γ [see their expression (51)].

In this case, for l = 100 and for three values of γ given

by Bernstein and Book, γ = 5/3, 4/3 and 2, the variations

between the two growth rates do not exceed 8% (see Table 1).

Therefore, the instantaneous expression, ω = τ√w0k0

gives a good approximation of the growth rate of RTI, even

for γ = 5/3. On the other hand, there is a significant un-

certainty, up to a factor five, on the value of the amplifi-

cation of the perturbation, since one takes the exponential

variation.

This simple analytical model can be used successfully

to compute the early evolution of the Crab nebula (Ribeyre

et al., 2006). Starting with Jun’s (Jun, 1998) input data, the

main properties of the Crab nebula are recovered (mass of the

filament, size of PWN and time corresponding to the early

fragmentation of the shell∼400 years). In addition, it is found

that the most probable mode, ldisup, leading to the disruption

of the shell is ldisrup = 60. This result is in agreement with

others studies (Bernstein and Book, 1978; Bucciantini et al.,

2004).

5 Conclusion

We examined the dynamics of a shell (representing a type II

SN remnant) blown by the strong wind emitted by a central

pulsar. Assuming that the shell mass is constant and evolves

in vacuum, we developed a simplified model describing a

non-stationary shell evolution. Moreover, we derive a sim-

ple dispersion relation for the RTI growth rate for an ideal

polytropic gas (γ = 5/3). We compare our approach with

the work achieved by Bernstein and Book (1978) and show

that the RTI growth and the growth rate of perturbation are

comparable even for over values of γ . Finally, this model

can be applied to the Crab nebula and results are in pretty

agreement with others studies (Jun, 1998; Bucciantini et al.,

2004) and with observations as well (Hester et al., 1996).

Acknowledgements The authors thank Dr T. Foglizzo for the fruitfuldiscussion and useful comments. This work is partly supported by theAquitaine Region Council.

References

Atzeni, S., Meyer-Ter-Vehn, J.: The Physics of inertial fusion. OxfordUnversity Press (2004)

Bernstein, I.B., Book, D.L.: ApJ 225, 633 (1978)Blondin, J.M., Chevalier, R.A., Frierson, D.M.: ApJ 563, 806 (2001)Bouquet, S., Feix, M., Munier, A.: ApJ 393, 494 (1985)Bucciantini, N., Amato, E., Bandiera, R., Blondin, J.M., Del Zanna, L.:

A&A 423, 253 (2004)Chevalier, R.A.: ApJ 619, 839 (2005)Davidson, K., Fesen, R.A.: ARA&A 23, 119 (1985)Hester, J.J., Stone, J.M., Scowen, P.A., Jun, B.-I., Gallacher III, J.S.,

Norman, M.L., Ballester, G.E., Burrows, C.J., Casertano, S.,Clarke, J.T., Crisp, D., Griffiths, R.E., Hoessel, J.G., Holtzman,J.A., Krist, J., Mould, J.R., Sankrit, R., Stapelfeldt, K.R., Trauger,J.T., Watson, A., Westphal, J.A.: ApJ 456, 225 (1996)

Jun, B.-I.: ApJ 499, 282 (1998)Kull, H.J.: Physics Reports 206, 5 (1991)Rayleigh, Lord: Proc. Lond. Math. Soc. 14, 170 (1883)Reynolds, S.P., Chevalier, R.A.: ApJ 278, 630 (1984)Ribeyre, X., Tikhonchuk, V.T., Bouquet, S.: Astrophys. Space Sci. 298,

75 (2005)Ribeyre, X., Hallo L., Tikhonchuk, V.T., Bouquet, S., Sanz, J.: Publi-

cation to be submitted to Astron. Astrophysics (2006)Sanz, J., Betti, R.: Phys. Plasmas 12, 042704 (2005)Trimble, V.: Astron. J. 73(7), 535 (1968)

Springer


DOI 10.1007/s10509-006-9233-6


Phenomenological Theory of the Photoevaporation FrontInstability

D. D. Ryutov · J. O. Kane · A. Mizuta · M. W. Pound ·

B. A. Remington


Abstract The dynamics of photoevaporated molecular

clouds is determined by the ablative pressure acting on the

ionization front. An important step in the understanding of

the ensuing motion is to develop the linear stability theory

for an initially flat front. Despite the simplifications intro-

duced by linearization, the problem remains quite complex

and still draws a lot of attention. The complexity is related

to the large number of effects that have to be included in the

analysis: acceleration of the front, possible temporal vari-

ation of the intensity of the ionizing radiation, the tilt of

the radiation flux with respect to the normal to the surface,

and partial absorption of the incident radiation in the ab-

lated material. In this paper, we describe a model where all

these effects can be taken into account simultaneously, and a

relatively simple and universal dispersion relation can be ob-

tained. The proposed phenomenological model may prove to

be a helpful tool in assessing the feasibility of the laboratory

experiments directed towards scaled modeling of astrophys-

ical phenomena.

Keywords HII regions . Ablation front instability . Eagle

nebula . Laboratory astrophysics

PACS Numbers: 98.38.Dq, 98.38.Hv, 52.38.Mf, 5257.FG,

52.72.+v

D. D. Ryutov () · J. O. Kane · B. A. RemingtonLawrence Livermore National Laboratory, Livermore, CA 94551

A. MizutaMax-Planck-Institut fur Astrophysik, Garching 85741, Germany

M. W. PoundAstronomy Department, University of Maryland, College Park,MD 20742, USA

1 Introduction

The shape of photoevaporated molecular clouds (e.g., Hester

et al., 1996; Pound et al., 2003) is most probably caused by

a variety of hydrodynamical processes occurring under the

action of the ablation force. Some of the models relate the

observed structures to the existence of large initial density

perturbations (see, e.g., Bertoldi, 1989; Bertoldi and McKee,

1990; Williams et al., 2001). The others (see below) attribute

the shape to the development of instabilities that grow from

small perturbations at the ablation (photoevaporation) front.

In the present paper, we consider several aspects of this sec-

ond approach, concentrating on the linear stage of instability.

We present a simple phenomenological model that allows one

to describe, in a unified way, all the stabilizing and destabi-

lizing factors which have been studied thus far in a piecemeal

fashion.

The linear analyses of the ablation front instability can

be traced back to the papers by Spitzer (1954) and Frieman

(1954) where the instability was identified as the Rayleigh-

Taylor (RT) instability of an accelerating interface. Kahn

(1958) has argued that the partial absorption of the ionizing

radiation in the ablated material should lead to a stabilization

of the RT instability. Vandervoort (1962) developed a de-

tailed theory of the ionization front instability, with radiation

tilt included, at zero acceleration (i.e., this was an instability

different from the RT instability). In the limit of negligible

density in the ablated plasma, η ≡ ρa/ρ → 0 (where ρa and

ρ are the density in the ablation flow and in the molecular

cloud, respectively), the instability is present only for non-

zero tilt; it can be called the “tilted radiation” (TR) insta-

bility. Axford (1964) and Sysoev (1997) included the effect

of absorption into the stability analysis of a non-accelerating

front and radiation at normal incidence and have found a gen-

erally stabilizing effect. Williams (2002) included both the

Springer


radiation tilt and absorption (but no acceleration) and con-

cluded that radiation tilt makes the system more unstable at

all wavelengths. Ryutov et al. (2003) considered the TR insta-

bility in the presence of acceleration (but without absorption

in the ablation flow). In numerical simulations by Kane et al.

(2005) and Mizuta et al. (2005a, b), which contained both the

linear and nonlinear stages of evolution, acceleration and ab-

sorption were present, but no radiation tilt. It was found that,

in such a situation, the absorption has a strong stabilizing ef-

fect on the linear RT instability but non-linear perturbations

would grow (Mizuta et al., 2005a).

In the present paper, which is limited entirely to the linear

theory, we include in the analysis all three factors: acceler-

ation, radiation tilt, and absorption in the ablation flow. We

discuss also the “impulsive acceleration” instability. By the

latter we mean the situation where the intensity of the photo-

ionizing radiation comes as a short pulse, with the time-scale

shorter than the dynamical time of the system. This scenario

is of some interest because the light curves of the young OB

stars may indeed have a substantial spike early in time (Iben

and Talbot, 1966; Cohen and Kuhi, 1979).

We call our model “phenomenological” because we use a

simplified description of several processes affecting the insta-

bility: the absorption of the incident radiation in the blow-off

plasma is described just by a constant absorption coefficient

κ; the ablation pressure is assumed to depend only on the

intensity of radiation at the ablation front, and the specific

shape of this dependence is not considered. Therefore, some

parameters that enter the final dispersion relation have to be

found either experimentally, or from more detailed analysis.

This is an obvious drawback of the phenomenological ap-

proach. On the other hand, the benefit of this approach is

related to the possibility of describing, in a unified fashion, a

broad variety of factors affecting the instability both in astro-

physics and in possible laboratory experiments (of the type

described in Remington et al., 1993), and identify effects that

have the strongest influence on the instability.

2 Basic assumptions

We assume that the radiation comes from a direction that

forms an angle θ with the normal to the unperturbed planar

surface (Figure 1). The absorption coefficient along the ray is

assumed to be a constant κ , so that the intensity along the ray

varies according to the equation d I ∗/ds = −κ I ∗, where s is

a coordinate along the ray. We use an asterisk to designate

the energy flux at a plane normal to the direction of the rays.

We denote this intensity at the unperturbed surface of the

cloud as I ∗0 . When radiation reaches the molecular cloud,

the absorption is assumed to occur in a zero-thickness layer.

In this last respect, our model is identical to that used in

Vandervoort (1962) and Ryutov et al. (2003).

x

y

z

k

g

h

Fig. 1 The geometry of the problem

In astrophysics, where absorption in the ablated low-

density material is caused by the presence of the neutral hy-

drogen produced by the recombination process, the absorp-

tion coefficient is proportional to the square of the density of

this material. In the laboratory experiments of the type (Rem-

ington et al., 1993), the absorption can be caused by a variety

of factors and, in particular, by the presence of the higher-Z

admixtures. In our phenomenological model we have to ad-

just the coefficient κ so as to fit absorption properties of a

particular system.

Following the model used in Kahn (1958), we assume

that the ablation pressure is some growing function of I, the

energy flux through the surface of the cloud:

pa = pa(I ) (1)

In the unperturbed state this energy flux is I0 = I ∗0 cos θ .

In what follows, we do not need the specific depen-

dence of pa vs I, just that this is a smooth dependence with

(I/pa)(dpa/d I ) ∼ 1. This allows us to cover in a unified

fashion not only an instability of the photoevaporated clouds,

but also the instability of ablation fronts in possible laboratory

experiments with intense lasers, where details of formation

of the ablation flow are different from astrophysical setting

(e.g., Ryutov et al., 2003).

In this brief communication we discuss only the simplest

model of the cloud, within which the cloud material is con-

sidered an incompressible fluid. As was shown in Ryutov

et al. (2003), the model of an incompressible fluid yields

the results that are very close to a more sophisticated model

treating the cloud as a compressible ideal gas.

The ablation pressure accelerates the cloud in the negative

direction of the axis z (Fig. 1). The absolute value g of the

acceleration is equal to

g = pa

ρh, (2)

where h is the cloud thickness. The effective gravity force

in the frame of the unperturbed ablation front is directed

towards z>0.

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We assume that the density ρa in the ablation flow is much

smaller than ρ, and present results corresponding to the limit

η ≡ ρa/ρ → 0. We work in the frame moving with the un-

perturbed ablation front. In this frame, the cloud material

flows through the surface of the ablation front with velocity.

v ∼ η√

pa/ρa (3)

As shown in Lindl (1995) and Takabe et al. (1999), the flow

through the interface does not have a significant effect on

the instability if the condition |k|v < Ŵ holds, with k and

Ŵ being the wave number and the growth rate of unstable

perturbations, respectively. For the typical RT growth rate

Ŵ ∼ (kg)1/2, this means that the flow through the interface is

unimportant if kh < 1/η. In the limit η ≪ 1 this condition

is not very restrictive and we neglect the flow through the

interface in the rest of the paper.

3 Equations for perturbations

Perturbation of the interface between the cloud and the ab-

lation flow leads to the perturbation of the energy flux I

through the perturbed surface. There are two sources for

this perturbation. First, if the surface gets tilted with respect

to its original orientation, the angle between the rays and

the surface changes, thereby leading to a change of I. If the

plane of incidence of the incoming radiation is the xz plane,

as shown in Fig. 1, then the corresponding change of I is

δ I = I ∗0 sin θ∂ξ/∂x , where ξ (x, y) is the displacement of

the surface in the z direction. Second, if a certain element

of the surface is displaced, the intensity changes because of

the change of the absorption along the ray. This contribu-

tion is, obviously, δ I = I ∗0 κξ , so that the total perturbation

of intensity is δ I = I ∗0 (sin θ∂ξ/∂x + κξ ). This leads to the

perturbation of the ablation pressure,

δpa = Cpa

(

sin θ∂ξ

∂x+ κξ

)

, (4)

where C>0 is a coefficient of order of unity: C =(I ∗

0 /pa)[∂pa(I0)/∂ I0].

At this point, it is convenient to perform a Fourier trans-

form in the xy plane, and separate the spatial and temporal

variables. In other words, the perturbation will have the fol-

lowing dependence on x, y, and t: exp(−iωt + ikx x + iky y).

An instability would correspond to Ŵ ≡ Imω > 0. We use

also the notationα for the angle between the two-dimensional

wave vector k and the axis x (Figure 1), so that kx = kcosα.

For such perturbations, according to Eq. (4),

δpa = Cpa(ik cos α sin θ + κ)ξ. (5)

The linearity constraint includes not only the smallness of

the amplitude compared to 1/k, but also the constraint that

no shadowing effects are present. The latter constraint reads

as kξ ≪ cos θ and becomes dominant at grazing incidence

(θ close to π /2).

By considering the dynamics of perturbations inside the

slab, one can relate the pressure perturbation at the inner

(molecular cloud) side of the perturbed interface to the dis-

placement of the interface. This can be done in a standard

way (in particular, see the corresponding derivation in Ryu-

tov et al., 2003). As the pressure perturbation at the inner side

of the perturbed surface has to be equal to δpa , we obtain that

(cf. Eq. (10) in Ryutov et al., 2003)

ξ = kδpa

ρ

×[

1

(1 − e2kh)(ω2 − kg)− 1

(1 − e−2kh)(ω2 + kg)

]

.

(6)

Then, from Eqs. (5) and (6), one obtains the following dis-

persion relation, that contains the effects of radiation tilt,

radiation absorption, and acceleration:

ω4 − ω2 kghC (ikh sin θ cos α + κh)

coth kh − k2g2[1 − C(ikh sin θ cos α + κh)] = 0. (7)

4 The analysis of the dispersion relation

It is instructive to see what this dispersion relation predicts

in the limiting cases that have been analyzed in the past. To

consider a situation of a semi-infinite cloud with no acceler-

ation (as it was done in Vandervoort (1962), Axford (1964),

Sysoev (1997), and Williams (2002)), one has to replace g

in Eq. (7) by its expression (2) and take the limit of large h.

One then obtains

ω2 − k(pa/ρ)C (ik sin θ cos α + κ) = 0. (8)

In the limit of no absorption (κ = 0), we essentially recover

the results by Vandervoort (for a low-density ablation flow,

η → 0): no instability for radiation at normal incidence (θ =0), and instability in the presence of radiation tilt, with the

growth rate proportional to the wave number,

Imω = ±k

√

Cpa sin θ | cos α|2ρ

. (9)

If we include absorption, then, for normal incidence, one

obtains non-damped oscillations, whereas in the presence of

tilt, the instability is present at arbitrarily large absorption

coefficient. The latter result corresponds to that obtained in

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0 0.25 0.5 0.75 1

h

Im/(

kg)1

/2

0.2

50

.50

.75

1.0

kh=0.1

kh=1

kh=10

Fig. 2 The normalized growth rate at a zero tilt vs. the normalizedabsorption coefficient. At κh > 1 the linear RT instability ceases toexist

the linear analysis by Williams (Williams, 2002). In the limit

of large absorption, κ ≫ k, the growth rate is equal to:

Imω = ±k sin θ cos α

√

Cpak

ρκ. (10)

Development of perturbations in the presence of accelera-

tion and absorption, was studied numerically in Mizuta et al.

(2005) for normal incidence. In this case, our Eq. (7) yields:

ω4 − ω2 kκgh2 C coth kh − k2g2(1 − Cκh) = 0. (11)

For large-enough absorption coefficients such that Cκh > 1,

the system becomes stable. This agrees with the results of

Mizuta et al. (2005). Dependence of the growth rate on the

absorption coefficient in the unstable domain (Cκh < 1) is

illustrated in Fig. 2 (for C = 1). The real part of the frequency

of the unstable modes is equal to zero, i.e., in this regard, they

behave as standard RT perurbations.

Finally, if we include all the ingredients, absorption, tilt,

and gravity (i.e., return to the general equation (7)), we find an

instability that exists at any absorption coefficient (for non-

zero tilt). This is illustrated in Fig. 3, where the normalized

growth rate is presented for the case of κh = 2, where the

system would be stable at a normal incidence (θ = 0). Unlike

the “standard” RT instability, perurbations here have a finite

real frequency (i.e., a finite phase velocity along the surface)–

a feature that can be exploited to experimentally identify this

mode in possible laboratory experiment (Ryutov et al., 2003).

5 Impulsive irradiation

It was noted by Pound (1998) that the dynamical time of evo-

lution of the Eagle Nebula is much shorter than the evolution-

0 /4 /2 /2 3 /4

0.2

50.5

0.7

5

h=2,

Im /(g/h)1/2

Re /2(g/h)1/2

Fig. 3 The growth rate (solid line) and real frequency (dashed line)for κh = 2, C = 1, and kh = 1 vs. the tilt angle θ . Note the differentnormalization of the real and imaginary parts

ary time of the typical O-type stars, the ones that produce the

ionizing radiation. This circumstance points at a possibility

that the stars are still in a transient stage of their formation,

and their luminosity may have varied significantly during

the past years. Such variations, including non-monotonous

variations, with the luminosity passing through a maximum,

is a common phenomenon in the evolution of very young

stars (e.g., Iben and Talbot 1966; Cohen and Kuhi, 1979)).

To get some insights into the possible implications of this

effect, we consider the following simple model: that the ab-

lative pressure “turns on” at t = 0, reaches the maximum

and “turns off” at some t = t0, which is much shorter that

the growth time of perturbations. This model corresponds to

the model of “impulsive acceleration,” which is sometimes

used to imitate the Richtmyer-Meshkov instability. In order

for our model of absorption to work, the time t0 should, on

the other hand, exceed the transit time of the ablated gas over

the distance of the order of 1/k. We will assume that this con-

dition is satisfied, i.e., our results would not be applicable to

very short bursts of radiation.

For simplification, we consider only perturbations with kh

> 2–3, so that we can neglect the feed-through to the back

surface of the cloud and concentrate on what is going on at

the front surface. For the time-dependent ablative pressure,

one can no longer consider the exp(−iωt + ikx x + iky y) de-

pendence of perturbations on time. We have to seek perturba-

tions of the form f (t)exp(ikx x + iky y). Quite analogously

to Eq. (8) but with the acceleration effects included, one then

obtains for the function ξ (t):

∂2ξ

∂t2= k

pa(t)

ρ(h−1 − ikC sin θ cos α − κC)ξ. (12)

Assuming that the initial conditions are ξ (t = 0) = ξ0, ξ (t =0) = 0, one readily finds that for a very short pulse

Springer


ξ (t) = ξ0kvt (1 − ikhC sin θ cos α − κhC) + ξ0, (13)

where v = (1/ρh)∫ t0

0padt . With no tilt and no absorption,

one finds a standard result for the impulsive acceleration.

For normal incidence, large absorption (κhC > 1) causes

the front inversion. If a substantial (θ ∼ 1) tilt is present, the

second term in parentheses in Eq. (13) becomes dominant. It

causes a 90 phase shift in the x direction.

The impulsive acceleration just after the “lighting up”

of the OB-type stars may be an additional mechanism for

launching a subsequent evolution of molecular clouds.

Acknowledgements Work performed under the auspices of the U.S.DoE by UC LLNL under contract No. W-7405-Eng-48; M.W. Pound issupported by NSF Grant No. AST-0228974.

References

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Cohen M., Kuhi L.V.: ApJ. Suppl. 41, 743 (1979)Frieman, E.: ApJ 120, 18 (1954)Hester, J.J., Scowen, P.A., Sankrit, R., et al.: Astron. J. 111, 2349 (1996)Iben, I., Jr., Talbot, R.J.: ApJ 144, 968 (1966)Kahn, F.D.: Rev. Mod. Phys. 30, 1058 (1958)Kane, J.O., Mizuta, A., Pound, M.W., et al.: Astrophys. Space Sci. 298,

261 (2005)Lindl, J.D.: Phys. Plasmas 2, 3933 (1995)Mizuta, A., Kane, J.O., Pound, M.W., et al.: ApJ 621, 803

(2005)Mizuta, A., Takabe, H., Kane, J.O., et al.: Astrophys. Space Sci. 298,

197 (2005)Pound, M.W., Reipurth, B., Bally, J.: Astron J. 125, 2108 (2003)Pound, M.W.: ApJ 493, L113 (1998)Remington, B.A., Weber, S.V., Haan, S.W., et al.: Phys. Fluids B5, 2589

(1993)Ryutov, D.D., Kane, J.O., Pound, M.W., Remington, B.A.: Plasma Phys.

Contr. Fusion 45, 769 (2003)Spitzer, L.: ApJ 120, 1 (1954)Sysoev, N.E.: Astr. Lett. 23, 409 (1997)Takabe, H., Nagamoto, H., Sunahara, A., et al.: Plasma Phys. Contr.

Fusion 41, A75 (1999)Vandervoort, P.O.: ApJ 135, 212 (1962)Williams, R.J.R:. MNRAS 331, 693 (2002)Williams, R.J.R., Ward-Thompson, D., Whitworth, A.P.: MNRAS 327,

788 (2001)

Springer


DOI 10.1007/s10509-006-9257-y


Photoionized Flows from Magnetized Globules

R. J. R. Williams


Abstract Low mass star formation may be triggered by the

dynamical effects of radiation fields and winds from massive

stars on nearby molecular material. The columns of neutral

material observed at the edges of many HII regions may be

the tracers of this process. Magnetic fields are dynamically

important in the molecular clouds from which new stars form,

but their effect on the development of molecular columns has

not been studied in detail. In this paper, I present initial MHD

simulations of this process.

Keywords MHD . Shock waves . ISM: Clouds

1 Introduction

In the region around a young massive star, intense ultraviolet

fields ionize the molecular gas. Columns of neutral gas are

frequently found at the boundaries of these regions, for exam-

ple in the Eagle nebula. As well as the overall morphology of

the columns, observations of optical and molecular line emis-

sion give information about flow velocities in the ionized and

neutral gas, respectively (Pound, 1998). Hydrodynamic mod-

els for the development of these columns have been presented

by Williams et al. (2001), while recent three-dimensional

modelling has demonstrated the formation of columns in tur-

bulent flows (Dale et al., 2005; Mellema et al., 2005). The

radiation-driven collapse of the molecular material at the tips

of these columns is a mechanism by which high-mass star

formation may catalyse further low-mass star formation.

The British Crown reserves the right to retain a non-exclusive, royalityfree licence in and to any copyright.c© British Crown Copyright 2006/MOD.

R. J. R. WilliamsAWE plc, Aldermaston, RG7 4PR, UK

These models, however, neglect the influence of magnetic

fields on the flow. Strong magnetic fields are a common

feature of star forming regions (Crutcher, 1999): the mag-

netic energy density is typically ten times greater than the

thermal energy of the gas, and comparable with the kinetic

energy in unresolved velocity fields.

Bertoldi (1989) discussed the effect of magnetic fields on

the evaporation of dense globules with an approximate an-

alytic approach. He describes the structures expected in the

initial collapse, but notes that numerical calculations will be

necessary to follow the evolution to late time. Ryutov et al.

(2005) have discussed the influence of magnetic fields on the

development of ionized columns. They suggest that magne-

tostatic turbulence may explain the support of the columns

which appear to have internal pressures significantly less than

those in the surrounding ionized nebula. Note, however, that

molecular line widths suggest the gas is in fact moving with

sufficient energy density to provide support (Williams et al.,

2001), although the rapid decay of turbulence in MHD sim-

ulations (Mac Low, 1999) means it is unclear how these high

velocities can be maintained. Ryutov et al. also argue that

magnetic tension in the swept-back field may significantly

suppress the development of columns, unless strong recon-

nection processes are present. While magnetic field support

may suppress the initial formation of a cool thin shell be-

tween the ionization front and leading shock (Ryutov et al.,

2005), one-dimensional analysis suggests that the subsequent

emission of a slow-mode shock from the interface can al-

low the material to collapse to high density (Williams et al.,

2000). While it has been argued that plasmas with high mag-

netic pressures are intrinsically unstable (Falle and Hartquist,

2002), it is not, however, clear that this organized loss of mag-

netic field support will be possible in multiple dimensions.

In this paper, we model this photoionization process us-

ing a two-dimensional magnetohydro-dynamic code. As a

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Fig. 1 Structure of flow at latetime, with a weak initialmagnetic field perpendicular tothe impinging radiation. Greyscale is log density, and thecontours are magnetic fieldlines. The location of theionization front is at x ≃ 2,where the density changesrapidly

simple initial condition, we assume that a uniform density

field contains one or more clumps of dense, cool gas. These

clumps have an initial density of nH = 2 × 105 cm−3, ten

times higher than the surrounding diffuse material, and are

initially in hydrostatic equilibrium. The pressure in the ma-

terial corresponds to an initial temperature of 10 K in the

clumps and 100 K in the gas which surrounds it. A simple

γ = 5/3 equation of state is used, as the material is pre-

dominantly either ionized or cool enough to suppress rota-

tional and vibrational modes. Once ionized, the material is

assumed to relax to a temperature between 100 K and 104 K

as a function of ionization fraction on a timescale given by

the ionization rate. This artificial prescription avoids numer-

ical problems which can result from the explicit treatment

of thermal source terms in regions with unresolved heating

fronts, as the emphasis here is on dynamical processes rather

than detailed modelling.

The numerical grid covers 2 pc × 4 pc, with a uniform nu-

merical resolution of 800 × 1600 cells. All of the boundaries

are set to allow free flow of material out of the grid, but no

inflow. The use of these boundary conditions maintains the

initial conditions in equilibrium, but allows material to ex-

pand freely from the ionized regions of the flow, to emulate

the effect of large-scale divergent geometry characteristic of

blister H II regions.

The region as a whole is threaded by a uniform magnetic

field. There is evidence for that the magnetic fields are or-

dered on large scales in molecular clouds (Ward-Thompson

et al., 2000). An alternative would be to use the results of

a turbulent flow simulation to initialize the flow. However,

due to the uncertainty about how the observed flow speeds

are maintained, the results here would remain dependent on

uncertain assumptions. For the present work, a quiet start is

appropriate.

2 Numerical methods

We have modelled the photoevaporated flows from mag-

netized globules using the two-dimensional MHD code

ATHENA (Gardiner and Stone, 2005).

We assume that the flow is governed by the equations

of MHD, with additional ionization balance terms. The

coupling distances between ions and neutral species are

far smaller than those which characterize the dynamics

(Williams, 2006), which is generally taken to be a sufficient

condition for the validity of these equations. Note, however,

that Tytarenko et al. (2002) found that in these circumstances,

accelerated flows are subject to strong instabilities which ini-

tially segregate the ionized and neutral components of the gas

and eventually result in the generation of highly turbulent

flows.

The direct component of the ionizing continuum, of in-

tensity 1012 cm−2 s−1, is plane parallel and incident parallel

to a co-ordinate axis. Diffuse radiation is treated using the

case B assumption, as is usual. Transfer of the direct contin-

uum, and its coupling to the ionization balance of the gas, is

treated using an implicit scheme similar to that detailed by

Williams (2002). In the present work, we have also used an

improved model of the thermal balance. Instead of assuming

the material heats (or cools) to an equilibrium temperature

corresponding to its ionization fraction immediately, as as-

sumed in Williams (2002), we limit the rate of heating to that

provided by the absorption of ionizing radiation.

3 Simulation results

3.1 Weak field

We first consider the flow from a single large clump where

the initial magnetic pressure is ∼10× the thermal pressure,

rather than ∼10× as observed. The clump is initially at (0.5,

0.5) pc and has radius 0.15 pc.

In Fig. 1, we show the flow structure which develops.

Compression and shear flow driven by the increased pressure

in the photoevaporating gas soon increase the magnetic field

strength in parts of the flow. The field in the ionized wind

is dragged perpendicular to the surface of the column, with

the magnetic field concentrated into a tulip-shaped region

surrounding the barrel of the column. The flow from the tip

Springer


Fig. 2 Initial field at 45 to theimpinging radiation, andmultiple initial clumps

Fig. 3 Structure of flow at latetime, with a strong initialmagnetic field parallel to theimpinging radiation

is supersonic, while around the barrel, significant variations

in flow speed occur along streamers following field lines from

the surface. The velocity field within the column is structured,

with several 2–4 km s−1 shocks.

The main surface of the ionization front has a more irreg-

ular structure, with numerous small clumps pointing into the

ionized region.

Behind the clump, the magnetic field is swept back into a

concentrated core, with its strength limited by reconnection

(as inferred by Ryutov et al., 2005). No explicit resistivity is

included in these simulations – instead, the reconnection is

caused entirely by the effects of numerical resisitivity. How-

ever, the reconnection occurs in regions of the flow where

we would expect that, for smaller but finite resistivity, even

higher small-scale field line curvature will result in the large-

scale effects of reconnection being similar to those found

here.

The results are strikingly similar for other initial angles

for the field.

3.2 Multiple clumps

Figure 2 shows the structure of the flow found for a model in

which the dense gas was initially in a regular array of small

clumps with radius 0.02 pc with a centre–centre separation

of 0.1 pc. The large-scale form of the flow is similar to that at

the head of one of the simulations seeded by a single dense

clump, as might be expected.

On smaller scales, the tension of the field swept back by

the movement of the shocked diffuse field tends to aggregate

the dense clumps. As the shock progresses, the initial com-

pression by the shock tends to make the clumps longer in

the direction of the radiation field, unlike the hydrodynam-

ical case where the initial compression produces an oblate

structure. While the velocity field in the neutral gas is highly

variable, the influence of the magnetic field means that it is

organized rather than turbulent.

3.3 Strong field

With an initial field with magnetic pressure 10 times the gas

pressure, comparable to the situation in observed molecular

clouds, magnetic effects dominate the dynamics, and the in-

creased pressure in the region heated by photoionization is

far less significant for the flow.

For a magnetic field parallel to the surface of the molecu-

lar cloud, the field pressure cushions the gas, and the dense

clump can be accelerated with little disruption. The surface of

the molecular cloud is only weakly perturbed by the presence

of the condensation.

More interesting is the case where the initial field is

perpendicular to the ionization front, shown in Fig. 3. In this

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case, the gas is constrained to move close to the initial field

direction. The overall development of the flow is similar to

that which would be expected for a set of one-dimensional

simulations. A plug of low-density gas is trapped between

the pressure of the ionized gas and the inertia of the dense

core, with higher density than the surrounding material

flowing from the unperturbed front.

Looking in more detail at the simulations, however, deep

fingers of ionized gas intrude into the neutral material. These

propagate across the surface of the main ionization front away

from the dense core, and consist of slow-mode shocks fol-

lowed by oblique ionization fronts.

The leading shock has an overall switch-on/switch-off in-

ternal structure. The second, slow-mode, sub-shock also has

a corrugated surface, suggestive of the instability mechanism

discussed by Stone and Edelman (1995).

For an oblique initial magnetic field, the early-stage evo-

lution is similar to that for parallel field. However, in this

gas the trapped plug of neutral gas is offset, and does not

entirely shield the dense clump from the radiation. When

eventually the ionization front reaches the dense material, a

weakly-magnetized transonic wind forms at the surface.

4 Conclusions

Magnetic fields are seen to have a significant influence on the

development of the columns around H II regions, for strengths

comparable with those observed.

With weak initial fields, shocks and shear flows amplify

the field until it becomes dynamically important. For single

clumps, the flows are remarkably similar, whatever the initial

orientation of the magnetic field. Once an ablation flow has

established, the magnetic field is oriented perpendicular to

surface of the neutral gas and is concentrated into intermit-

tent tubes. These tubes correlate with velocity and density

modulations in the ionized wind from the clump. The wider

surface of the cool material has a turbulent structure, with

the magnetic field concentrated into sheets.

For multiple clumps, the flows are more complex, with

highly structured ablation flows. In the neutral region, the

interaction between the leading shock and the globules again

leads to the formation of a highly turbulent flow threaded by

intense sheets of magnetic flux.

For stronger initial fields, the pressure increase due to ion-

ization becomes a less significant influence, but the develop-

ment of the flows is more strongly influenced by the initial

magnetic field. Ionization fronts and shocks are subject to

strong instabilities, as might be expected for such low flows

(Stone and Edelman, 1995; Falle and Hartquist, 2002).

Important questions remain, such as the form of the

flows in fully three-dimensional situations, and the processes

which control reconnection in the multicomponent molecular

material.

References

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(eds.), Diffuse Matter from Star Forming Regions to ActiveGalaxies. Springer, in press (2006)

Springer


DOI 10.1007/s10509-006-9252-3


Nonlinear Dynamics of Ionization Fronts in HII Regions

Akira Mizuta · Jave O. Kane · Marc W. Pound ·

Bruce A. Remington · Dmitri D. Ryutov ·

Hideaki Takabe


Abstract Hydrodynamic instability of an accelerating

ionization front (IF) is investigated with 2D hydrodynamic

simulations, including absorption of incident photoionizing

photons, recombination in the HII region, and radiative

molecular cooling. When the amplitude of the perturbation is

large enough, nonlinear dynamics of the IF triggered by the

separation of the IF from the cloud surface is observed. This

causes the second harmonic of the imposed perturbation

to appear on the cloud surfaces, whereas the perturbation

in density of ablated gas in the HII region remains largely

single mode. This mismatch of modes between the IF and

the density perturbation in the HII region prevents the strong

stabilization effect seen in the linear regime. Large growth of

the perturbation caused by Rayleigh-Taylor-like instability

is observed late in time.

Keywords HII regions . ISM: molecules . ISM: kinematics

and dynamics . Hydrodynamics . Instabilities . Methods:

numerical . ISM individual object: M16

A. Mizuta ()Max-Planck-Institute fur Astrophysik, Karl-Schwarzschild-Str. 1,85741 Garching, Germanye-mail: [email protected]

J. O. Kane · B. A. Remington · D. D. RyutovUniversity of California, Lawrence Livermore NationalLaboratory, 7000 East Ave., Livermore, CA 94551, USA

M. W. PoundDepartment of Astronomy, University of Maryland, College Park,MD 20742 USA

H. TakabeInstitute of Laser Engineering, Osaka University, 2-6 YamadaOka, Suita, Osaka, 565-0871, Japan

1 Introduction

Columns or pillars are common structures seen in HII re-

gions. One of the most popular examples of this structure

is the Eagle Nebula, which has three famous pillars beside

some O stars (Hester et al., 1996). The boundary between the

cloud and HII region is an ionization front (IF) where pho-

toevaporation occurs, resulting in photoevaporated flow. The

HII region consists of almost fully ionized hydrogen and is

isothermal at T ∼ 104 K. The inside of the pillars consists of

dense and cold molecular hydrogen. The hydrogen number

density in the pillar in the Eagle Nebula is ∼104–105 cm−3

and the temperature is about a few tens of Kelvins (Pound,

1998). Pound found the velocity gradient along pillar from

the head to the bottom (see also Pound et al., 2006; Kane et al.,

2006 in this volume). Dense clumps, some of which will be-

come young stars, are observed in the pillar (McCaughrean

and Andersen, 2002). Since this cloud is optically thick for

incident photons, the thickness of the IF is very thin. Because

of the similarity of this phenomena with laser ablation, scaled

laboratory experiments using laser ablation are proposed to

study IF dynamics (Kane et al., 2005).

It is still not fully understood how the pillars form, though

several hypotheses have been proposed. For example, some

models are based on hydrodynamic instability of the IF.

Spitzer (1954) proposed a model that the shape of pillars

are at due to nonlinear phase of the Rayleigh-Taylor instabil-

ity which occurs when a light fluid accelerates a denser fluid.

Vandervoort (1962) theoretically found unstable modes at an

IF without acceleration, but the important role recombina-

tion plays in the HII region was not included. Axford (1964)

extended Vandervoort’s work, including recombination and

found that recombination in the HII region works to stabi-

lize the perturbation, as suggested by Kahn (1958). Sysoev

(1997) did more complete analysis and found the growth

Springer


of long-wavelength instabilities for normally incident radi-

ation. Williams (2002) confirmed this with 2D simulations.

He also included effects of the radiation tilt in the analysis.

The effect of tilted rays of the incident radiation to the IF

with acceleration was theoretically studied by Ryutov et al.

(2003, 2006). Recently (Mizuta et al., 2005) have numer-

ically studied an accelerating IF and concluded that stabi-

lization due to recombination works even with acceleration

in the linear regime (i.e. small amplitude of the perturba-

tions). This paper discusses cases with larger initial imposed

perturbation.

2 Model

We studied the dynamics of an accelerating IF by varying

the initial amplitude of the imposed perturbations. The 2D

hydrodynamic equations are solved using the code described

in Mizuta et al. (2005). The energy equation includes sources

of cooling and heating due to absorption of the incident pho-

tons, recombination in the HII region, and radiative molecular

cooling. The transport equation for the incident photoioniz-

ing photons is also solved, considering the photon absorption

by neutral hydrogen and recombination of ionized hydrogen.

On the other hand, we do not consider far ultraviolet photons

which will act to heat gas in the photo dissociation region

behind the IF. Recombination to the ground state is ignored,

assuming that the diffusive photon is locally absorbed (on

the spot approximation). See Mizuta et al. (2005) for more

details.

A 0.46 pc × 3 pc computational box with 184 × 1200

uniform grid points is used. Periodic boundary conditions

are employed at x = 0 and x = 0.46 pc. Outflow boundary

condition is imposed at y = 0 and y = 3 pc. A quarter pc

thickness finite cloud is located at a distance of 0.5 pc from

the boundary at y = 3 pc, where the incident photon flux

comes in. The hydrogen number density n(H) of the cloud

and other regions are 105 and 10 cm−3, respectively. After

the compression through a shock, the density of the cloud

is a few times 105 cm−3 which is comparable to current ob-

served number density at the pillar in the Eagle Nebula. The

region y < 2.5 pc is isothermal with T = 40 K and the region

y > 2.25 pc is pressure matched. A constant incident photon

number flux of 5 × 1011 cm−2 s−1 is taken to be parallel to

the y axis. When a neutral hydrogen atom absorbs an incident

photon and becomes ionized, an internal energy increment of

1.73 × 10−12 erg is locally deposited into the gas. An isother-

mal state whose temperature is about 104 Kelvin is achieved

in the HII region as a result of the balance between this heat-

ing by photon absorption and cooling by the recombination

of the ionized hydrogen. The molecular cloud is very cold

(∼40 Kelvin) due to the strong radiative molecular cooling,

even if shocks cross in the cloud.

A sinusoidal surface perturbation is initially imposed.

Here we have studied four different initial amplitude per-

turbations: 3.8 × 10−3 pc (case S038), 9.0 × 10−3 pc (case

S090), 1.4 × 10−2 pc (case S140), and 1.9 × 10−2 pc (case

S190), respectively. Figure 1(a) shows the number density

contour of the initial condition, where the initial amplitude

is 9.0 × 10−3 pc.

3 Results and discussions

Figure 2 shows the amplitude of the perturbation as a function

of time, where the amplitude is defined as half the peak-to-

valley amplitude of the contour corresponding to an ioniza-

tion fraction of f = 0.5. Since the IF sometimes separates

from the cloud surface, this amplitude shown in Fig. 2 does

not always correspond to the amplitude of the perturbation at

the cloud surface. Case S038 shows a small amplitude which

oscillates with time, and does not grow, as shown in Mizuta

et al. (2005). When the incident photon flux comes in, the

flow of the photoionized plasma begins from cloud surface.

Since this flow is normal to the cloud surface, the density of

the photoevaporated flow around the bubble region (surface

concavities at x = 0, 0.46 pc) becomes higher than around

spike region (x = 0.23 pc), due to focusing effect. Higher

number density results in stronger absorption of the incident

photon flux in the HII region. As a result, the ablation pressure

around the bubble region becomes lower than that around the

spike region. The local difference of the ablation pressure

works to strongly stabilize the perturbation in linear regime.

The amplitude of the other cases, however, increase, with

time, in striking contrast to the strong stabilization observed

in the linear regime. The reason is due to the ‘separation of

the IF’ from the cloud surface. Such cases were theoretically

studied by Newman and Axford (1968) and Beltrametti et al.

(1982) in other contexts, assuming spherical symmetry. The

strong absorption of incident photon around the bubble oc-

curs, reducing the ablation pressure there, also in the larger

initial amplitude case. The surface perturbation inverts phase

completely during the shock propagates in the dense cloud

(Fig. 1(b) and (c)). The number density around the bubble

region becomes higher as the IF becomes strongly concave.

The increasing density around the bubble in the HII region

causes absorption of all incident photons before the cloud

surface as shown in Fig. 1(d), since the number of incident

photons is finite. This causes the separation of the IF from the

cloud surface, which has not been considered before. When

the separation of the IF from the cloud surface occurs, a tiny

and warm HI region (neutral hydrogen) appears between the

cloud surface and the IF, since the gas was once ionized and

recombined to neutral hydrogen. The cloud surface locally

does not feel any ablation pressure and expands in the y di-

rection, when the separation of the IF appears.

Springer


Fig. 1 Number densitycontours (color) and incidentphoton number flux contours(solid white) at intervals of1 × 1011 cm−2 s−1, startingfrom the IF where ionizationfraction (f) goes to zero. Thetimes shown correspond to (a) 0kyr, (b) 30 kyr, (c) 60 kyr, (d) 90kyr, (e) 120 kyr, (f) 200 kyr, (g)340 kyr and (h) 480 kyr,respectively. As an example, theIF is indicated with an arrow in(f), a case where there is clearseparation with the cloudsurface (ablation front). Theseparation of the IF can be seenin (d), resulting the appearanceof the tiny and warm HI region(neutral hydrogen atom).Figures are taken from Mizuta etal. (2006) and reproduced bypermission of the AAS

0.001

0.01

0.1

0 100 200 300 400 500

Am

plit

ud

e (

pc)

Time (kyr)

S190S140

S038S090

Fig. 2 Time evolution of the“amplitude” of the perturbationfor cases S038, S090, S140, andS190. Figure is taken fromMizuta et al. (2006) andreproduced by permission of theAAS

Springer


After a few tens of kyr, the IF corresponds to the cloud

surface again. But the perturbation on the cloud surface is not

longer single mode; a second harmonic of the imposed per-

turbation has appeared (Fig. 1(e)). To the contrary the density

perturbations in the HII regions above the cloud surface are

still single mode (Fig. 1(e)). This mismatch between the per-

turbation modes on the cloud surface and those in the density

of the ablated plasma mean that the stabilization observed in

the linear regime will not occur. (Recall, this stabilization

requires that these two perturbations be “mode locked” 180

degrees out of phase.) As a result, a Rayleigh-Taylor-like in-

stability evolves and large growth of the second harmonic of

the imposed perturbation appears (Fig. 1(f)–(h)). There still

remains another possibility for the growth of the perturba-

tions. That is thin-shell instability for the ionization shock

front which caused by the unbalanced forces between the

ram pressure to the shock front and thermal pressure by the

HII region (Garcia-Segura and Franco, 1996).

We can measure the velocity gradient along the y axis

as Pound observed in the pillars in the Eagle Nebula. We

find the velocity gradient to be about 12 km s−1 pc−1 in the

central column at t = 480 kyr. This is good agreement with

observed one (an average magnitude of 8.3 km s−1 pc−1).1

Although we fixed the wavelength of the initially im-

posed perturbations in this paper, the essentially nonlinear

dynamics is observed from the beginning of the simulation

in cases which show the evolution of the columns. The case

of multi mode perturbation should be studied in the near

future.

The dynamics described in this paper is good target to

model the scaled laboratory experiments, since theoretical

analysis is difficult to do because of the nonlinear dynamics.

We will demonstrate some numerical simulations to seek

the possible laboratory experiments to study the instability

presented in this paper.

4 Conclusion

We present a new type of the instability for the accelerat-

ing IF triggered by the separation of the IF from the cloud

surface. When the initial amplitude of the perturbation is

small enough, the strong stabilization is observed. When

1 This value does not include the effect of the inclination angle.

the surface becomes concave, the number density increases.

Strong absorption through this region locally reduces the ab-

lation pressure as theoretically discussed by Axford (1964)

for the non-accelerating IF.

To the contrary, when the initial amplitude of the pertur-

bation exceeds a critical value, roughly when the ratio of

the initial amplitude to wavelength is greater than 0.02, the

nonlinear dynamics is triggered by the separation of the IF.

The separation of the IF causes the appearance of the second

harmonic of the imposed perturbation. The perturbation in

density in the HII region, however, remains largely single

mode. This mismatched of the modes between the perturba-

tions on the cloud surface and in the density of the ablated

flow prevents the stabilization effect seen in the linear regime.

A kind of Rayleigh-Taylor instability takes over. The large

growth of the second harmonic of the imposed perturbation

is observed in the later phase.

Acknowledgments Work performed under the auspices of the U.S.Department of Energy by the Lawrence Livermore National Labora-tory under Contract No. W-7405-ENG-48 and with support from NASAGrant NRA 00-01-ATP-059 and from National Science Foundation un-der Grant No. AST-0228974. MWP supported by NSF Grant No. AST-0228974.

References

Axford, W.I.: ApJ 140, 112 (1964)Beltrametti, M., Tenorio-Tagle, G., Yorke, H.W.: A&A 112, 1 (1982)Garcia-Segura, G., Franco, J.: ApJ 469, 171 (1996)Hester, J.J., et al.: AJ 111, 2349 (1996)Kane, J.O., et al.: Ap&SS 298, 261 (2005)Kane, J.O., et al.: Ap&SS, this issue (2006)Kahn, F.D.: Rev. Mod. Phys. 30, 1058 (1958)McCaughrean, M.J., Andersen, M.: A&A 389, 513 (2002)Mizuta, A., et al.: ApJ 621, 803 (2005)Mizuta, A., et al.: ApJ 647, 1151 (2006)Newman, R.C., Axford, W.I.: ApJ 151, 1145 (1968)Pound, M.W.: ApJ 493, L113 (1998)Pound, M.W., et al.: Ap&SS, this volume (2006)Sysoev, N.E.: Astronomy Letters 23, 409 (1997)Ryutov, D.D., et al.: Plasma Physics and Controlled Fusion 45, 769

(2003)Ryutov, D.D., et al.: Ap&SS, this volume (2006)Spitzer, L.J.: ApJ 120, 1 (1954)Vandervoort, P.O.: ApJ 135, 212 (1962)Williams, R.J.R.: MNRAS 331, 693 (2002)

Springer


DOI 10.1007/s10509-006-9214-9


Pillars of Heaven

Marc W. Pound · Jave O. Kane · Dmitri D. Ryutov ·

Bruce A. Remington · Akira Mizuta


Abstract Sometimes the most beautiful things are the hard-

est to understand. Pillars like those of the Eagle Nebula form

at the boundary between some of the hottest (10000 K) and

coldest (10 K) gas in the Galaxy. Many physical processes

come into play in the birth and growth of such gaseous pillars:

hydrodynamic instability, photoionization, ablation, recom-

bination, molecular heating and cooling, and probably mag-

netic fields. High-quality astronomical observations, quan-

titative numerical simulations, and scaled laser experiments

provide a powerful combination for understanding their for-

mation and evolution.

We put our most recent hydrodynamic model to the test,

by creating simulated observations from it and comparing

them directly to the actual radioastronomical observations.

Successfully reproducing major characteristics of the obser-

vations in this manner is an important step in designing ap-

propriate laser experiments.

Keywords Eagle Nebula . Radio astronomy .

Hydrodynamic models . Aperture synthesis

1. Introduction

The pillars of the Eagle nebula are the most spectacular exam-

ple of a phenomenon that is commonly seen wherever molec-

M.W. Pound ()Astronomy Department, University of Maryland, College Park,MD 20742

J.O. Kane . B.A. Remington . D.D. RyutovLawrence Livermore National Laboratory, Livermore, CA 94551

A. MizutaMax-Planck-Institut fur Astrophysik, Garching, 85741, Germany

ular clouds are situated near O stars. Proposed formation

mechanisms for such pillars generally fall into two broad cat-

egories: (i) instabilities at the boundary between the cloud and

the ionized region which grow with time (e.g. Spitzer, 1954;

Frieman, 1954; Williams et al., 2001; Mizuta et al., 2005a,b)

and (ii) pre-existing density enhancements (i.e., clumps)

which locally retard the ionization front creating “cometary

globules” (Reipurth, 1983; Bertoldi and McKee, 1990).

We have developed a comprehensive, 2-D hydrodynamic,

cometary globule model of pillar formation (Mizuta et al.,

2005; Kane et al., in this volume) that includes energy depo-

sition and release due to the absorption of UV radiation, re-

combination of hydrogen, radiative molecular cooling, mag-

netostatic pressure (Ryutov et al., 2002), and geometry/initial

conditions based on Eagle observations. Pillar formation by

both ionization-front instability (Mizuta et al., 2005b and

dense cores (Kane et al.) have been examined; both meth-

ods can grow a pillar a few tenths of a parsec long in a few

hundred thousand years. This timescale is comparable to the

dynamic time measured for the Eagle Pillars (Pound, 1998).

The CO(J = 1−0) observations, taken with the Berkeley-

Illinois-Maryland interferometer are those of Pound (1998),

with the addition of more recent higher spatial resolution

data (see Figure 1b of Pound et al., 2005). To facilitate com-

parison between model and observations, we create “syn-

thetic observations” from the model by filtering it through

the known telescope response function and processing the

resultant data using identical methods as for the observa-

tions to produce maps. The details of this technique are de-

scribed in Pound et al. (2005). In that paper, we compared

synthetic integrated intensity maps of the Mizuta et al. (2005)

instability- model pillars to the observed maps. Here, we take

the Kane et al. cometary-model pillar (Figure 1) and perform

a similar analysis but with a significant improvement: the new

model allows us to create full synthetic data cubes (position,

Springer


Fig. 1 Phases of pillar growth from Kane et al. model; (left) initial con-dition of a dense core embedded in a molecular cloud, (middle) after125,000 years, (right) after 250,000 years with total length of about 2 pc.

The model at 250,000 years is taken as the basis for creating syntheticobservations

Fig. 2 (left) The syntheticintegrated intensity map derivedfrom processing the model.(right) The actual integratedintensity map from Pound(1998). Maps are to samephysical scale and use identicalintensity range. Lines indicatelocations of position-velocitycuts shown in Figures 3 and 4

position, velocity). The addition of velocity information pro-

vides a powerful test of the model, as it allows detailed com-

parison of the model and observed gas dynamics.

2. Results

Figure 2 shows the synthetic and actual integrated intensity

maps. Integrated intensity is a measure of the amount of gas

along the line of sight. Brighter intensity means higher molec-

ular hydrogen column density and, since the pillars are as-

sumed to be roughly cylindrically symmetric, also indicates

higher particle density. The size and shape of the model pillar

reasonably match that of Eagle Pillar II (the middle of the 3

Eagle pillars). Furthermore, the final average particle density

of the model volume density is close the that inferred from

the observations However, there does not appear to be enough

material in the model “tail”; it is underdense compared with

Pillar II.

Figure 3 shows the position-velocity (p-v) diagrams along

the length of the synthetic pillar and Eagle Pillar II. Both show

a velocity gradient from “head” to “tail”, with the observed

gradient being slightly larger. However, since any measured

velocity gradient (synthetic or observed) is a function of in-

clination angle, decreasing the inclination angle of the model

pillar can make the gradients match. The extent of the emis-

sion on the velocity axis is a measure of the internal velocity

dispersion of the gas. One can see that the observed velocity

dispersion is about 2.5 times greater than the synthetic one.

This should not be surprising since no turbulent support was

put into the model; this is an area where the model could be

improved.

Figure 4 shows the p-v diagrams across the head of the

synthetic and observed pillars. There are two features of

the synthetic p-v diagram (upper panel) worth noting. The

first is overall shape: rounded contours on the top and flat-

tened contours on the bottom. This is what would be ex-

pected of “inside-out” velocity shear—that is, material inte-

rior to the pillar is flowing more slowly than its surface (see

inset). The second feature is the two symmetrically-placed

bright spots indicative of limb-brightening. Both these fea-

tures make sense in light of the dynamics of the model: the

massive core in the head resists motion and the lighter mate-

rial gets pushed around it, sweeping back a cometary shape

with a dense outer shell. Material directly behind the head is

less affected and flows more slowly. The observed p-v dia-

gram (lower panel of Figure 4) shows neither of these fea-

tures. There are two bright spots, but they are not symmetric

and are likely two individual cores. There is no evidence of

inside-out velocity shear.

Springer


Fig. 3 (top) The syntheticposition-velocity diagram alongthe length of the model pillar.The “head” of the pillar is on theleft, the “tail” on the right.(bottom) Actual length-wiseposition-velocity diagram ofEagle Pillar II

Fig. 4 (top) The syntheticposition-velocity diagram acrossthe head of the model pillar.Double-peak is indicative oflimb-brightening and overallshape (curved outer contours ontop, flat contours on bottom)indicates velocity shear. Inset attop right shows velocity flowpattern which would producesuch features in a p-v diagram(bottom) Actual cross-wiseposition-velocity diagram ofEagle Pillar II

3. Conclusions

The cometary model can produce a large pillar with roughly

the correct size and shape within the measured dynamical

timescale. The model starts with an isolated 30 M⊙ core in

a lower density envelope. This produces a pillar head with

properties like the Eagle’s but without enough material in

the tail. However, the structure of the Eagle pillars is clearly

not as simple as a single core, and that suggests the next

step to enhance the simulation. Models with multiple cores

of different size and mass should increase both the amount

of tail material as well as the internal velocity dispersion.

Acknowledgements Work performed under the auspices of the U.S.DoE by UC LLNL No W-7405-Eng-48. MWP supported by NSF GrantNo. AST-0228974.

References

Bertoldi, F., McKee, C.F.: ApJ 354, 529 (1990)Frieman, E.A.: ApJ 120, 18 (1954)Mizuta, A., Kane, J.O., Pound, M.W., Remington, B.A., Ryutov, D.D.,

Takabe, H.: ApJ 621, 803 (2005)Mizuta, A., Takabe, H., Kane, J.O., Pound, M.W., Remington, B.A.

Ryutov, D.D.: Astrophysics & Space Science 298, 197 (2005)

Springer


Pound, M.W.: ApJ 493, L113–L116 (1998)Pound, M.W., Reipurth, B., Bally, J.: AJ 125, 2108 (2003)Pound, M.W., Kane, J.O., Remington, B.A., Ryutov, D.D., Mizuta, A.,

Takabe, H.: Astrophysics & Space Science 298, 177 (2005)Reipurth, B.: A&A 117, 183 (1983)

Ryutov, D., Kane, J., Mizuta, A., Pound, M., Remington, B.: APS Meet-ing Abstracts, 1004P (2002)

Spitzer, L.: ApJ 120, 1 (1954)Williams, R.J.R., Ward-Thompson, D., Whitworth, A.P.: MNRAS 327,

788 (2001)

Springer


DOI 10.1007/s10509-006-9182-0


The Evolution of Channel Flows in MHD Turbulence Drivenby Magnetorotational Instability

T. Sano

Received: 14 April 2006 / Accepted: 22 May 2006C© Springer Science + Business Media B.V. 2006

Abstract MHD turbulence driven by magnetorotational in-

stability (MRI) in accretion disks is investigated using the

local shearing box calculations. The growth of many short-

wavelength MRI modes, which are called “channel flows”, is

found in the spatial distribution of the current density. These

small channel flows can be regarded as a unit structure of

MRI driven turbulence. Nonlinear evolution of the channel

flow affects the saturation amplitude and time variability of

the Maxwell stress. Exponential growth of a channel mode

is stopped by the Kelvin-Helmholtz type instability which

triggers the subsequent magnetic reconnection. The charac-

teristics of the magnetic reconnection are consistent with the

Sweet-Parker model. These studies of the nonlinear evolution

of the channel flow are required to understand the saturation

mechanism of the MRI.

Keywords Accretion disks . MHD . Turbulence

1 Introduction

Magnetorotational instability (MRI) is the most promising

source of angular momentum transport in accretion disks

(e.g., Balbus & Hawley, 1998). Local and global simulations

of magnetized accretion disks have revealed that the Maxwell

stress in MHD turbulence driven by the MRI can transport

angular momentum significantly (e.g., Hawley et al., 1995;

Hawley, 2000). However the nonlinear saturation mechanism

of the MRI have not been understood yet, so that what deter-

mines the saturation amplitude of the stress, or the size of the

α parameter of Shakura & Sunyaev (1973), is still unclear.

T. Sano ()Institute of Laser Engineering, Osaka University, Suita,Osaka 565-0871, Japane-mail: [email protected]

It is important to investigate throughly the nature of MHD

turbulence in accretion disks to understand the saturation

processes of the MRI.

In this paper, we focus on the nonlinear evolution of chan-

nel flow in MRI driven turbulence. The channel flow is an

unstable mode of axisymmetric MRI whose wavevector is

parallel to the rotation axis. The channel mode is impor-

tant because it is the equal fastest-growing of all linear MRI

modes, even including those with non-zero radial and az-

imuthal wavenumbers. The eigenfunctions of this mode sat-

isfy not only the linearized MHD equations but also the non-

linear equations in the incompressible limit (Goodman & Xu,

1994). MRI driven turbulence can be treated as incompress-

ible because the turbulent velocity is about the Alfven speed

and much smaller than the sound speed. Thus the amplitude

of channel flow can grow exponentially even in the nonlinear

regime. The nonlinear growth of channel modes is the most

efficient mechanism of field amplification in the disks.

In two-dimensional axisymmetric simulations under the

ideal MHD approximation, a two-channel flow appears at

the nonlinear stage and continues to grow without saturation

(Hawley & Balbus, 1992). When the vertical wavelength of a

channel flow is fitted to the height of computational domain

or the disk thickness, it is called a two-channel flow. The ver-

tical length scale of MRI modes increases as the magnetic

field is amplified, and finally becomes comparable to the box

size. A two-channel flow consists of two streams moving ra-

dially inward with a sub-Keplerian rotational velocity and

radially outward with a super-Keplerian rotational velocity.

As the amplitude of two-channel flow increases, strong ver-

tical shear is developed which can generate and amplify the

oppositely directed horizontal fields. Thus the growth of the

two-channel mode is associated with both the development

of the vertical shear in the horizontal flow and the formation

of a pair of current sheets.

Springer


Fig. 1 Spatial distributions of (a) the current density |J| and (b) the az-imuthal component of the current Jy in MRI driven turbulence drawn ona radial-vertical slice. Snapshot data are obtained from a local shearingbox simulation including the ohmic dissipation. The initial field geom-etry is a zero net flux Bz and the plasma beta is β0 = 104. The magneticdiffusivity is assumed to be uniform and the initial Lundquist number

is SMRI = 20. Model parameters are identical to those of model S52r inSano et al. (2004) except for the grid resolution (128 × 512 × 128). Thedata are taken at t/trot = 100. The gray colors are linearly scaled (a)from 0 (black) to 814 J0 (white) and (b) from −250J0 (black) to 250J0

(white) where J0 ≡ cB0/4πL z . The growth of many channel modeswith short wavelengths can be recognized in this figure

If the ohmic dissipation is taken into account, magnetic

reconnection could break up the structure of channel flow. On

the other hand, the strong shear brings the Kelvin-Helmholtz

type instability (Goodman & Xu, 1994) which could also in-

terrupt the growth of a channel mode. The saturated turbulent

state is achieved by a balance between the growth of the MRI

(channel modes) and the field dissipation through magnetic

reconnection. Then the breakup of the channel flow must be

relevant to the saturation mechanism.

In the following sections, we demonstrate the importance

of the channel flow in MRI driven turbulence and examine

the nonlinear evolution of a channel mode. The local shear-

ing box calculations including the ohmic dissipation are most

appropriate for this purpose. Low Lundquist number simu-

lations are inevitable to resolve the dissipation scale. Such

simulations would be quite useful to compare the properties

of turbulence with various theories of incompressible MHD

turbulence (e.g., Schekochihin et al., 2004) and applicable di-

rectly to the dynamics of protoplanetary disks (Stone et al.,

2000; Sano et al., 2000; Inutsuka & Sano, 2005).

2 Presence of Channel Flows

When accretion disks are penetrated by a uniform verti-

cal field Bz , a two-channel flow appears quasi-periodically

during the turbulent phase in three-dimensional simulations

(Sano & Inutsuka, 2001). The channel flow is related to large

time variability of the magnetic energy and the Maxwell

stress. Exponential growth of the two-channel mode and

exponential decay of the magnetic field by reconnection are

the origin of spike-shaped variations in the magnetic energy.

The joule heating contributes almost all the increase of the

thermal energy. Thus the nonlinear evolution of the MRI is

characterized by the two-channel flow for this case.

On the other hand, if the net magnetic flux of the shearing

box is zero, the two-channel flow never appears in the nonlin-

ear regime. Instead, many small-scale structures can be seen

in the spatial distribution of the magnetic field. We find that

these small fluctuations have the similar field geometries to

the channel modes. Figure 1shows snapshots of the current

densities |J| and Jy during the turbulent phase. Model pa-

rameters are identical to those of model S52r in Sano et al.

(2004) except that the grid resolution is 128 × 512 × 128 in-

stead of 32 × 128 × 32. The initial Lundquist number for the

MRI is SMRI ≡ v2A/η = 20, where vA = Bz/(4πρ)1/2 is the

Alfven speed, η is the magnetic diffusivity, and is the an-

gular velocity. Numerical resolution of this case is sufficient

to capture most of the heating by the ohmic dissipation term.

Note that the number SMRI is called the magnetic Reynolds

number in our previous papers (Sano & Miyama, 1999; Sano

et al., 2004). We would emphasize that this number SMRI is

specified for the MRI, because the most unstable wavelength

of the MRI vA/ is adopted as a typical length scale. In fact,

both the linear and nonlinear evolution of the MRI can be

well-characterized by this number SMRI.

As seen from Figure 1, many pairs of current sheets fill

the domain. The current directions of each pair are oppo-

site. These structures are stretched in the azimuthal direction

as a result of the Keplerian shear motions. At each pair of

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Fig. 2 Correlation between the radial and azimuthal components of themagnetic field. We use snapshot data taken at 100 orbits for the samemodel as in Figure 1. The solid line denotes By/Bx = 2.7, which is thetypical ratio of their rms values in MRI driven turbulence

current sheets, both the radial and azimuthal components of

the field are sinusoidal with respect to z but their signs are

opposite. These properties are similar to the channel mode.

The correlation between Bx and By in the saturated turbu-

lence is shown by Figure 2.Anti-correlation of the horizontal

fields is one of the most important features of MRI driven

turbulence, because this ensures outward transport of angular

momentum. The ratio of the horizontal components is about

〈〈B2y 〉〉1/2/〈〈B2

x 〉〉1/2 ≈ 2.7, which is interestingly independent

of any initial conditions (Sano et al., 2004). The kurtosis of

the magnetic field is 〈B4〉/〈B2〉2 ≈ 4.0, which denotes the

intermittency of the field. The origin of these small struc-

tures may be the growth of the channel modes, although the

directions of the current sheets are not always horizontal ow-

ing to the turbulent motions. Therefore each small channel

flow can be regarded as a characteristic unit of the structure

in MRI driven turbulence for the zero net flux cases.

Both the saturation level and time variability of the

Maxwell stress for zero net flux cases are much smaller than

those in the uniform vertical field runs (Sano et al., 2004).

The difference can be explained by the characteristics of the

channel mode. When the shearing box is dominated by a

two-channel flow, the magnetic field is amplified effectively

and a spike-shaped excursion is expected in the time evolu-

tion. However if there are many channel modes with small

amplitude in the box and each mode develops independently,

the time variability in the volume average of the stress must

be small.

Note that the saturation level of the Maxwell stress is

slightly lower as the grid resolution is higher for the zero net

flux cases. The time- and volume-averaged Maxwell stress

in the model shown by Figure 1 is 〈〈−Bx By/4π〉〉/〈〈P〉〉 ≈3.2 × 10−4 which is about half of that for the lower resolu-

tion run in Sano et al. (2004). The channel mode can be an

exact solution of the nonlinear MHD equations only when

the amplitudes of all the other modes are negligible. Thus,

small fluctuations impede the growth of a large-scale channel

flow and reduce the efficiency of field amplification. Here we

define the typical thickness of current sheets as λJ = 2π/kJ

where k2J = 〈(∇ × B)2〉/〈B2〉. The thickness of the current

sheets is about λJ/L z ≈ 9.3 × 10−2 for this model, but we

find the grid size dependence of λJ at least within the limit of

our calculations. The current sheets are thinner in the higher

resolution cases, so that the timescale of magnetic reconnec-

tion can be shorter. This could also cause the smaller satura-

tion amplitude of the field. Further results on the resolution

dependence will be reported elsewhere.

3 Nonlinear Evolution of a Channel Mode

Next we perform two-dimensional simulations of the nonlin-

ear evolution of a single channel mode. Consider a small box

in the disk threaded by a uniform vertical field B0. We ignore

the vertical gravity so that the initial density ρ = ρ0 and pres-

sure P = P0 are spatially uniform. The box height is set to be

the most unstable wavelength of the MRI, L z = 2πvA0/,

where vA0 = B0/(4πρ0)1/2. The initial pressure and field

strength are assumed to be P0 = 5 × 10−3 and vA0 = 10−3.

We use ρ0 = 1 and = 10−3 as the normalizations, and thus

the initial plasma beta is β0 = 104 and the box height cor-

responds to L z ≈ 0.63H where H is the scale height of the

disk. The ohmic dissipation terms are included in the energy

and induction equations. The magnetic diffusivity is assumed

to be uniform η = 10−3 which means the initial Lundquist

number for the MRI SMRI ≡ v2A0/η = 1. We use a rectan-

gular box with a wide width L x = 2L z and the grid resolution

of 256 × 128. As for the initial perturbations, small random

fluctuations are added to the velocity with the maximum am-

plitude |δv|max = vA0.

3.1 Parasitic Instability

At the beginning, the most unstable mode evolves dominantly

and then the wavelength of the channel flow is equal to the

box height, λch = L z . The evolution of this mode is rep-

resented by a Fourier mode amplitude of the radial velocity

Vx (k) whose radial wavenumber is zero, kx ≡ kx L x/2π = 0,

and vertical wavenumber is unity, kz ≡ kz L z/2π = 1. This

is shown by the solid curve in Figure 3as a function of time

normalized by the rotation time trot = 2π/. The growth

rate obtained from our simulation is ωMRI ≈ 0.20 and this

is exactly the same as the prediction by the linear analysis

for the case of SMRI = 1 (Sano & Miyama, 1999).

However, the exponential growth is stopped and turned

to decrease when the amplitude of the channel flow exceeds

the strength of the vertical field Vx/vA0>∼1. Goodman & Xu

(1994) have shown that the channel mode is unstable for the

Kelvin-Helmholtz type instability (parasitic instability). The

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Fig. 3 Time evolution of Fourier mode amplitudes of the velocitiesnormalized by the initial Alfven speed. The growth of a two-channelflow is represented by a mode amplitude of the radial velocity withthe radial wavenumber kx ≡ kx L x/2π = 0 and the vertical wavenum-ber kz ≡ kz L z/2π = 1 (solid curve). A mode amplitude of the verticalvelocity with kx = 1 and kz = 0 is shown by the dashed curve, whichindicates the growth of the parasitic instability. The growth rates ofthe MRI and the parasitic instability expected from the linear analysis(ωMRI/ = 0.2 and ωPI/ = 0.6) are also plotted by thin lines in thisfigure

maximum growth rate is given by ωPI ∼ 0.2b where b ≡Bh/B0 is the amplitude of the channel flow and Bh = (B2

x +B2

y )1/2 is the horizontal component of the field. Thus the

growth rate of the parasitic instability increases exponentially

with time. The most unstable growth is expected when the

vertical wavenumber is zero and the radial wavelength is

about twice the length of the channel flow. Note that the

unstable growth of the parasitic instability requires the longer

wavelength of disturbances than λch.

The evolution of the parasitic mode is given by a Fourier

mode amplitude of the vertical velocity Vz(k) with kx = 1

and kz = 0. When the mode amplitude b becomes unity, the

growth rate of the parasitic mode is of the order of the angu-

lar velocity and then catches up with the channel mode.

The amplitude of the channel mode can be estimated ap-

proximately by using a mode amplitude of the radial velocity,

b ∼ Vx (kx = 0, kz = 1)/vA0. The amplitude is b ≈ 3.0 at the

peak (t ≈ 6.4trot), and thus the maximum growth rate of the

parasitic instability is ωPI ∼ 0.6. The growth rate obtained

numerically around the peak is consistent with this theoret-

ical prediction (see Fig. 3). The magnetic energy begins to

decrease just after the fast growth of the parasitic instability,

so that this instability breaks up the channel flow and triggers

magnetic reconnection.

3.2 Sweet-Parker Reconnection

After the breakup of the channel flow, magnetic reconnec-

tion takes place and the magnetic energy decreases expo-

nentially. The gas pressure increases rapidly by the joule

heating. Figure 4shows the time history of the volume-

averaged magnetic energy. This simple calculation of a chan-

Fig. 4 Time history of the volume-averaged magnetic energy in asimple simulation of a channel mode. The nonlinear evolution of thechannel flow reproduces a spike-shaped time variation. The dashedline denotes the decay rate of the Sweet-Parker reconnection model;B2 ∝ exp(−ωSPt) = exp(−2π t/tSP)

nel mode can reproduce a spike-shaped variation success-

fully.

We find that the decay timescale and the reconnection

rate are consistent with the Sweet-Parker reconnection model

(Sweet, 1958; Parker, 1957), although the reconnection in our

calculation is unsteady. The decay timescale of the magnetic

energy obtained from this simulation is about τdec ≈ 3.9trot,

while the model prediction is τSP = (τAτη)1/2 ≈ 4.1trot. The

Alfven timescale is τA = L/vA ≈ 0.67trot and the diffusion

timescale τη = L2/η ≈ 25trot, where we assume the typical

length scale as L ∼ 2λch and the Alfven speed is vA ∼ bvA0.

The reconnection rate is given by Mi = vi/vA where vi is the

inflow velocity to the diffusion region. We can extract the in-

flow velocity from the simulation data by means of a mode

amplitude of the vertical velocity with kx = kz = 1. The in-

flow velocity obtained numerically is vi/vA0 ≈ 0.43 at the

peak of the channel mode. Then the reconnection rate be-

comes Mi ≈ 0.15 which is consistent with the Sweet-Parker

model Mi = (η/LvA)1/2 ∼ 0.16.

When we use a square box L x = L z , the magnetic dis-

sipation proceeds gradually without topological change of

the magnetic field. For this case, the parasitic instability can-

not grow because the radial width is smaller than the critical

wavelength. This suggests the importance of the parasitic

mode as a triggering mechanism of magnetic reconnection.

4 Discussion

The Maxwell stress in MRI driven turbulence is proportional

to the magnetic pressure (e.g., Hawley et al., 1995; Sano et al.,

2004). However the relation between the gas and magnetic

pressure in the saturated turbulence has still many uncertain-

ties. The local shearing box calculation would be a useful

tool to investigate MHD turbulence in accretion disks. In

this paper, we have shown that the evolution of the channel

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flow determines the characteristics of MRI driven turbulence.

The channel flow like structures can be seen also in the global

disk simulations (Machida & Matsumoto, 2003).

The nonlinear saturation level of the MRI has depen-

dences on some physical quantities, such as the gas pres-

sure and initial field strength (Sano et al., 2004). Using our

simple simulations of a channel flow, we find that the recon-

nection rate is unaffected by the gas pressure. We change

the initial gas pressure over 3 orders of magnitude fixing

the other parameters, but the reconnection rate Mi is al-

ways around 0.1. We also perform the same calculations of a

channel flow in three-dimension. The reconnection rate and

timescale are almost the same as those in two-dimensional

simulations.

The growth rate of the parasitic instability is comparable

to the MRI when the amplitude of the channel mode b is

about 5. The ratio of the horizontal field to the vertical one

in MRI driven turbulence is about 〈〈B2h 〉〉1/2/〈〈B2

z 〉〉1/2 ≈ 5.2

(Sano et al., 2004). This is consistent with a picture that the

parasitic instability constrains the field amplification in MRI

driven turbulence. For the zero net flux cases, the typical

wavelength of channel modes is small because of the weak

vertical field. The saturated stress is much smaller than that

in the uniform Bz runs. Therefore, the net flux of the vertical

field may be essential for the efficient transport of angular

momentum in accretion disks.

Acknowledgements Numerical computations were carried out onVPP5000 at the National Astronomical Observatory of Japan and onSX-6 and SX-8/6A at the Institute of Laser Engineering, Osaka Uni-versity. This work was also supported by the Grant-in-Aid (16740111,17039005) from the Ministry of Education, Culture, Sports, Science,and Technology of Japan.

References

Balbus, S.A., Hawley, J.F.: Rev. Mod. Phys. 70, 1 (1998)Goodman, J., Xu, G.: ApJ 432, 213 (1994)Hawley, J.F.: ApJ 528, 462 (2000)Hawley, J.F., Balbus, S.A.: ApJ 400, 595 (1992)Hawley, J.F., Gammie, C.F., Balbus, S.A.: ApJ 440, 742 (1995)Inutsuka, S., Sano, T.: ApJ 628, L155 (2005)Machida, M., Matsumoto, R.: ApJ 585, 429 (2003)Parker, E.N.: J. Geophys. Res. 62, 509 (1957)Sano, T., Inutsuka, S.: ApJ 561, L179 (2001)Sano, T., Inutsuka, S., Turner, N.J., Stone, J.M.: ApJ 605, 321 (2004)Sano, T., Miyama, S.M.: ApJ 515, 776 (1999)Sano, T., Miyama, S.M., Umebayashi, T., Nakano, T.: ApJ 543, 486

(2000)Schekochihin, A.A., Cowley, S.C., Taylor, S.F., Maron, J.L.,

McWilliams, J.C.: ApJ 612, 276 (2004)Shakura, N.I., Sunyaev, R.A.: A&A 24, 337 (1973)Stone, J.M., Gammie, C.F., Balbus, S.A., Hawley, J.F.: in Protostars &

Planets IV, ed. V. Mannings, A.P. Boss, S.S. Russell (Tuscon: Univ.of Arizona Press), 589 (2000)

Sweet, P.A.: in IAU Symp. 6: Electromagnetic Phenomena in CosmicalPhysics, ed. B. Lehnert (Cambridge Univ. Press), 123 (1958)

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DOI 10.1007/s10509-006-9225-6


Laboratory Exploration of Solar Energetic Phenomena

David Alexander


Abstract The solar atmosphere displays a wide variety of

dynamic phenomena driven by the interaction of magnetic

fields and plasma. In particular, plasma jets in the solar

chromosphere and corona, coronal heating, solar flares and

coronal mass ejections all point to the presence of magnetic

phenomena such as reconnection, flux cancellation, the for-

mation of magnetic islands, and plasmoids. While we can

observe the signatures and gross features of such phenom-

ena we cannot probe the essential physics driving them, given

the spatial resolution of current instrumentation. Flexible and

well-controlled laboratory experiments, scaled to solar pa-

rameters, open unique opportunities to reproduce the relevant

unsteady phenomena under various simulated solar condi-

tions. The ability to carefully control these parameters in

the laboratory allows one to diagnose the dynamical pro-

cesses which occur and to apply the knowledge gained to the

understanding of similar processes on the Sun, in addition

directing future solar observations and models. This talk in-

troduces the solar phenomena and reviews the contributions

made by laboratory experimentation.

Keywords Laboratory astrophysics . Solar physics .

Dynamic phenomena . Magnetic reconnection . Plasma jets

1 Introduction

The Sun is often quoted as being a “Laboratory for Astro-

physics” and in many ways it lives up to this sobriquet. A

wide range of physical phenomena involving the interaction

of magnetic fields and plasma occur throughout the solar at-

D. AlexanderDepartment of Physics and Astronomy, Rice University,6100 Main St, Houston, TX 77005

mosphere and provide a unique perspective into the physics

governing astrophysical processes. Observations of the Sun

from ground- and space-based observatories provide a wealth

of data, often continuous, with high resolution in space, time

and energy (or wavelength or frequency). Yet, despite its

proximity and the array of instrumentation available it is still

150 million km away and is a ‘laboratory’ where the experi-

ments are uncontrolled and where the physics occurs on spa-

tial scales far smaller than we can resolve. To fully understand

the observed solar phenomena and the physics driving them,

carefully designed terrestrial laboratory experiments, scal-

able to solar conditions, are necessary. In this paper, I intro-

duce some of the solar phenomena of particular interest and

illustrate some of the relevant laboratory investigations cur-

rently underway. By the very nature of the subject, this cannot

be considered a comprehensive survey but it should give a fla-

vor of some of the issues involved. Some key energetic solar

phenomena are discussed in Section 2 although we consider,

in a little more depth, the specific phenomena of plasma jets

on the Sun in Section 3. Scaling issues, associated with the

connection between the laboratory experiments and the Sun,

are discussed in Section 4, and we conclude in Section 5.

2 Solar phenomena

The Sun exhibits a wide array of energetic phenomena, often

resulting from the interaction between plasma and magnetic

field. The very generation of the Sun’s magnetic field is under

intense study with observational and theoretical advances in

heliosiesmology and solar dynamo physics being one of most

exciting areas of solar physics research over the last decade.

Solar transient phenomena and their role in driving space

weather is another area of intense interest involving a syner-

gistic approach between observation, theory, and modeling.

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Fig. 1 (left) Post-flare loop arcade seen in the 195A channel of theTRACE telescope for an X281 class flare (soft X-ray intensity of 2.8mW/m2) which occurred at the west limb of the Sun on 2003 Nov4. The plasma in the magnetic loops shown has a temperature of or-der 1.5 MK, having cooled from 20–40 MK. The image size repre-sents 175,000 km × 233,000 km at the Sun. (right) Associated CME

seen in Thomson scattered white light by the SOHO/LASCO telescope.The white circle in the center of the LASCO image marks the lo-cation of the Sun, the grey disk is the occulter, and the scale of theimage is ∼30 solar radii or 21 million km. The CME is clearly evi-dent, erupting from the south-west limb of the Sun (bottom right on thefigure)

In this paper, we focus on the transient phenomena and leave

the interested reader to explore the physics of helioseismol-

ogy and solar dynamo physics from better and more compre-

hensive treatments (see Christensen-Dalsgaard, 2004 for an

excellent review).

2.1 Solar flares and coronal mass ejections

Some of the most energetic phenomena in the Universe occur

on the Sun. A solar flare can generate power as high as 1022

W from as little as 1014 kg of mass (10−16 solar masses), or

approximately 108 W/kg. Similar numbers for blazars, with

an energy output from 109 solar masses and 1042 W, and

gamma-ray bursts, ∼1 solar mass and 1045 W, are 103 W/kg

and 1015 W/kg, respectively. A gamma-ray burst is clearly

more powerful than a solar flare but it is interesting to note

that while gamma-ray bursts accelerate particles to the order

of 1 MeV, solar flares can generate ions and electrons with

energies in excess of 100 MeV.

The largest solar flares tend to be associated with a quite

distinct but no less impressive phenomenon known as a

Coronal Mass Ejection (CME). CMEs are large expulsions

of mass and magnetic field from the Sun. Their velocities

can exceed 2000 km/s, some 3–5 times faster than the am-

bient solar wind. The CME drives a shock in interplane-

tary space which subsequently accelerates ions to 10–100

MeV/nucleon. Figure 1 shows an event from 2003 Nov 4

(one of the famous Hallowe’en storms: Gopalswamy et al.,

2005) which included a very large flare (GOES class X28), a

1 The X28 designation was based on saturated X-ray detectors. it is nowthought this flare, the largest on record, may have been as high as anX45 (4.5m W/m2), see Brodrick et al., (2005).

fast CME, and a proton storm at 1 AU. The coronal EUV im-

age on the left of the figure was taken in the 195A channel of

the Transition Region and Coronal Explorer (TRACE), while

the CME white light image was taken by the Large Angle

and Spectrometric Coronagraph (LASCO) on the Solar and

Heliospheric Observatory mission (SOHO).

The initiation of these events is an ongoing problem in

the solar physics community with distinct efforts focusing

on flares and CMEs separately, although the commonality of

some of the physics is understood. Several models have been

put forward to explain how the solar corona builds up and

then releases magnetic energy in the form or large-scale erup-

tions, localized heating, and the ejection of mass and mag-

netic flux. Loss-of equilibrium (Lin and Forbes, 2000), mag-

netic breakout (Antiochos, DeVore, and Klimchuk, 1999)

and tether-cutting (Moore et al., 2001) models have been the

most prevalent in recent years, although see also Chen and

Krall (2003). In many of these cases, magnetic reconnection

is critical to the initiation or evolution of the energy release.

The breakout model requires reconnection in the overlying

arcade field to occur in order for the energized sheared field

to erupt (‘break-out’), whereas the loss-of-equilibrium model

requires fast reconnection in a current sheet formed below a

rising magnetic fluxrope to enable the fluxrope to erupt as a

CME: for a review of all of these models see Lin, Soon, and

Baliunas (2003). While there is no direct evidence that mag-

netic reconnection occurs in the solar atmosphere (gyro-radii

are on the cm scale while observations are on the hundreds

of km scale), many of the expected signatures of the recon-

nection process (counter-flowing jets, particle acceleration,

topological changes in the magnetic field, etc) have been ob-

served. A better understanding of the consequences of mag-

netic reconnection under conditions relevant (or scaleable)

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to the solar corona would significantly enhance our under-

standing of CME initiation.

The largest flares/CMEs are often associated with coronal

destabilization triggered by the eruption of a solar filament.

2.2 Solar filaments and prominences

Solar filaments (or prominences) are structures in which cool,

dense chromospheric plasma is embedded in the hotter solar

corona (see Gilbert et al., 2001). The typical length scale of a

solar prominence is 104–105 km and the magnetic geometry

is thought to contain regions of upward concavity in the local

magnetic field, either helical structures such as fluxropes (e.g.

Rust and Kumar, 1994) or in a series of dips on individual

fieldlines (e.g. Kuperus and Raadu, 1974) where the substan-

tial mass of the filaments plasma can collect (although see

Karpen et al., 2001). Despite the presence of large amounts of

dense plasma, the dynamics and evolution of the filament are

governed by the dominant magnetic forces: the plasma beta

in filaments is on the order 10−3 to 10−1. Eruptions of these

filaments are commonly associated with energy release in

the corona resulting in a range of phenomena including solar

flares and CMEs. Determining how these eruptions are driven

is a primary goal in the physics of CME initiation. Factors

which influence the eruption process include; the role of mag-

netic reconnection (where it occurs, whether it is a driver or a

consequence of the eruption, how fast it proceeds), magnetic

topology (whether the filament is a helical fluxrope prior to

eruption or develops into a fluxrope as a consequence of the

eruption), whether an MHD instability occurs (Fig. 2 shows

an example where the kink instability is thought to be re-

sponsible for the observed kinking of the filament structure),

whether the eruption is triggered by a global destabilization

of the corona or by a local change in magnetic field via the

emergence of new flux or the cancellation of old flux. Models

exist which explore all of these issues. For a comprehensive

review see Lin et al. (2003).

2.3 Magnetic fluxtubes on the Sun

A fundamental component of magnetic fields in astrophysics

and in the laboratory is the magnetic fluxtube. In the Sun,

fluxtubes are generated in the solar interior by solar dynamo

processes occurring at the base of the convective zone, the

strong magnetic fluxtubes then buoyantly rise through the

convective zone to emerge through the solar surface to gen-

erate the coronal loops which characterize the key build-

ing blocks of the solar corona. It is the coronal magnetic

structures which, when stressed, generate the array of ener-

getic transient phenomena discussed above. Understanding

the physical processes which govern the creation, transport,

and evolution of magnetic fluxtubes throughout the various

regimes of the solar atmosphere and interior is necessary to

understand solar variability, both on the long- and the short-

term. As an example of theoretical progress in this area, Lin-

ton et al. (2001) have recently performed a series of numerical

simulations focused on understanding the interaction of flux-

tubes. They find a wide range of behavior from the merging

of fluxtubes, fluxtubes ricocheting off each other, reconnec-

tion and slingshot-type dynamics, and even the tunneling of

one fluxtube through another. The various behaviors result

from the degree of twist in the interacting fluxtubes, their

angle of approach, and the value for the Lundquist number

adopted.

Fig. 2 A dynamic solarfilament occurring on 2002 May27 seen in the 195A channel ofTRACE. The maximum heightattained by the filament (see lastframe) is ∼80,000 km. Thecontours show locations of hardX-ray emission associated withthe filament activation (seeAlexander et al., 2006 fordetails)

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Laboratory experimentation is the only means by which

we can study fluxtube behavior in detail. However, to under-

stand the full range of consequences of fluxtube interaction

in regimes pertinent to astrophysical systems closer collabo-

ration between the laboratory astrophysics, solar theoretical,

and solar observational communities is essential.

3 Laboratory simulation of solar plasma jets

There are many examples of laboratory simulations of solar

phenomena. I would like to call out for particular attention,

given the focus of solar transients in this paper, the work

of the Princeton MRX team (Yamada, 2004; Ji et al., 1998)

on driven reconnection, magnetic helicity, wave generation

and ion heating, all of which are important in the solar at-

mosphere, and that of Bellan and Hansen (1998), Hansen

and Bellan (2001), and Hansen, Tripathi, and Bellan (2004)

on simulating solar prominences and their interaction in the

laboratory.

Much of the astrophysical focus of the HEDLA 2006 con-

ference centered on jet phenomena and I would like to bring a

solar perspective to those discussions by focusing on jet phe-

nomena in the Sun and what we might learn from laboratory

simulations. Figure 3 shows an example of a solar plasma

jet studied by Alexander and Fletcher (1999). Such jets are

commonly observed and take a number of forms depending

on the nature of the magnetic field interactions driving them.

The basic characteristics of these jets include:

• plasma velocities of order, 150–200 km/s

• the presence of adjacent hot (T∼100 eV) and cool

(T∼1 eV) jets

• evidence for twisting and bifurcation of the jet with time

The observed jets are mostly associated with the emer-

gence or cancellation of magnetic flux with modeling efforts

suggesting that they result from the coalescence of magnetic

islands and require enhanced resistivity to generate the neces-

sary velocities and energy release rates (Shibata, Yokoyama,

and Shimojo, 1996; Karpen et al., 1995). Multi-thermal solar

plasma jets have also been detected out to 3 solar radii from

the Sun (Fig. 4) and traveling as fast as 500 km/s (Ko et al.,

2005).

Some preliminary work has been performed at the

Lawrence Livermore National Laboratories using a

Spheromak-like Compact Torus (SCT) formed by the Com-

pact Torus Injection Experiment (CTIX; Hwang et al., 2000).

The internal field of the SCT is of order 1 kG with a tem-

perature of 50–75 eV and a plasma beta around 10−3. The

idea here is that the interaction of two SCTs focused through

an appropriately shaped exit aperture results in the creation

of oblique shocks which interact to accelerate the plasma

(Ryutova and Tarbell, 2000) and thereby mimic chromo-

spheric and transition region shocks observed at the Sun

(Tarbell et al., 2000).

4 Scaling issues

One of the major issues in the applicability of laboratory ex-

periments to astrophysical phenomena is the markedly differ-

ent regimes in which the various physical interactions occur.

Fig. 3 Plasma jet in the solarcorona. Left panel: The 171Aimage from the TRACEtelescope at 02:55:20 UT on1998 Aug 19 with the jet clearlyvisible extending out of an‘anemone’ like kernel. Rightpanel: A difference image from7 minutes later(03:02:20–03:01:20 UT)showing that the jet bifurcatedwith evidence of twistingmotions from the alternatingblack and white striations:white/black impliespositive/negative change inemission (from Alexander andFletcher, 1999)

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Table 1 Physical parameters inlaboratory and solar regimes(SCT parameters from Howard,2006)

Parameter Solar photosphere Solar corona CTIX/SCT

Plasma density (cm−3) 1017 109 1015

Plasma temperature (eV) 0.5 100 10

Magnetic field strength (G) 500 10 1000

Characteristic length (cm) 4 × 106 109 20

Alfven timescale (s) 14.3 12 5.8 × 10−7

Resistive timescale (s) 8.5 × 105 2 × 1012 1.54 × 10−3

Lundquist number 5.9 × 104 1.7 × 1011 2.6 × 103

Spitzer resistivity (s) 5.3 × 10−13 4.8 × 10−16 4.2 × 10−15

Plasma frequency (GHz) 2.8 × 103 0.284 2.8 × 102

Debye length (cm) 1.5 × 10−4 23.5 4 × 10−4

Fig. 4 Solar plasma jetoccurring on 1999 Aug 26 foundto extend out to 3 solar radii(from Ko et al., 2005). The arcstarting in the top left corner ofeach frame is the solar limb. Thescale of each frame is 175,000km × 124,000 km

Typical scales in astrophysics can be 10–20 orders of magni-

tude greater than achievable in the laboratory. Consequently,

the issue of the scalability of laboratory phenomena is crucial

to their application to astrophysics (Ryutov and Remington,

2002).

Table 1 details the various parameters in the solar photo-

sphere and solar corona (two very different regimes in the

Sun separated by only 2000 km). For comparison the equiv-

alent parameters for the CTIX SCT are also listed. The key

difference is in the relative strengths of the plasma and the

magnetic field and the consequences this has for dynamical

scale-heights and transient phenomena.

Clearly, the physical scales are widely disparate, particu-

larly in density, characteristic length scales, and the relevant

timescales. However, more important is how the various di-

mensionless parameters of the system compare (see Ryutov,

this volume). We see that for the case of the SCT the

Lundquist number is within an order of magnitude of that

in the solar photosphere while several orders of magnitude

smaller than in the corona.

5 Conclusions

We can learn a lot about the physics behind energetic solar

phenomena from well-tailored laboratory experiments. How-

ever, the inclusion of magnetic field is crucial. The range

of dynamic phenomena occurring across a wide array of

Springer


physical regimes on the Sun can act as a “laboratory for

laboratory astrophysics”, especially for providing guidance

on the role of magnetic fields. Consequently, the comparison

of laboratory experiments and simulations with detailed solar

observations may provide insight into the physical processes

at work in astrophysical plasmas. Scaling is still a major issue

but the range of solar phenomena and wealth of data increase

the chances that scalability can achieved.

References

Alexander, D., Fletcher, L.: Sol. Phys. 190, 167 (1999)Alexander, D., Liu, R., Gilbert, H. R.: ApJ 653, 720 (2006)Antiochos, S.K., DeVore, C.R., Klimchuk, J.A.: ApJ 510, 485 (1999)Bellan, P.M., Hansen, J.F.: Phys. of Plasmas, 5, 1991 (1998)Brodrick, D., Tingay, S., Wieringa, M.: JGR 110, A09S36 (2005)Chen, J., Krall, J.: J. Geophys. Res. 108(A11), 1410 (2003)Christensen-Dalsgaard, J.: Equation-of-State and Phase-Transition

in Models of Ordinary Astrophysical Matter, Celebonovic, V.,Gough, D., Dappen W. (eds). New York, 18 (2004)

Gilbert, H.R., Holzer, T.E., Low, B.C., Burkepile, J.T.: ApJ 549, 1221(2001)

Gopalswamy, N., et al.: J. Geophys. Res. 110, A09S00 (2005)Hansen J.F., Bellan, P.M.: ApJ 563, 183 (2001)Hansen, J.F., Tripathi, S.K.P., Bellan, P.M.: Phys. Plasmas 11, 3177

(2004)Howard, S.J.: PhD Thesis, UC Davis (2006)Hwang, D.Q. et al.: Nuc. Fusion 40, 897 (2000)Ji, H., Yamada, M., Hsu, S., Kulsrud, R.: Phys. Rev. Lett. 80, 3256

(1998)Lin, J., Forbes, T.H.: J. Geophys. Res. 105, 2375 (2000)Lin, J., Soon, W., Baliunas, S. L.: New Astron. Rev. 47, 53 (2003)Linton, M.G., Dahlburg, R.B., Antiochos, S.K.: ApJ 553, 905 (2001)Karpen, J.T., Antiochos, S.K., Devore, C.R.: ApJ 450, 422 (1995)Karpen, J.T., Antiochos, S.K., Hohensee, M., Klimchuk, J.A.,

MacNeice, P.J.: ApJ 553, 85 (2001)Ko, Y.-K. et al.: ApJ 623, 519 (2005)Kuperus, M., Raadu, M.A.: A&A 31, 189 (1974)Moore, R.L., Sterling, A.C., Hudson, H. S., Lemen, J. R.: ApJ 552, 833

(2001)Rust, D.M., Kumar, A.: Solar Phys. 155, 69 (1994)Shibata, K., Yokoyama, T., Shimojo, M.: Adv. Sp. Res. 17, 197 (1996)Yamada, M.: Proc. 35th COSPAR Scientific Assembly, p. 4411 ( 2004)Tarbell, T.D., Ryutova, M., Shine, R.A.: Sol. Phys. 193, 195 (2000)Ryutov, D.D., Remington, B.A.: Plasma Phys. Control. Fusion 44, 407

(2002)Ryutova, M., Tarbell, T.D.: ApJ 541, 29 (2000)

Springer


DOI 10.1007/s10509-006-9230-9


Explosion Mechanism of Core-Collapse Supernovae andCollapsars

S. Nagataki


Abstract We have performed 2-dimensional MHD simula-

tions of collapsars with magnetic fields and neutrino cool-

ing/heating processes. It is found that explosion energy of

a hypernova is not obtained from the neutrino heating pro-

cess. However, strong jet is found when magnetic fields are

included, and total energy of the jet component can be of

the order of 1052 erg, which is comparable to the one of a

hypernova.

Keywords Supernova . Collapsar . GRB

1. Introduction

There has been growing evidence linking long gamma-ray

bursts (GRBs; in this study, we consider only long GRBs,

so we call long GRBs as GRBs hereafter for simplicity) to

the death of massive stars. The host galaxies of GRBs are

star-forming galaxies and the positions of GRBs appear to

trace the blue light of young stars. Also, ‘bumps’ observed in

some afterglows can be naturally explained as contribution

of bright supernovae. Moreover, direct evidences of some

GRBs accompanied by supernovae have been reported such

as the association of GRB 980425 with SN 1998bw and that

of GRB 030329 with SN 2003dh.

It should be noted that these supernovae are catego-

rized as a new type of supernovae with large kinetic en-

ergy (∼1052 ergs), nickel mass (∼0.5 M⊙), and luminosity,

so these supernovae are sometimes called as hypernovae.

Also, since GRBs are considered to be jet-like phenomena,

S. NagatakiYukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho Kitashirakawa Sakyo-ku, Kyoto 606-8502, Japane-mail: [email protected]

it is natural to consider the accompanying supernovae to be

jet-induced explosions.

The central engine of GRBs accompanied by hypernovae

is not known well. But it is generally considered that normal

core-collapse supernovae can not cause an energetic explo-

sion of the order of 1052 erg. So another scenario has to be

considered to explain the system of GRBs associated with hy-

pernovae. One of the most promising scenario is the collapsar

scenario (MacFadyen and Woosley, 1999). In the collapsar

scenario, a black hole is formed as a result of gravitational

collapse. Also, rotation of the progenitor plays an essential

role. Due to the rotation, an accretion disk is formed around

the equatorial plane. On the other hand, the matter around the

rotation axis falls into the black hole. It was pointed out that

the jet-induced explosion along to the rotation axis occurs

due to the heating through neutrino anti-neutrino pair anni-

hilation that are emitted from the accretion disk. MacFadyen

and Woosley (1999) demonstrated the numerical simulations

of the collapsar, showing that the jet is launched ∼7 s af-

ter the gravitational collapse and the duration of the jet is

about 10 s, which is comparable to the typical observed du-

ration of GRBs. However, detailed neutrino heating process

is not included in MacFadyen and Woosley (1999). Also, it

is pointed out that effects of magnetic fields may be so im-

portant (Proga et al., 2003; Fujimoto et al., 2005). So in this

study, we solved the dynamics of collapsars with neutrino

cooling/heating processes and magnetic fields.

2. Models and numerical methods

Our models and numerical methods of simulations in this

study are shown in this section. First we present equations

of ideal MHD, then initial and boundary conditions are ex-

plained. Micro physics included in this study, equation of

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state (EOS), nuclear reactions, and neutrino processes are

also explained.

2.1. Magnetohydrodynamics

We have done two-dimensional MHD simulations taking ac-

count of self-gravity and gravitational potential of the central

point mass. The calculated region corresponds to a quar-

ter of the meridian plane under the assumption of axisym-

metry and equatorial symmetry. The spherical mesh with

150(r ) × 30(θ ) grid points is used for all the computations.

The radial grid is nonuniform, extending from 5.0 × 106 cm

to 1.0 × 1010 cm with finer grids near the center, while the

polar grid is uniform. The location of the inner most radius

is same with MacFadyen and Woosley (1999).

The basic equations in the following form are finite dif-

ferenced on the spherical coordinates:

Dρ

Dt= −ρ∇ · v (1)

ρDv

Dt= −∇ p − ρ∇ + 1

4π(∇ × B) × B (2)

ρD

Dt

(

e

ρ

)

= −p∇ · v − L−ν + L+

ν + Lnucl (3)

∂B

∂t= ∇ × (v × B), (4)

where ρ, v, P , , e, L±ν , Lnucl, and B are density, veloc-

ity, pressure, gravitational potential, internal energy density,

heating/cooling rates due to neutrino processes, energy gain

(loss) rate due to nuclear reaction, and magnetic field. The

Lagrangian derivative is denoted as D/Dt . The gravitational

potential of the central point mass is modified to account for

some of the effects of general relativity, φ = −G M/(r − rs)

where rs = 2G M/c2 is the Schwartzshild radius, G is the

Gravity constant, and M is the mass of the black hole. The ini-

tial mass of the black hole is set to be 1.69 M⊙ and M becomes

larger along with time since mass accretes from the inner

boundary. The ZEUS-2D code developed by Stone and Nor-

man (1992) has been used to solve the MHD equations with

second order accurate interpolation in space. Heating/cooling

rates due to neutrino processes and energy gain (loss) rate

due to nuclear reaction are described in Subsections 2.3.2

and 2.3.3.

2.2. Initial and boundary conditions

We adopt the Model E25 in Heger et al. (2000). The star in this

model has 25 M⊙ initially with solar initial metallicity, but

lose its mass and becomes to be 5.45 M⊙ as a Wolf-Rayet star

at the final stage. This model seems to be a good candidate

as a progenitor of a GRB since losing their envelope will

be suitable to make a baryon poor fireball. The mass of iron

core is 1.69 M⊙ that is covered with Si layer whose mass is

0.55 M⊙. So we assume that the iron core has collapsed and

formed a black hole at the center.

Angular momentum was distributed so as to provide a

constant ratio of 0.04 of centrifugal force to the component

of gravitational force perpendicular to the rotation axis at all

angles and radii, except where that prescription resulted in j16

greater than a prescribed maximum value, 10. This treatment

is exactly same with MacFadyen and Woosley (1999). Total

initial rotation energy is 5.7 × 1048 erg that corresponds to

initial ratio of the rotation energy to the gravitational energy,

T/W = 8.3 × 10−3.

Configuration and amplitude of the magnetic fields in a

progenitor prior to collapse are still uncertain. So in this study

we choose a simple form of the initial configuration and the

amplitude is changed parametrically. Initial configuration of

the magnetic fields is chosen as follows:

B(r ) = 1

3B0

(

r0

r

)3

(2 cos θ er + sin θ eθ ) for r ≥ r0 (5)

= 2

3B0(cos θ er − sin θ eθ ) for r < r0. (6)

This configuration represents that the magnetic fields are uni-

form in a sphere (r < r0), while dipole at outside of the

sphere. We set r0 to be the boundary between CO core/Si

layer. B0 corresponds to the strength of the magnetic field

in the sphere. We have chosen B0 to be 0, 108G, 109G,

and 1010G. Initial ratios of the magnetic energy relative

to the gravitational energy are 0, 1.1 × 10−8, 1.1 × 10−6,

1.1 × 10−4, respectively. The initial lowest plasma beta,

which are realized at the outer boundary, are ∞, 5.8 × 105,

5.8 × 103, and 5.8 × 101, respectively.

As for the boundary condition in the radial direction, we

adopt the outflow boundary condition for the inner and outer

boundaries. That is, the flow from the central black hole is

prohibited at the inner boundary and the inflow from the

surface of the progenitor is prohibited at the outer boundary.

Of course, the mass of the central black hole becomes larger

due to the mass accretion from the inner boundary. As for

the boundary condition in the zenith angle direction, axis

of symmetry boundary condition is adopted for the rotation

axis, while reflecting boundary condition is adopted for the

equatorial plane.

2.3. Micro physics

2.3.1. Equation of state

The equation of state (EOS) used in this study is the one

developed by Blinnikov et al. (1996). This EOS contains an

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Fig. 1 Density contour (from1 – 1012 g cm−3) with velocityfields at t = 2.5 s after thecollapse for the regionr ≤ 108 cm. The case withoutmagnetic fields is shown in theleft panel, while the case withmagnetic fields (109G) is shownin the right panel

electron-positron gas with arbitrary degeneracy, which is in

thermal equilibrium with blackbody radiation and ideal gas

of nuclei.

2.3.2. Nuclear reactions

Although the contribution of ideal gas of nuclei to the to-

tal pressure is negligible, effects of energy gain/loss due

to nuclear reactions are important. In this study, nuclear

statistical equilibrium (NSE) was assumed for the region

where T ≥ 5 × 109 [K] is satisfied as Nagataki et al. (2003),

while no nuclear reaction occurs for the region where T <

5 × 109 [K]. This treatment is based on the assumption that

the timescale to reach and maintain NSE is much shorter

than the hydrodynamical time. Note that complete Si-burning

occurs in explosive nucleosynthesis of core-collapse super-

novae for the region T ≥ 5 × 109 [K]. The hydrodynamical

time in this study, ∼ sec, is comparable to the explosive

nucleosynthesis in core-collapse supernovae, so the assump-

tion adopted in this study seems to be farely well. 5 nuclei,

n, p,4 He,16 O,56 Ni was used to estimate the binding energy

of ideal gas of nuclei in NSE for given (ρ, T , Ye). Ye is elec-

tron fraction that is obtained from the calculations of neutrino

process in Section 2.3.3.

2.3.3. Neutrino processes

Neutrino cooling processes due to pair capture on free nu-

cleons, pair annihilation, and plasmon decay are included

in this study. Since photoneutrino and bremsstrahlung pro-

cesses are less important ones at 109 < T < 1011 [K] and

ρ < 1010 [g cm−3] where effects of neutrino cooling are im-

portant in our calculations, we do not include these processes.

Neutrino heating processes due to νe and νe captures on

free nucleons and neutrino pair annihilation with blocking

factors of electrons and positrons are included in this

study. The νe and νe captures on free nucleons are inverse

processes of electron/positron captures. As for the neutrino

pair annihilation process, the formulation of Goodman et al.

(1987) is adopted. We assume that the matter is optically

thin against neutrinos to obtain the neutrino heating rate as

mentioned above.

3. Results

In Fig. 1, density contour with velocity fields at t = 2.5 s af-

ter the collapse. The case without magnetic fields is shown

in the left panel, while the case with magnetic fields (109G)

is shown in the right panel. It is clearly shown that a jet prop-

agates along to the rotation axis for the case with magnetic

fields. The total energy of the jet component can be of the

order of 1052 erg, which is comparable to the one of a hy-

pernova. We found that the jet is launched by the magnetic

pressure of Bφ , which is amplified by the winding-up effect.

As for the amplitude of Br , Bθ , they are much smaller than

the amplitude of Bφ .

In Fig. 2, total emitted energy by neutrino processes as

a function of time (solid line), total absorbed energy by

Fig. 2 Solid line: total emitted energy by neutrino processes as afunction of time. Dashed line: total absorbed energy by neutrino anti-neutrino pair annihilation. Dotted line: total absorbed energy by neutrinocapture on nucleons

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neutrino anti-neutrino pair annihilation (dashed line), and

total absorbed energy by neutrino capture on nucleons (dot-

ted line) are shown. It is noted that total absorbed energy is

so little that this effect can not explain the explosion energy

of a hypernova.

As for the dependence of the strength of the initial mag-

netic fields, we found that strong jet is launched when strong

initial magnetic fields are assumed initially.

4. Summary and discussion

We have done 2-dimensional MHD simulation of collapsars

with magnetic fields and neutrino cooling/heating processes.

It is found that explosion energy of a hypernova is not ob-

tained from the neutrino heating process. However, strong jet

is found when magnetic fields are included, and total energy

of the jet component can be of the order of 1052 erg, which

is comparable to the one of a hypernova.

To tell the truth, we consider that neutrino heating process

can be a possible key process to drive a GRB jet when ef-

fects of general relativity are taken into consideration. Since

our code is Newtonian at present. So we are developing a

GRMHD code now. We hope to present new results in the

very near future.

Acknowledgement The author is grateful to M. Watanabe and S.Yamada for useful discussion. The computation was partly carried outon NEC SX-5 and SX-8, SGI Altix3700 BX2, and Compaq AlphaServerES40 at Yukawa Institute for Theoretical Physics, and Fujitsu VPP5000at National Astronomical Observatory of Japan. This work is partiallysupported by Grants-in-Aid for Scientific Research from the Ministry ofEducation, Culture, Sports, Science and Technology of Japan throughNo. 14102004, 14079202, and 16740134.

References

Blinnikov, S.I., Dunina-Barkovskaya, N.V., Nadyozhin, D.K.: ApJ 106,171 (1996)

Fujimoto, S., et al.: ApJ 644, 1040 (2006)Heger, A., Langer, N., Woosley, S.E.: ApJ 528, 368 (2000)MacFadyen, A.I., Woosley, S.E.: ApJ 524, 262 (1999)Nagataki, S., Mizuta, A., Yamada, S., Takabe, H., Sato, K.: ApJ 596,

401 (2003)Proga, D., MacFadyen, A.I., Armitage, P.J., Begelman, M.C.: ApJ 599,

L5 (2003)

Springer


DOI 10.1007/s10509-006-9236-3


Astrophysical Radiation Dynamics: The Prospects for Scaling

John I. Castor

Received: 14 June 2006 / Accepted: 10 August 2006C© Springer Science + Business Media B.V. 2006

Abstract The general principles of scaling are discussed,

followed by a survey of the important dimensionless pa-

rameters of fluid dynamics including radiation and magnetic

fields, and of non-LTE spectroscopy. The values of the pa-

rameters are reviewed for a variety of astronomical and lab-

oratory environments. It is found that parameters involving

transport coefficients – the fluid and magnetic Reynolds num-

bers – have enormous values for the astronomical problems

that are not reached in the lab. The parameters that mea-

sure the importance of radiation are also scarcely reached in

the lab. This also means that the lab environments are much

closer to LTE than the majority of astronomical examples.

Some of the astronomical environments are more magnet-

ically dominated than anything in the lab. The conclusion

is that a good astronomical environment for simulation in a

given lab experiment can be found, but that the reverse is

much more difficult.

Keywords Hydrodynamics . Radiation . Scaling

PACS Nos: 95.30.Jx, 95.30.Lz, 97.10.Ex, 97.10.Gz,

98.62.Mw

1 Introduction

Radiation hydrodynamics is the discipline in which not only

the material fluid but also the radiation (photons or neutrinos)

The U.S. Government’s right to retain a non-exclusive, royalty-freelicense in and to any copyright is acknowledged.

J. I. CastorLawrence Livermore National Laboratory, L-16, Livermore,CA 94550, USAe-mail: [email protected]

must be treated dynamically. Since the speed of light is so

large, it is tempting and often successful to neglect the fluid

velocity in the dynamical equation for the radiation. Account-

ing for the effects thereby ignored is the business of radiation

hydrodynamics. Chief among these are the advective flux of

radiation energy and the subtraction of momentum and en-

ergy from the radiation when it exerts a force on the material.

From a computational point of view, the proper accounting

for the velocity effects is one of two main challenges in ra-

diation hydrodynamics; the other challenge is meeting the

requirement of a full transport solution with all the spectral

and angular detail that the radiation field possesses. The latter

challenge is faced even when the fluid velocity is negligible.

The computational complexity of radiation hydrodynamics

is the motivation for seeking laboratory analogues of astro-

nomical environments for which, owing to the radiation hy-

drodynamic effects, numerical simulations are very difficult;

the analogue experimental results can provide benchmarks

for the simulations.

In this paper I will give a quick review of the princi-

ple of scaling for physical systems described by a small

set of partial differential equations. The central point is

the non-dimensionalization of the equations, which leads

to a minimum set of non-dimensional parameters the val-

ues of which must all match for two physical systems in

order for one system to be the scaled version of the other.

Next I provide a list of possibly relevant dimensionless pa-

rameters that arise in describing various astronomical en-

vironments. After a brief explanation of the parameters I

provide a table of the parameter values for several astro-

nomical environments and also several laboratory environ-

ments that may be proposed as scaling candidates for the

astronomical ones. The discussion of this table is the main

point of this paper, and after the discussion I offer a short

conclusion.

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2 The principle of scaling

The scaling concept is described as follows: It is assumed

that our physical system is described fully and with suffi-

cient accuracy by providing the values of a few fields, such

as the density, fluid velocity, perhaps magnetic induction,

etc., as functions of a few independent variables, such as

coordinates x , y, z and time t . It is also assumed that to

sufficient accuracy the fields obey a certain set of partial

differential equations over these coordinates. These equa-

tions may be put into non-dimensional form by expressing

each field or coordinate as the product of a representative

value and a dimensionless function. When the transformed

equations are simplified, the dimensional representative val-

ues can be grouped together into dimensionless combina-

tions, which are the fundamental parameters of the problem.

The two systems can be scaled versions of each other if the

non-dimensional partial differential equations that describe

them are identical, and, in particular, if the dimensionless

parameters are identical. This means that the same set of

physical processes is an accurate description of both sys-

tems, and that the relative magnitudes of the different pro-

cesses that are included are also identical between the two

systems.

The test for scaling is therefore this: identify the relevant

dimensionless parameters and test each of them for equality.

Each dimensionless parameter can be expressed as the

ratio of two physical quantities that appear in the governing

equations and which have the same dimensions. When the

dimensionless ratio is either extremely large or extremely

small, it means that one of the physical quantities is negligible

compared with the other. In this case the equations could

be simplified by discarding the negligible term(s). So when

we test for equality of the dimensionless parameters for two

systems, we can ignore a parameter that is different for the

two systems if it happens to be extremely large or extremely

small in both; that parameter involves physics that is not

actually relevant for these systems.

3 The astronomical environments

Astronomical bodies have characteristic length and time

scales that are huge compared with terrestrial laboratories,

of course, but there is also a great dynamic range among

them. But just to pick one example, consider the interstel-

lar medium. The typical length scale is roughly one parsec,1

and the typical time scale is very roughly 1000 years. Each of

these numbers is 3 × 1019 larger than laser experiment scales

of 1 mm and 1 nanosecond. The density may be 1–1000 par-

1 1 parsec (pc) is 3.08568 × 1018 cm

Table 1 Selected astronomical environments

Environ Length Velocity # density Temp B

Warm ISM 3 × 1018 107 1 104 10−5

Dense cld 3 × 1018 5 × 105 103 102 10−4

Stellar atm 109 107 1015 104 102

Stellar env 1010 107 1018 106 102

AGN disk 2 × 1013 3 × 108 1012 107 106

XRB disk 106 3 × 107 3 × 1021 107 106

NS acc col 4 × 104 3 × 109 1023 108 1012

Stellar wnd 1012 108 1011 105 5 × 101

Table 2 Selected laboratory environments

Environ Length Velocity # density Temp B

Burn thru 10−3 106 1024 6 × 105 106

hohlraum 10−2 107 1022 106 106

NIF hohl 3 × 10−2 2 × 107 1022 3 × 106 106

Z expt 10−1 107 1022 106 106

Short pulse 10−3 108 1024 107 108

ticles per cubic centimeter, which is at least 1019 times less

than the typical laser target density 1022 cm−3. It is a stringent

test of scaling to span nineteen orders of magnitude!

For the present discussion I have selected eight astronom-

ical environments as candidates for scaling to the lab: (1)

warm interstellar medium; (2) a dense interstellar cloud; (3)

a stellar photosphere; (4) an interior point in a stellar enve-

lope; (5) the accretion disk around an active galactic nucleus;

(6) an x-ray binary accretion disk; (7) in a neutron star ac-

cretion column; and (8) a point in the wind of a hot star.

Table 1 shows the characteristic properties and dimensions

of the environments. The units are cgs, kelvins and Gauss,

and the particle density is atoms per cubic centimeter.

4 Some laboratory environments

I will consider here a few selected laboratory environments

that have been employed or proposed for laboratory astro-

physics experiments. These are: a burn-through foil that

might be inserted in an Omega hohlraum wall; plasma at criti-

cal density in a modest-temperature hohlraum; the same thing

but sized for a NIF hohlraum; the same thing on the Z pulsed-

power machine; conditions produced by a short-pulse laser.

Table 2 lists the characteristics chosen to represent the differ-

ent experiments. As above, the units are cgs-kelvin-Gauss.

5 Relevant dimensionless parameters

The physical processes that dominate the behavior in an as-

tronomical environment that can be scaled to the laboratory

are necessarily simple: ideal gas dynamics with radiation

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flow and perhaps MHD. The gas dynamics by itself in-

troduces one parameter, the Mach number M = u/cs , in

which cs = (γ p/ρ)1/2 is the adiabatic sound speed. Viscos-

ity might be significant, in which case the Reynolds number

Re = ρuL/µ is a relevant parameter.

We suppose that molecular heat conduction is negligible

compared with the radiative heat flux, so we need not be

concerned with the Prandtl number. The radiative flux is de-

scribed with the Boltzmann number, which is the ratio of

the convective heat flux to the 1-way radiative flux σB T 4:

Bo = ρuC p/(σB T 3). In some of the environments the radi-

ation mean free path λp is short and in some it is long; the

optical depth parameter τ = L/λp is the measure of it. The

mean free path is λp = 1/(κρL) in terms of the opacity κ ,

which may be the Rosseland mean or some other fiducial

value. When the optical depth is large, the radiation is said

to be in the diffusion limit, and the net radiative flux can be

computed from a heat-conduction-like formula. The ratio of

the convective flux to the diffusive radiative flux is the Peclet

number Pe = (3/4)τρuC p/(σB T 3) = (3/4)τBo.

If the characteristic optical depth of the plasma, κρL , is

small, the optical depth may in fact not be a useful con-

cept. In such plasmas, e.g., a galactic nebula, the opacity

and therefore the optical depth varies dramatically between

the cores of strong lines and the continuum regions. The to-

tal cooling rate per unit mass by radiative emission is given

formally by the expression C = 4πκP B in LTE, in which

κP is the Planck-weighted opacity and B = σB T 4/π is the

Planck function. A similar expression applies in non-LTE,

but it can only be evaluated by treating in detail the cool-

ing processes of collisional excitation, radiative recombina-

tion, etc. The cooling function has been studied by Raymond

et al. (1976) and Dalgarno and McCray (1972), among oth-

ers. We may regard κP as formally accounting for all those

processes. The optically-thin cooling time based on C is

tcool = C pT/ C, and the relevant dimensionless number is

the ratio of tcool to the characteristic flow time, L/u. This is

Cool = utcool/L = uC pT/(L C ). For those cases in which

it is appropriate to identify C with 4πκP B, the parameter be-

comes utcool/L = uC pT/(4πκP L B). In this last form Cool

is the Boltzmann number defined above divided by 4κPρL;

the last has the form of a characteristic optical depth, which

is expected to be very small in the case being discussed.

With the qualification that C = 4πκP B is not always ap-

propriate, the manipulation to express the cooling time pa-

rameter in terms of the Boltzmann number is justified by the

parallel between this relation in the optically thin case and

that between the Peclet number and the Boltzmann number

in the diffusive case. It goes without saying that quantitative

studies must account for all the detailed physical processes

rather than using the crude expressions presented here.

When the magnetic field is significant the equations of

MHD replace the Euler equations. An additional parameter

appears for ideal MHD, the plasma beta, β = 8πp/B2. If

the electrical conductivity is not effectively infinite then

the equations of resistive MHD must be used, and an ad-

ditional parameter is the magnetic Reynolds number, Rm =µ0uL/η = 4πσuL/c2, in which η is the electrical resistivity

in SI units, and σ is the conductivity in Gaussian units (s−1).

These are the relevant parameters in a collisional plasma. In

the weak-collision regime there are additional parameters,

such as the Larmor radius divided by L and the collision

frequency times L/u, as discussed in these proceedings by

Ryutov (2006).

6 Scaling parameters for astronomical

and lab environments

I have evaluated the eight parameters discussed earlier, τ , M,

Re, Rm, Bo, Pe, Cool and β, for the various astronomical and

laboratory environments. These are shown in Table 3. For

the warm ISM, dense cloud and stellar wind environments

the effective value of the opacity was based on the cooling

Table 3 Scaling parameters for astronomical and laboratory environments

Environ τ M Re Rm Bo Pe Cool β

Warm ISM 10−4 101 107 1019 2 × 10−3 2 × 10−7 5 3 × 10−1

Dense cld 2 × 10−5 6 7 × 1013 6 × 1014 10−9 2 × 10−14 2 × 10−5 3 × 10−2

Stellar atm 101 101 5 × 1012 4 × 109 6 × 10−2 6 × 10−1 10−3 3

Stellar env 3 × 103 1 5 × 1011 4 × 1013 3 × 10−5 7 × 10−2 2 × 10−9 3 × 105

AGN disk 5 101 7 × 107 6 × 1019 2 × 10−12 6 × 10−12 8 × 10−14 3 × 10−8

XRB disk 103 1 109 4 × 1011 5 × 10−4 4 × 10−1 10−7 8 × 101

NS acc col 103 4 × 101 6 × 108 5 × 1013 8 × 10−2 8 × 101 2 × 10−5 3 × 10−8

Stellar wnd 2 × 10−2 4 × 101 1010 1015 3 × 10−6 5 × 10−8 3 × 10−5 10−2

Burn thru 7 × 101 2 × 10−1 2 × 104 2 × 10−1 3 × 101 2 × 103 10−1 2 × 103

Omega hohl 9 × 10−3 1 3 × 103 5 × 101 4 × 10−1 2 × 10−3 101 3 × 101

NIF hohl 10−3 1 103 2 × 103 10−2 10−5 3 9 × 101

Z expt 9 × 10−2 1 3 × 104 5 × 102 2 × 10−1 10−2 5 × 10−1 3 × 101

Short pls 10−2 4 103 103 3 × 10−1 3 × 10−3 5 3

Springer


function (T ) given by Dalgarno and McCray (1972), and

for the warm ISM, stellar wind and accretion column envi-

ronments a dilution factor for the radiation field was included

in the definition of the Boltzmann number. The way in which

we would like to use these tables is to look up the astronomi-

cal environment we want to simulate in Table 3, and then find

a laboratory environment in the table that has similar values

of the scaling parameters. But we see at a glance that the

scaling parameters for the astronomical environments have a

huge dynamic range while the laboratory parameters do not.

Some of the notable differences between the astronomi-

cal and laboratory environments that are seen in Table 3 are

these: The transport parameters Re and Rm, the ordinary and

magnetic Reynolds numbers, are much larger in all the as-

tronomical environments. In view of the comments earlier

about very large parameters, it seems that viscosity and re-

sistivity can quite generally be neglected in the astronomical

environments, while this may not be true in the laboratory.

The Boltzmann number is generally very small for the as-

tronomical environments; this means that they are radiation-

dominated. The derived values of the astronomical Peclet

and cooling numbers are also small, with one exception in

each case. In the laboratory, because the density is so much

higher, radiation is generally not dominant. This is a major

impediment to simulating astronomical radiation hydrody-

namics problems in laboratory experiments. Also because of

the high density, the lab environments have a great difficulty

achieving a low β. Some, but not all, astronomical environ-

ments are very highly magnetized, which is not true of the

laser and pulsed-power experiments considered here.

Leaving aside the radiation and magnetic field effects, in

other words just looking at the Mach number, we see a bet-

ter overlap between astronomical and laboratory parameter

values. So pure gas dynamics looks promising for scaling

astronomical problems to the laboratory.

7 Non-LTE and astronomical spectra

Another area in which we would hope to simulate an astro-

nomical problem in the lab is in plasma spectroscopy: Can

we create a radiation source in the lab of which the spectrum

would be a good match to that of the astronomical object?

Scaling spectroscopy is harder than scaling hydrodynam-

ics. Replacing one element by another does not work very

well, since complex spectra are unique; hydrogenic spectra

are the exception. So we suppose that the ions of interest are

not hydrogen-like, and that the same element will be used

in the simulation that occurs in the astronomical problem.

Since atomic excitation and ionization depend on the ratio of

the ionization potential to kT , we conclude that T will also

not be scaled.

The competition between collisional excitation and de-

excitation processes and their radiative counterparts is the

heart of non-LTE excitation and ionization equilibrium. The

scaling parameters that express this competition are the ε

values defined by

ε = NeCuℓ

Auℓ

,

in which u → ℓ is an atomic transition forming a spectral

line, Cuℓ is the rate coefficient for collisional de-excitation

and Auℓ is the spontaneous radiative decay rate. In or-

der for the emitted spectrum to match, all the values of

ǫ should match. This expression for ε can also be written

as ε = Ne/(Ne)crit, where (Ne)crit ≡ Auℓ/Cuℓ is called the

critical density for this transition. Many of the lines in the

spectra of nebulae are electric-dipole forbidden, and the val-

ues of Auℓ are of order 1 s−1 or less, which makes the critical

density of order 102–106 cm−3. Obviously these lines are not

candidates for laboratory studies. For permitted lines Van

Regemorter’s (1962) semi-empirical formula may used to

approximate the collisional rate, then the dependence on Auℓ

cancels out and the result is ε ∝ Neλ3ph, where λph is the pho-

ton wavelength. This explicit dependence on Ne means that

the line formation process is not scalable.

In nebulae in our galaxy, and in emission-line regions of

active galactic nuclei and elsewhere, we have conditions very

far from LTE in which the plasma is strongly photoionized

by a diluted but energetic radiation field. The plasma tem-

perature comes to an equilibrium in which photoionization

heating balances radiative cooling, mostly in line emission.

The ionization balance and the temperature then depend, for

a given shape of the ionizing spectrum, on the ionization pa-

rameter defined by Krolik et al. (1981), Tarter et al. (1969)

= U

ρC pT.

Here U is the diluted radiation energy density for photon

energies above the photoionization threshold; U is related to

the radiative flux F by U = F/c. (Different authors have de-

fined the ionization parameter in different ways; the citations

above give two; for a third definition see (Netzer, 1990). The

definitions are simply related to each other, and are ways of

parameterizing the ionization ratio in a photoionized plasma,

namely, the ratio of the photoionization rate to the radiative

recombination rate. The ionization ratio depends on atomic

properties in addition to the environmental characteristics.)

We see that is roughly the same as u/(cBo), except that

here the Boltzmann number must be defined using the ac-

tual radiative flux F rather than the thermal flux σB T 4; the

relation is approximately F = WσB T 4, in which W is the ge-

ometrical dilution factor. The Boltzmann number is again the

Springer


important scaling parameter for non-LTE. The typical values

of in nebulae and active galactic nuclei are 102–103, while

the values in the lab environments are closer to 10−1.

8 Prospects

We have seen that the dynamic range of the scaling parame-

ters for the astronomical environments is very large indeed,

much larger than the range among the available laboratory

experiments. A large part of the range covered by the as-

tronomical environments is inaccessible in the laboratory.

This means that the the odds that a given astronomical en-

vironment can be simulated in the laboratory are not good.

However, the odds that a given laboratory environment has

an analogue in astronomy are much better.

Some processes do not scale very well – the viscosity and

resistivity effects are generally much smaller in the astronom-

ical environments, and both radiation and magnetic fields

tend to be stronger (small Bo and β) in the astronomical

cases. Scaling appears to be most successful for pure gas

dynamics. Within these limitations the prospects for scaling

are good.

Acknowledgements This work was performed under the auspices ofthe U.S. Department of Energy by the University of California LawrenceLivermore National Laboratory under Contract No. W-7405-Eng-48. Iwould like to thank B. A. Remington and R. P. Drake for an educationin laboratory astrophysics on Nova, and D. Ryutov for his insightfulcomments on scaling. I am grateful to the organizers of HEDLA 2006for the opportunity to present this work. The article was improved bythe referee’s helpful comments.

References

Dalgarno, A., McCray, R.A.: ARA&A 10, 375–426 (1972)Krolik, J.H., McKee, C.F., Tarter, C.B.: ApJ 249, 422–442 (1981)Netzer, H.: In: Courvoisier, T.J.-L., Mayer, M. (eds.), Swiss Society for

Astrophysics and Astronomy, Springer, New York (1990)Raymond, J.C., Cox, D.P., Smith, B.W.: ApJ 204, 290–292 (1976)Ryutov, D.: In: Lebedev, S. (ed.), Proceedings of HEDLA (2006)Tarter, C.B., Tucker, W.H., Salpeter, E.E.: ApJ 156, 943–951 (1969)Van Regemorter, H.: ApJ 136, 906–915 (1962)

Springer


DOI 10.1007/s10509-006-9235-4


Experiments to Study Radiation Transport in Clumpy Media

P. A. Rosen · J. M. Foster · M. J. Taylor · P. A. Keiter ·

C. C. Smith · J. R. Finke · M. Gunderson · T. S. Perry


Abstract Clumpiness of the interstellar medium may play

an important role in the transfer of infrared continuum radi-

ation in star forming regions (Boisse, 1990). For example,

in homogeneous models, C II emission should be confined

to the cloud edge (Viala, 1986). However, in star formation

regions (such as M17SW, M17 and W51), it is observed to

extend deep into the molecular cloud (Stutzki et al., 1988;

Keene et al., 1985). One plausible interpretation of these ob-

servations is that, due to their clumpiness, the clouds are

penetrated by UV radiation far deeper than expected from

simple homogeneous models.

The interaction of H II regions around young massive stars

with a clumpy medium is another area of interest. Molecular

clouds are well established to be clumpy on length scales

down to the limits of observational resolution. Clumps can

act as localized reservoirs of gas which can be injected into

the surroundings by photoionization and/or hydrodynamic

ablation (Dyson et al., 1995; Mathis et al., 1998).

The calculation of radiation transport in hot, clumpy mate-

rials is a challenging problem. Approximate, statistical treat-

ments of this problem have been developed by several work-

ers, but their application has not been tested in detail. We

describe laboratory experiments, using the Omega laser to

test modelling of radiation transport through clumpy media

in the form of inhomogeneous plasmas.

P. A. Rosen () · J. M. Foster · M. J. Taylor · C. C. SmithAWE Aldermaston, Reading, RG7 4PR, UKe-mail: [email protected]

P. A. Keiter · J. R. Finke · M. GundersonLos Alamos National Laboratory, Los Alamos, NM 87545, USA

T. S. PerryLawrence Livermore National Laboratory, Livermore, CA 94550,USA

Keywords Radiation transfer . Radiation flow . Clumpy

media . Inhomogeneous plasma

1 Introduction

Calculations of radiation or particle transport are greatly

complicated by the presence of regions in which two (or

more) materials are randomly and inhomogeneously mixed

(that is, where discrete chunks of random size are randomly

dispersed in a host material). A variety of statistical meth-

ods has been developed to treat this problem, and these are

discussed extensively in the literature (Henke et al., 1984;

Vanderhagen, 1986; Pomraning, 1988, 1991; Haran et al.,

2000; Smith, 2003). In brief summary, it is assumed that at

any point in the inhomogeneously mixed material, the proba-

bility of occurrence of each of the constituents is known, and

the goal is to calculate the mean particle or radiation inten-

sity, averaged over all possible configurations of the ensem-

ble, and thus the effective opacity of the mixture. Theoretical

treatments differ according to the methods they employ and

the statistical distribution assumed for the components of the

mixture. Our aim is to carry out laboratory investigations

of radiation transport through inhomogeneously mixed plas-

mas, that can be analysed using these statistical methods.

2 Experiment design

We have developed a laboratory experiment to study ra-

diation transport through inhomogeneous media. Figure 1

shows the experimental concept. A sample of inhomoge-

neous, gold-loaded hydrocarbon foam is contained within a

gold tube, driven from one end by thermal radiation from

a hohlraum target heated by the Omega laser (Soures et al.,

Springer


Fig. 1 Experimental configuration showing the laser-heated hohlraumand the experimental package. The gold tube is 1 mm length and 0.8 mmdiam

1996). Radiation burnthrough of the foam is diagnosed using

an x-ray framing camera viewing through a longitudinal slit

in the wall of the gold tube. We compare the radiation burn-

through times of samples of pure foam, and foam containing

either uniformly distributed gold particles, or an equal mass

of atomically mixed gold. Differences of opacity result in dif-

ferences of burnthrough time, and the experimental data are

compared with calculational models for the inhomogeneous

and homogeneous mixtures.

In our present experiments, a peak radiation drive tem-

perature of 200 eV is obtained in a 1.6 mm diam., 1.2 mm

length hohlraum that is heated by 15 beams of the Omega

laser, with a total energy of ∼6.5 kJ in a 1 ns duration laser

pulse of 0.35µm wavelength. The experimental package is

mounted over a 0.8-mm-diam. hole in the end wall of the

hohlraum; it comprises a 0.8-mm-diam., 1-mm-length gold

tube containing the inhomogeneous gold/foam mixture.

In the design of the experiment, we consider various can-

didate gold/foam mixtures under different conditions of tem-

perature and density. We assume that the gold particles are

uniformly heated, and that they expand until pressure equi-

librium with the surrounding host material is attained. For

particles of micron size appropriate to our experiments at

the Omega laser, the timescale for this expansion is approx.

0.5 ns. We calculate separately the (spectrally dependent)

opacities of the heated hydrocarbon foam and gold parti-

cles, using the IMP opacity code (Rose, 1992). We then ap-

ply the inhomogeneous-mixture opacity models discussed

above, to obtain the effective opacity of the mixture, and from

this calculate the mean opacity. Figure 2 shows calculations

of Rosseland mean opacity, as a function of initial particle

size and temperature, for a triacrylate (C15H20O6) foam of

0.05 g cm−3 density containing 15% by weight of admixed

gold. Particle size following pressure equilibrium is approx.

a factor of three larger than initial particle size. We note that

Fig. 2 Rosseland mean opacity of an inhomogeneous, pressure-equilibrated gold/foam mixture (0.05 g cm−3 total density, 15% byweight gold, various initial particle sizes)

the mean opacities of pure foam, and foam containing 15%

by weight admixed gold, differ by approx. a factor of five.

The mean opacity in the particulate mix case falls midway

between these two limits, for particles of 2-µm initial size.

For our present experiments, triacrylate foams have been

manufactured at the University of St. Andrews, UK (Falconer

et al., 1995), and have been successfully loaded with parti-

cles with a narrow distribution of diameters (1.5–3µm), or

with an organo-metallic gold compound. Pre-shot character-

isation of these materials is accomplished using a variety of

techniques including weighing (to determine bulk density),

x-ray radiography (to determine uniformity of bulk density

and gold loading), and examination by scanning electron mi-

croscopy (SEM) (to determine particle size distribution and

to detect agglomeration of particles). X-ray fluorescence and

neutron activation measurements are also carried out in sam-

ples contained in polymer cylinders, to establish gold content.

Figure 3 shows data from a multi-channel, time-gated x-

ray pinhole camera viewing the outside of the foam-filled

tube. This camera employs absorption-edge x-ray filters to

limit its response to narrow spectral regions close to 300 and

450 eV. The multiple images shown in Figure 3 (obtained

using a pure foam sample) were recorded at 0.5, 1.5, 2.5 and

3.5 ns after the onset of radiation drive. These data clearly

show the progression of the radiation-driven heat wave as

well as the closure (at late time in the 450-eV images) of the

diagnostic slit. Comparison of the burnthrough times of the

different foams (pure, particle-loaded, and atomically mixed)

is the basis of our experiment.

3 Modelling

Our modelling of the experiment has adopted two ap-

proaches: design studies using a radiation-temperature drive

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Fig. 3 X-ray framing cameradata from the 300-eV channel(top) and 450-eV channel(bottom) showing progressionof the radiation-driven heat frontalong the foam-filled tube. The(white) shadow of a verticalfiducial wire is visible in someimages

prescription; and a more detailed, fully integrated calcula-

tional model of a limited number of specific experiments.

Both originate from a detailed model of the laser-heated

hohlraum, but the radiation-temperature drive prescription

uses an equivalent Planckian drive spectrum for simplicity

in subsequent simulations, whereas the detailed model in-

cludes the hohlraum in each and every simulation and thus

provides a better approximation of multi-frequency effects in

the radiation transport. In all cases, we use the Lagrangian ra-

diation hydrocode NYM (Roberts et al., 1980), multi-group

implicit Monte-Carlo radiation transport, and opacities gen-

erated using the IMP code (Rose, 1992).

In the design simulations, we use a drive prescription

identical to that described by Foster et al. (2002), and ap-

ply this (by means of appropriate boundary conditions) to

a simulation of the experimental package (foam-filled tube)

alone. The inhomogeneous gold-loaded foam mixture is sim-

ulated approximately by means of opacity multipliers, using

as a basis opacity data for both pure foam (required opac-

ity multiplier >1) and atomically mixed, gold-loaded foam

(opacity multiplier <1). The appropriate opacity multipli-

ers are obtained from calculations of the effective opac-

ity of the mixture, using the methods of Pomraning et al.

(1988, 1991) and Vanderhaegen (1986). The radiation hy-

drocode simulation is post-processed by ray tracing to gen-

erate spatially resolved x-ray emission intensity for com-

parison with the x-ray framing camera data. Figure 4 sum-

marises the position of the emission front (in the 300-eV

channel of the x-ray framing camera), from simulations of

both pure and atomically mixed foam, together with these

same calculations in which the radiative opacity has been

scaled by factors of 2 and 5 (pure foam case) and 0.5 and

0.2 (atomically mixed case). It is evident that these simula-

tions are consistent with the factor-of-5 difference of opac-

ity between the pure-foam and atomic-mix cases (Figure 2),

and that the use of an opacity multiplier thus provides

a good first approximation to the inhomogeneous-mixture

case.

Fig. 4 Emission (in the range 200–300 eV) front position versus timefrom post-processed NYM simulations of pure foam and atomically-loaded foams using nominal and scaled radiative opacities

The fully detailed calculations of specific experimental

shots include both the hohlraum and experimental package

in each simulation, and so better represent the detailed drive

spectrum (including the long-range preheat originating in the

gold M-band emission from the hohlraum). Nevertheless,

their results are not greatly different from simpler, pre-shot

models of the experiment. Again, ray-tracing post-processing

is used to generate synthetic diagnostic signals for compar-

ison with experiment. In the fully detailed calculations, we

have considered only the case of pure foam and atomically

mixed gold-loaded foam; we have not yet carried out opacity

multiplier simulations as a representation of the particulate-

mix case.

4 Comparison of experimental data with simulation

Figure 5 compares the measured and calculated positions of

the x-ray emission front (300-eV x-ray images) with pre-shot

Springer


Fig. 5 Emission front position as a function of time for experimentaldata and simulation of pure foam. A multiplier to the radiative opacityhas been applied

simulations (including ray-tracing post-processing) in which

the particulate- and atomic-mix cases are approximately rep-

resented by opacity multiplier of 2 and 5 (applied to simula-

tions of the pure-foam case). Although the density and gold

mass fraction of the foam samples differ somewhat from their

nominal values (50 mg cm−3, and 15% by weight, respec-

tively), it is clear that the experimental data demonstrate the

trends evident in simulation: data from experiments in 2002

and 2005 are self consistent (within the timing uncertainty

of the x-ray framing camera), and the radiation burnthrough

time of the particle-mix case falls between that of the pure

foam and atomic-mix cases.

Figure 6 compares these experimental data with both pre-

shot and detailed simulations of the experiment. In both cases,

simulations are for foams of nominal density (50 mg cm−3),

and gold mass fraction (15% by weight, in the case of the

atomic mixture). The pre-shot and detailed simulations show

differences of emission front position that arise from differ-

ences of detail in the drive prescription and spectrum (overall

energetics, and multi-group radiation transport effects), but

these differences are comparable to the uncertainty inherent

in the experimental measurement. An arbitrary (but plausi-

ble) offset of 300 ps of the 2002 data results in very close

agreement of the 2002 and 2005 data sets, and a detailed

simulation of the pure foam experiment for 60–65 mg cm−3

foam density (not shown) lies very close (within experimen-

tal error) to the experimental data. Experimental data from

the particulate-mix foams appears to lie close to simulations

assuming atomic mix, and we speculate that this may be as

a result of break up of the gold particles following radiation

(M-band) preheating. This speculation can only be resolved

in further experiments using larger diameter gold particles.

Data from the single shot using atomically mixed gold dif-

Fig. 6 Measured and calculated positions of the x-ray emission front,for the 300-eV x-ray images. The experimental data (points) are forpure, particle-loaded and atomically loaded foams. All foams were of0.05 g cm−3 nominal density, although the densities apparent in pre-shotcharacterisation measurements (figures in parentheses) differ somewhatfrom this value. The simulations (lines) are for foams of 0.05 g cm−3

density and 15% by weight gold (atomic mix case)

fers from both the pure-foam and particulate-mix cases. This

difference implies a foam of nominal density, although pre-

shot characterisation of this single case implied a rather lower

(than nominal) foam density.

5 Summary

We have reported the results of experiments carried out at

the Omega laser in 2002 and 2005, to study radiation trans-

port through an inhomogeneous mixture. Good experimental

data have been obtained, and it has been possible to mea-

sure radiation-burnthrough times that are correlated with the

composition of the pure-foam, and inhomogenous or atomic

gold-foam mixture, that was used. Modelling successfully

reproduces both the burnthrough time of the pure foam sam-

ples, and the trend of burnthrough time with material com-

position. Differences between the measured and modelled

burnthrough times of the particulate and atomic mixtures

may point to pre-heat induced break-up of the gold particles

(in which case, the particulate mix behaves as an atomically

dispersed mixture), or to uncertainties in characterisation of

the material samples. Further experiments using better char-

acterised materials, and a wider range of particle sizes, are

planned.

Acknowledgements It is our pleasure to thank the target fabricationteams at AWE and LANL, Chris Bentley (AWE) for his work on foamcharacterisation, and the laser-operations team at LLE.

Springer


References

Boisse, P.: Astron. Astrophys. 228, 483 (1990)Dyson, J.E., Williams, R.J.R., Redman, M.P.: Mon. Not. R. Astron. Soc.

277, 700 (1995)Falconer, J.W., Nazarov, W., Horsfield, C.: J. Vac. Sci. Technol. A 13,

1941 (1995)Foster, J.M., Wilde, B.H., Rosen, P.A., Perry, T.S., Fell, M., Edwards,

M.J., Lasinski, B.F., Turner, R.E., Gittings, M.L.: Phys. Plasmas,9, 2251 (2002)

Haran, O., Shvarts, D., Thieberger, R.: Phys. Rev. E 61, 6183 (2000)Henke, B.L., et al.: J. Opt. Soc. Am. B 1, 818 (1984)Keene, J., Blake, J.A., Phillips, T.G., Huggins, P.J., Beichman, C.A.:

Astrophys. J. 299, 967 (1985)

Mathis, J.S., Torres-Peimbert, S., Peimbert, M.: Astrophys. J. 495, 328(1998)

Pomraning, G.C.: J. Quant. Spectros. Radiat. Transfer 40, 479 (1988)Pomraning, G.C.: Linear Kinetic Theory and Particle Transport in

Stochastic Mixtures. World Scientific, Singapore (1991)Roberts, P.D., Rose, S.J., Thompson, P.C., Wright, R.J.: J. Phys. D 13,

1957 (1980)Rose, S.J.: J. Phys. B 25, 1667 (1992)Smith, C.C.: J. Quant. Spectros. Radiat. Transfer 81, 451 (2003)Soures, J., McCrory, R.L., Verdon, C.P., et al.: Phys. Plasmas 3, 2108

(1996)Stutzki, J., Stacey, G.J., Genzel, R., Harris, A.I., Jaffe, D.T., Lugten,

J.B.: Astrophys. J. 332, 379 (1988)Vanderhagen, D.: J. Quant. Spectros. Radiat. Transfer 36, 557 (1986)Viala, Y.: Astron. Astrophys. Suppl. 64, 391 (1986)

Springer


DOI 10.1007/s10509-007-9292-3


Laboratory Observation of Secondary Shock Formation Aheadof a Strongly Radiative Blast Wave

J. F. Hansen · M. J. Edwards · D. H. Froula ·

A. D. Edens · G. Gregori · T. Ditmire

Received: 24 April 2006 / Accepted: 18 December 2006C© Springer Science + Business Media B.V. 2007

Abstract We have previously reported the experimental dis-

covery of a second shock forming ahead of a radiative shock

propagating in Xe. The initial shock is spherical, radiative,

with a high Mach number, and it sends a supersonic radiative

heat wave far ahead of itself. The heat wave rapidly slows

to a transonic regime and when its Mach number drops to

two with respect to the downstream plasma, the heat wave

drives a second shock ahead of itself to satisfy mass and

momentum conservation in the heat wave reference frame.

We now show experimental data from a range of mixtures of

Xe and N2, gradually changing the properties of the initial

shock and the environment into which the shock moves and

radiates (the radiative conductivity and the heat capacity).

We have successfully observed second shock formation over

the entire range from 100% Xe mass fraction to 100% N2.

The formation radius of the second shock as a function of Xe

mass fraction is consistent with an analytical estimate.

Keywords Radiative . Shock . Heat wave . Supernova

J. F. Hansen () · M. J. Edwards · D. H. FroulaLawrence Livermore National Laboratory, Livermore, CA 94550,USAe-mail: [email protected]

A. D. EdensSandia National Labs, Albuquerque, NM 87185, USA

G. GregoriCCLRC – Rutherford Appleton Laboratory, Chilton, DidcotOX11 0QX, UK

T. DitmireUniversity of Texas at Austin, Austin, TX 78712, USA

1 Introduction

We previously reported on the experimental discovery of

secondary shock formation ahead of strongly radiative blast

waves in Xe (Hansen et al., 2005, 2006). The process can

be summarized as follows (c.f. Fig. 1): a shock (which

we will refer to as S1) is initially fast enough to radiate

very strongly, with an inverse Boltzmann number Bo−1 =σT 4/vsρ0cvT ∼ 50 (where T is the shock temperature, vs is

the shock speed, andρ0 is the density of the ambient gas). The

radiation mean free path in the cold ambient gas ahead of S1

is relatively short, resulting in the formation of a supersonic,

radiative heat wave (RHW), which propagates in advance of

S1. As S1 continually slows down, it radiates less and less,

and the radiated power soon drops below the rate at which S1

sweeps up energy from gas heated by RHW, i.e., the energy

loss rate becomes negative. At this time, most of the energy

that was originally in S1 has been radiated to RHW, and RWH

is far ahead of S1. However, the velocity of RHW has also

been diminishing rapidly because of expansion and a rapidly

weakening driving source. Eventually RHW becomes tran-

sonic and gives birth to a second shock wave (S2). RHW then

falls behind S2, which itself is too slow to be radiative. S1

continues to weaken as it propagates in the downstream ma-

terial of S2 and soon dissipates. After S2 has roughly doubled

its radius, it is no longer influenced by the details of how it

was formed, and the shock trajectory closely assumes that of

a self-similar Sedov-Taylor blast wave (Taylor, 1950; Sedov,

1959; Zeldovich and Raizer, 1966; Liang and Keilty, 2000).

The motivation behind our experiments [and behind many

other experiments in laboratories around the world (Bozier

et al., 1986; Remington et al., 1999; Ryutov et al., 1999;

Shigemori et al., 2000; Robey et al., 2001; Keiter et al., 2002;

Fleury et al., 2002; Bouquet et al., 2004)] is an interest in

astrophysical shocks which have high Mach numbers and

Springer


Fig. 1 Sequence of events (time increasing from left to right): (a) Aspherical shock, S1, radiatively drives a heat wave far ahead of itself. (b)As S1 expands it slows down and it radiates less and less. The radiativeheat wave, RHW, also slows down and its expansion rate can be estimatedusing Barenblatt’s solution. (c) S1 gains on RHW, but before it catchesup, RHW becomes transonic and gives birth to a second shock wave, S2.RHW immediately falls behind S2, which itself is too slow to be radiative.(d) S1 continues to weaken as it propagates in the downstream materialof S2 and soon dissipates. In the experiment, shocks that are drawn herewith solid curves are visible in the schlieren images, features drawnwith dashed curves are not

which may be radiative (Blondin et al., 1998), e.g., shocks

originating in supernova (SN) explosions (Muller et al., 1991;

Reed et al., 1995; Sonneborn et al., 1998; Remington et al.,

1999; Bartel et al., 2000). The nature of these shocks is im-

portant to understand as the shocks mix up interstellar matter

and thus affect mass-loading, stellar formation (McKee and

Draine, 1991; Allen and Burton, 1993; Klein and Woods,

1998), and the history of the Milky Way and other galaxies.

Although the motivation for our experiments was an interest

in astrophysical shocks, we should point out that the charac-

ter of these laboratory shocks may be different from any of

astrophysical importance, and that secondary shock forma-

tion has not been observed in astrophysical shocks to date.

2 Background

The radiative nature of a shock, coupled with the optical

opacity and heat capacity of its surroundings, largely de-

termines the evolution of the shock and its rate of expan-

sion. The energy loss rate can be quantified (in units of how

much energy the expanding shock sweeps up) by the di-

mensionless number (Cohen et al., 1998; Liang and Keilty

, 2000) ε = − (dE/dt) (2πρ0)−1 r−2s (drs/dt)−3, where E is

the total energy content and rs the shock radius. In a fully

radiative case, in which radiation escapes to infinity, the in-

coming kinetic energy swept up by a shock is entirely ra-

diated away (ε → 1) and the shocked material collapses to

a thin shell directly behind the shock. For an adiabatic case

(ε = 0) once the shock has swept up more mass than what

was initially present, the shock could be regarded as with-

out characteristic length or time scales, and one would ex-

pect the well-known self-similar motion of a Sedov-Taylor

blast wave (Taylor, 1950; Sedov, 1959; Zeldovich and Raizer,

1966; Liang and Keilty , 2000), rs ∝ tα , where the exponent

α = 2/5. In a case where radiation removes energy from

the shock in an optically thin environment, analytical and

numerical studies predict a slower shock expansion, such

as α = 2/7 (the “pressure-driven snowplow”), α = 1/4 (the

“momentum-driven snowplow”; the shock is simply coast-

ing) (McKee and Ostriker, 1977; Blondin et al., 1998), and

2/7 < α < 2/5 (the thermal energy of the shocked gas is not

completely radiated away) (Cohen et al., 1998; Liang and

Keilty , 2000).

In a case where the environment is not optically thin –

which is the case for many experiments including ours –

radiation is reabsorbed in the upstream material and if the

shock is travelling fast enough a supersonic, radiative heat

wave (RHW) breaks away from the shock in a situation anal-

ogous to a supercritical shock wave (Zeldovich and Raizer,

1966). It has been shown that the shock and RHW will coexist

and eventually propagate as r ∝ tα where α is larger for the

shock (Reinicke and Meyer-ter-Vehn, 1991). This means that

the shock would eventually catch RHW, after which a second

state is obtained in which RHW is of the ablative type and

the shock moves in a classical Sedov-Taylor trajectory with

α = 2/5.

In our earlier experiments (Hansen et al., 2006) we came

across an additional possibility for optically thin environ-

ments, namely that prior to the shock catching RHW, the lat-

ter enters a transonic regime, stalls, and generates a second

shock (S2). We showed that an analytical estimate for the

formation radius of S2 can be obtained from a standard result

of heat front physics (Zeldovich and Raizer, 1966; Mihalas

and Weibel Mihalas, 1984; Hatchett, 1991), using the 1D

fluid equations for conservation of mass ρ1u1 = ρ2u2 and

momentum p1 + ρ1u21 = p2 + ρ2u2

2 in the frame of RHW,

where subscript 1 denotes the region ahead of RHW, and sub-

script 2 denotes the region behind RHW. Assuming an ideal

gas (so that p = ρc2 where c is the isothermal sound speed

(Mihalas and Weibel Mihalas, 1984)) we combine these to

obtain

ρ2

ρ1

=c2

1 + u21 ±

√

(

c21 + u2

1

)2 − 4c22u2

1

2c22

. (1)

A supersonic (u1 > c1) RHW and a real compression η ≡ρ2/ρ1 requires

u1 ≥ c2 +√

c22 − c2

1 ≈ 2c2 (2)

(where the approximation is valid because the temperature

behind RHW is much higher than the temperature before it),

i.e., requires the mixed Mach number

M ≡ u1

c2

≥ 2. (3)

Springer


Once the Mach number drops to 2, RHW can no longer fulfill

Equation (1), and S2 forms at RHW. S2 immediately moves

ahead of RHW and acts to slow down u1 so that RHW is now

subsonic, satisfying

u1 ≤ c2 −√

c22 − c2

1 ≈ c21

2c2

. (4)

In the limit of transition from super- to subsonic, ρ2/ρ1 =u1/c2 and u2 = c2.

To estimate the radius rh of RHW at the moment when its

Mach number is 2, and thus the formation radius of S2, we

can assume a radiative conductivity of the ambient gas of the

form

χ = χ0ρa T b (5)

and use Barenblatt’s solution for an instantaneous point re-

lease of energy (Barenblatt, 1979; Reinicke and Meyer-ter-

Vehn, 1991):

rh (t) = r0tδ (6)

where

r0 =(

K b1 K2

)δ(7)

δ = 1

3b + 2(8)

K1 = γ − 1

2π B

(

3

2, 1 + 1

b

)

E

ρ0

(9)

K2 = 2χ0 (γ − 1)

Ŵb+1ρ1−a0

1

bδ, (10)

where γ is the adiabatic index, m0 is the molecular weight,

B (x, y) is the beta function, and Ŵ is the gas constant from

the ideal gas equation of state

(γ − 1) e = ŴT, (11)

where e is the internal energy [i.e., the heat capacity cv =Ŵ/(γ − 1)]. The RHW Mach number is obtained from

u1 = drh

dt(12)

and

c2 (r = 0) =(

K 21 K −3

2 t−3)δ/2 = K δ

1 K−3δ/22 (rh/r0)−3/2 .

(13)

Fig. 2 Measured shock radius versus time in experimental images ofshocks in xenon (El ≈ 100 − 200 J). Note the step in radius around12 mm when both shocks are visible

Using the sound speed at the RHW center r = 0 is a reasonable

approximation as the temperature profile throughout RHW is

quite flat (Hansen et al., 2006; Barenblatt, 1979; Reinicke and

Meyer-ter-Vehn, 1991). We should also point out that using

Barenblatt’s solution to estimate the RHW radius assumes that

radiation can be treated in the diffusion approximation. In the

earliest expansion phase this is not the case, but Barenblatt’s

solution turns out to be reasonable approximation when the

wave has cooled somewhat and has a large enough optical

depth. For pure Xe this would be at t ≈ 20 ns; at this time

rh ≈ 4 mm, compared to rs ≈ 1.4 mm and only about one

eight the total energy still residing in the initial shock (the

rest is in the RHW). For pure Xe with ρ0 ∼ 10−5 g/cm3 and

E = 5 J, this analytical estimate says that the RHW Mach

number drops to Mach 2 when rh ≈ 10 mm, in reasonable

agreement with the experiment where we first observed S2

with r ≈ 12 mm (Hansen et al., 2006), see Fig. 2.

We report here on a new series of experiments using mix-

tures of Xe and N2. The goal of these experiments was to

further study the previously unreported second shock, to

see what the effect is of changing the radiative conductiv-

ity (or equivalently the opacity) and the heat capacity, and to

see if the above analytical estimate holds under a range of

conditions.

3 Experiment set-up and diagnostics

We create spherically expanding blast waves in the follow-

ing fashion: a high-energy infrared pulsed laser (1064 nm

wavelength) is focused onto the tip of a solid (stainless

steel) pin surrounded by a Xe/N2 mixture with a density

of ρ0 = 3.6 × 10−5 g/cm3. The laser pulse is 5 ns in dura-

tion with energy El ≈ 5 J (the exact energy fluctuates slightly

from shot to shot but is measured and recorded). The laser

Springer


energy is deposited in pin material which then becomes very

hot and expands rapidly, pushing at the surrounding gas, set-

ting up a strong, radiative initial shock (S1). At the end of the

laser pulse (t = 5 ns), S1 is traveling in excess of 60 km/s

and is (at least for mixtures with a significant fraction of

Xe) strongly radiative. When the radius rs ∼ 0.4 mm, S1 has

swept up enough material that the details of its initial condi-

tions are unimportant. Radiation from S1 heats the surround-

ing gas. The (inverse) Boltzmann number Bo−1 50 initially,

i.e., S1 is supercritical and drives a supersonic RHW that trav-

els rapidly outward, leaving a large separation between S1 and

RHW. With time, S1 slows and its ability to radiate efficiently

quickly decreases. Also, S1 is traveling into the counter pres-

sure of hot RHW plasma, which is becoming comparable to the

ram pressure; the Mach number of S1 drops rapidly, and the

post-shock compression reduces correspondingly. The Mach

number for RHW is also decreasing, and when it reaches ∼2,

RHW stalls and creates S2, with a radius measured in ear-

lier experiments in pure Xe of ∼12 mm. After this time, S1

continues to weaken until it dissipates, while S2 is essentially

non-radiative and once it has swept up enough mass (doubled

its initial radius), it propagates like rs ∝ t2/5, see Fig. 2.

To image S1 and S2 on spatial scales up to ∼5 cm, we used

two lenses in a telescope configuration and a gated, single-

frame, high-speed CCD camera (2 ns gate), along with a low

energy, green laser pulse (λ = 532 nm wavelength, 15 ns du-

ration) as a backlighter. We employed a schlieren technique

with a vertical knife edge at the telescope focal point to

remove light which had not been deflected by the plasma.

With this method, image brightness corresponds to the spa-

tial derivative of plasma electron density in the horizontal

direction, so that vertical structures in the plasma are read-

ily seen. A monochromatic filter was placed in front of the

camera to prevent damage to the CCD (damage occured in

our previous experiments), with the disadvantage that glow

from the heated plasma could not be seen in this experiment.

A spectrometer was used to obtain spectral line intensity

as a function of position (ahead of and behind the blast wave),

which was then Abel inverted to get spectral line intensity as

a function of radius. An estimate of temperature as a function

of radius was then calculated from pair-wise line ratios. This

was done in pure N2 using two NII lines (399.5 nm and 444.70

nm) and two NIII lines (451.485 nm and 463.413 nm) and in

pure Xe using three XeII lines, 441.48 nm, 446.22 nm, and

460.3 nm.

4 Results

Keeping the density constant (by keeping the partial pres-

sures pXem Xe + pN2m N2

= C where C is a constant), we

varied the composition of the ambient gas from 100% Xe

(by mass) to 100% N2 and tracked the formation of the

second shock. We found that the higher the fraction of N2,

the smaller the formation radius became (and the sooner the

second shock forms). This trend is shown in Fig. 3, where

each column represent a certain mixture of Xe and N2, and

time runs toward the bottom of each column. In the top row of

images, we only see the initial shock. As we follow each col-

umn down, the second shock forms, and the initial shock dis-

sipates; this should be particularly obvious around the middle

of each column. The bottom row shows images where only

the second shock can be seen.

It is worthwhile noting the experimental difficulty in ob-

serving the second shock. Previous experiments have not ob-

served secondary shock formation, and this may be because

of any of the following: (a) the experiment was studying ra-

diative shocks, so images were obtained only relatively early

in time, while S1 is still radiative, (b) the experiment was

studying blast waves, so images were only obtained rela-

tively late in time, to ensure that a stable Sedov-Taylor blast-

wave had formed, (c) images were too sparse in time, i.e., the

sequence of images shows S1 in the first few images, then

switches to S2 without capturing the moments when both

exist simultaneously, and the experimenter believed – quite

naturally – that the same shock was observed in all the im-

ages, (d) the schlieren technique was not sensitive enough.

The latter condition is one that we struggled with. When the

knife-edge position was not carefully calibrated, we obtained

images for the intermediate times (when both shocks exist)

that simply show no shock at all (both shocks are too weak

to perturb the plasma enough to overcome the crudely posi-

tioned knife-edge).

Figure 3 can be represented in a bar plot, where each

bar represents an uncertainty in the formation radius of the

second shock; the lower end of each bar is the radius of S1 in

an image where the second shock cannot yet be seen, while

the upper end is the radius of S2 in the earliest image for

each mixture in which we can see both S1 and S2. We have

measured all shock radii and are showing this data in Fig. 4.

Also shown in Fig. 4 are second shock formation radii from

our previous work in pure Xe (Hansen et al., 2006).

5 Comparison to analytical estimate

Barenblatt’s solution assumes a radiative heat conductivity

which is a function of temperature (and density), but the heat

capacity is implicitly assumed to be constant. This means

that, at best, Barenblatt’s solution will only be an approx-

imation to real gases with non-constant heat capacities. To

estimate when the RHW becomes transonic, we have used this

solution taking the heat capacity for the conditions around

where we expect the transition to occur, namely T ∼ 3.8 eV

based on emission spectroscopy data in the near-ultraviolet

Springer


Fig. 3 Images of shocks inXe/N2 mixtures with a densityρ0 = 3.6 × 10−5 g/cm3. createdby a laser focused on a pin(visible in most images). Themixture is the same within eachcolumn of images, and themass-fraction of each gas iswritten at the top of the column.The number at the top left ofeach image is the experimentaltime in nanoseconds (after thelaser pulse)

range in pure Xe and pure N2. The following steps illustrate

our method in this regard:

1. The adiabatic index γ for each gas mixture is assumed

unknown and is determined from Sedov-Taylor’s formula

for a self-similar blast wave:

rs =(

75

16π

(γ − 1) (γ + 1)2

3γ − 1

)

15(

E

ρ0

)15

t25 , (14)

where we use the measured blast-wave radius rs from an

image obtained at a very late time t so that the second

shock is well-developed past the point where its initial

conditions matter, in addition to the recorded laser energy

E and measured density ρ0. We typically find γ ≈ 1.05,

see Table 1.

2. The gas constant Ŵ is calculated from Equation (11) using

the adiabatic index γ from Step 1, an initial estimate of

a representative temperature T ∼ 3.8 eV, and an internal

Springer


Fig. 4 Calculated and experimentally observed second shock forma-tion radii as a function of Xe mass fraction in Xe/N2 mixtures. Thelower/upper end of each bar represents an image where the secondshock cannot/can be seen

energy

e = e0ρf T g (15)

with parameters e0, f , and g from Table 1.

3. The various parameters appearing in Barenblatt’s solution

[in Equations (7–10)] are calculated using the adiabatic

index γ from Step 1, the gas constant Ŵ from Step 2,

and a radiative conductivity from Equation 5 using the

parameters χ0, a, and b from Table 1.

4. The RHW radius rh can be solved for analytically us-

ing Equations (12–13) above (but the expression is too

complicated to cast any light on the physics and is not

included here). We then set the mixed mach number

M = u1/c2 = 2 to obtain the radius r2 = rh |M=2 when

the second shock forms. [Note that if the parameter b ≫ 1

(which is the case here) the simple estimate

r2 ≈(

K b1 K2

6b

)1/3b

(16)

can be used to 15% accuracy.]

As a final check one could calculate the temperature

T = c22

Ŵ. (17)

inside RHW and compare to the initial estimate of T ∼ 3.8 eV

obtained from our spectrometer data. We find that a temper-

ature calculated from Equation 17 agrees quite well with the

Table 1 Radiative conductivity and heat capacity parametersfor mixtures of nitrogen and xenon

χ0 a b e0 f g γ

5% Xe, 95% N2 1 × 10−32 −2.2 7 160 0.1 1.4 1.03

20% Xe, 80% N2 3 × 10−32 −2.2 7 50 0.1 1.5 1.05

40% Xe, 60% N2 5 × 10−32 −2.2 7 40 0.1 1.5 1.06

60% Xe, 40% N2 8 × 10−32 −2.2 7 30 0.1 1.5 1.05

80% Xe, 20% N2 1 × 10−31 −2.2 7 7 0.1 1.6 1.05

100% Xe 1 × 10−44 −2.2 10 2.6 0.1 1.65 1.05

spectrometer estimate. If we go back and use the temperature

from Equation 17 in Step 1 we get less than a 4% difference

in our final answer for r2.

The calculated values for r2 in six different mixes of Xe

and N2 are shown alongside the experimental data in Fig. 4.

The trend of smaller formation radii for higher fractions of

N2 is reproduced by analytical estimates, and generally the

agreement between analytical estimate and the experimental

data is quite good.

6 Summary

We have varied in a systematic way the opacity and heat ca-

pacity of the gas into which a spherical shock wave expands

(by mixing Xe with N2 keeping the mass density constant)

and measured when a second shock forms ahead of the initial

shock. The formation radius of the second shock as a function

of Xe mass fraction is consistent with an analytical estimate

where the expansion of a radiatively driven heat wave is es-

timated using Barenblatt’s solution and the heat wave then

stalls as its Mach number drops to ∼2.

Acknowledgements We thank Dwight Price and the staff at the Janusfacility (where the experiments were conducted) for their valuable as-sistance.

This work was performed under the auspices of the U. S. Departmentof Energy by the University of California, Lawrence Livermore NationalLaboratory under Contract No. W-7405-Eng-48.

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namics. Oxford University Press, Oxford (1984)

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Springer


DOI 10.1007/s10509-006-9226-5


FLASH Code Simulations of Rayleigh-Taylor andRichtmyer-Meshkov Instabilities in Laser-Driven Experiments

Nathan C. Hearn · Tomasz Plewa · R. Paul Drake ·

Carolyn Kuranz


Abstract We present two- and three-dimensional simula-

tions involving Richtmyer–Meshkov and Rayleigh-Taylor

instabilities run with the adaptive mesh refinement code,

FLASH. Variations in the rate of mixing layer growth due

to dimensionality, perturbation modes, and simulation reso-

lution are explored. These simulations are designed for de-

tailed comparisons with experiments run on the Omega laser

to gain understanding of the mixing processes and to prepare

for validation of the FLASH code.

Keywords Adaptive mesh refinement . Rayleigh-Taylor .

Richtmyer-Meshkov . Supernova . Code validation . Omega

laser

1 Introduction

Rayleigh-Taylor and Richtmyer-Meshkov instabilities are

thought to play a critical role in the final distribution of el-

ements, the spectra, and the light curve evolution of a su-

pernova explosion (Kifonidis et al., 2006; Haas et al., 1990;

Blinnikov, 1999). Therefore, the supernova calculations cur-

rently being run at the ASC Flash Center with its adaptive

mesh refinement code, FLASH (Fryxell et al., 2000), must be

able to model these processes accurately. To gain an under-

standing of the processes, and confidence in the models, we

are constructing a set of FLASH simulations that can be di-

rectly compared with experiments of shock-induced mixing

on the Omega laser.

N. C. Hearn () · T. PlewaASC Flash Center, University of Chicagoe-mail: [email protected]

R. P. Drake · C. KuranzSpace Physics Research Laboratory, University of Michigan

Such detailed comparisons are a critical part of FLASH val-

idation process (see (Calder et al., 2002; Weirs et al., 2005)).

As the Rayleigh-Taylor instability plays an important role in

these experiments, this work may help to resolve discrepan-

cies between experimentally measured and numerically es-

timated growth rates (see Dimonte et al. (2004)). Simulation

studies of similar experiments have been performed using

other codes (see, for instance, Miles et al. (2004)).

The aim of these experiments is to study these mixing pro-

cesses in a two-fluid medium (see the article by C. Kuranz

et al., this issue; (Drake et al., 2004; Kane et al., 1997; Miles

et al., 2004; Robey et al., 2001)). Here, a polyimide plastic

and a lower-density foam are formed into a cylinder, such that

the interface between the two media is planar (transverse to

the cylinder axis) with sinusoidal perturbations, where one

or more perturbation modes are present. A thin layer of poly-

imide (herein referred to as the “tube wall”) was also placed

around the perimeter of the cylinder. In the experiment, a

planar blast wave propagates across the interface from the

polyimide into the foam. Here, the Richtmyer-Meshkov in-

stability initiates the mixing, and the Rayleigh-Taylor insta-

bility continues it.

2 Methods and simulation setup

The simulations presented here have dimensions compatible

with those of the Omega laser experiments. Not including the

∼25 µm tube wall, the diameter of the cylinder is 900 µm,

and the amplitude of each interface perturbation mode is

2.5 µm. The primary perturbation mode has a wavelength of

71µm, applied along both transverse axes, with an additional

213 µm wavelength perturbation applied along one axis in

the double-mode cases. These perturbations are applied in

either two or three dimensions as described below.

Springer


Fig. 1 A section of the mixinglayer isosurface from a 3Ddouble-mode simulation. Thetime relative to the start of thesimulation is shown in thecorner of each panel. The initialperturbation is shown in theupper left panel, and thesubsequent panels show theshape of the mixing layer afterthe passage of the shock wave

The shape and magnitude of the initial pressure pulse

was computed with a one-dimensional HYADES (Larsen and

Lane, 1994) simulation that models the initial laser-driven

shock propagation in the cylinder. The equation of state for

these simulations is approximated by a ideal gas with an adi-

abatic index of 1.4. The effective adiabatic index remains

below 5/3 because the polyimide is not fully ionized.

The simulations discussed in this paper were run with

adaptive mesh refinement at three different resolutions. With

the lowest resolution (labeled L20), there are 256 elements

across the diameter of the tube, or roughly 20 elements per

wavelength of the primary perturbation. The intermediate

resolution (L40) has 40 elements per wavelength, and the

highest resolution (L80) has 80 elements per wavelength.

Reflective boundary conditions are used for the sides of the

cylinder, except for the case where the polyimide tube wall

enclosing the cylinder is present (Fig. 3), in which an outflow

boundary is used.

This set of simulations was run with a uniform polyimide

density distribution that does not include the enhanced den-

sity at the location of the pressure pulse. However, newer

simulations that include the correct density profile do not

show any qualitative differences in their evolution.

3 Dynamics overview

Comparing the dynamics of 2D and 3D simulations is impor-

tant for gaining an understanding of the growth of the insta-

bilities. Simulations in three dimensions allow us to explore

situations that are more like those found in the actual experi-

ments and in supernovae. By working in a three-dimensional

space, the fluid has an additional degree of freedom for shap-

ing its flow.

Abundance isosurfaces denoting equal polyimide and

foam composition from a double-mode, 3D simulation are

shown in Fig. 1 at four moments in time. The upper-left panel

shows the initial perturbation before the shock has passed.

Here, the interface has a 71 µm wavelength perturbation run-

ning in both directions, of the form cos y cos z, and a 213 µm

wavelength perturbation along the y-axis only, of the form

cos y. (In all cases, the x-axis is parallel to the axis of the

cylinder.) After the shock crosses the interface, the spike-and-

bubble structures are produced, which are illustrated in the

subsequent panels. The final time of this simulation (30 ns)

will be the reference time used in all of the simulations pre-

sented in this paper.

4 Self-convergence in 2D

The convergence properties of the simulations (with respect

to resolution) are depicted in Fig. 2. Here, the mixing layer of

a 2D, double-mode simulation is shown at the final time. The

plotted variable is the mass fraction of the polyimide material,

with the color-to-mass-fraction correspondence shown in the

inset legend. The convergence of the mixing layer thickness

with respect to resolution is illustrated in Fig. 6.

Springer


Fig. 2 Polyimide massfractions for the final times(30 ns) of three 2D simulationswith the same perturbation(double mode), but differentresolutions. From left to rightare resolutions L20, L40, andL80. (See the inset legend forcolors corresponding topolyimide mass fractions.) Notethe increase of structure withresolution

Fig. 3 Evolution of a flatinterface with and without a tubewall present. The top row offigures shows two times (10 and30 ns) from a simulation with aflat (perturbation-free) interface,without a tube wall, but withreflective boundary conditionsalong the sides of the cylinder.The second row shows the sametimes from a simulation with atube wall. In the presence of thetube wall, the interface showssome significant deformationnear the edges, but remainsreasonably flat in the center

5 Adding the tube wall

We have implemented a simple model of the tube wall by ex-

panding the computational domain and filling the additional

volume with polyimide. In this trial, an additional 100 µm

of polyimide material was added around the cylinder, with

outflow boundary conditions applied to the outside of the

tube wall. This tube is somewhat thicker than the real tube in

the experiments, but this arrangement allows us to study the

tube wall’s interaction with the interior media while avoiding

the need to model the external vacuum. The initial pressure

pulse was set in the tube wall in the same fashion as the rest

of the material, resulting in a shock wave that is planar prior

to crossing the interface. A comparison of the dynamics of

a flat (unperturbed) interface in 2D with and without a tube

wall at resolution L80 is shown in Fig. 3.

Once the shock wave crosses the material interface,

boundary effects are seen due to the different shock speeds

in the foam and the polyimide tube walls. A number

of important phenomena result from the presence of the

tube:

Fig. 4 Comparison of polyimide mass fractions between single anddouble perturbation modes. The final time of a 2D simulation with thesingle-mode perturbation is shown in the left panel, and an image fromthe double-mode simulation at the same time is displayed to the right.The position of the mixing layer is the same, but the thickness of themixing layer is 8% smaller in the single-mode case

1. A curved shock front (not visible here, but seen in density

plots),

2. Deposition of foam along the inner tube walls behind

the interface,

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Fig. 5 Polyimide mass fractions from two-dimensional slices throughthe three-dimensional simulations. The left and middle panels showthe final times for 3D simulations with single and double mode pertur-bations, respectively. For reference, the image from a 2D double-mode

simulation at the same resolution is displayed at the right. The 3D simu-lations have a much thicker interface compared to their 2D counterparts(by roughly 40%), with spikes that are narrower

Fig. 6 The growth of themixing layer for the variousperturbation modes (single anddouble) and resolutions (L20,L40, and L80) in 2D. A trendtowards faster growth withhigher resolution is clear. Herewe see evidence for convergencein the double-mode simulations,but more analysis is required forthe single-mode simulations.For comparison, the preliminarygrowth curves for single anddouble mode experiments areshown

3. A transverse displacement of the tube walls behind the

interface, and

4. Deformation of the interface due to lateral motion and

wave propagation along the interface.

At late times, the central regions of the interface tend to

flatten out, which agrees with simulations of the perturbations

and tube walls combined that have also been run. Only a small

deviation in the position of the interface along the cylinder

axis is seen. The other simulations shown in this paper do not

include the tube walls, so their analysis can only focus on the

properties of the mixing layer near the axis of the cylinder.

6 Single- versus double-modes

The polyimide mass fractions for the final time of a single-

mode 2D simulation is shown in the left panel of Fig. 4. For

comparison, the final state of the 2D double-mode run of the

same resolution (L80) is displayed to the right. The thickness

of the mixing layer is about 9% larger in the double-mode

simulation.

7 3D Models

Two 3D simulations are shown in Fig. 5, one with a sin-

gle perturbation mode (left panel), and one with the double

mode (middle panel). For reference, a double mode 2D sim-

ulation at the same resolution (L20) is shown in the right

panel. The double mode 3D simulation is the same as shown

in Fig. 1, with the longer wavelength perturbation varying

only along the y-axis. The slices shown in Fig. 5 are parallel

to the x-z plane, at a y-coordinate where the amplitude for

the short wavelength perturbation is at a maximum. Owing

to the greater coordinate freedom, the mixing layer in the 3D

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Fig. 7 The growth of themixing layer for the variousperturbation modes (single anddouble) at resolution L20 in 3D.The faster growth withdouble-mode perturbations isseen here, as well as reasonablygood agreement with theexperimental data (see Fig. 6).The non-monotonic behavior atselect points is likely due toinaccuracies in estimating themixing layer thickness

runs is thicker, and features narrower, more-elongated struc-

tures compared with the 2D counterparts. These results are

consistent with previous work on this subject (see Kane et

al. (2000)).

8 Discussion

We have explored the effects of resolution, dimensionality,

and perturbation modes on the growth of the mixing layer

for these simulations. We note three specific factors that can

increase the thickness of the mixing layer:

1. Increasing resolution, with some signs of convergence

at higher resolutions (as seen in comparisons between

L40 and L80; even higher resolutions may be needed for

convergence in 3D),

2. Increasing the number of perturbation modes, and

3. Greater dimensionality (2D versus 3D).

Of these factors, it would appear that the last (dimension-

ality) has the greatest effect in this sample, with a roughly

40% increase in mixing layer thickness between the 2D and

3D L20 runs. As the resolution is increased in the 2D runs

(from L20 to L80), the difference in mixing layer thickness

between the single and double-mode runs decreases from

about 25% to just under 10%.

Acknowledgements This work is supported in part by the U.S. De-partment of Energy under grant no. B523820.

References

Blinnikov, S.I.: Astron. Lett. 25, 359 (1999)Calder, A.C., Fryxell, B., Plewa, T., et al.: ApJS 143, 201 (2002)Dimonte, G., Youngs, D.L., Dimits, A., et al.: Phys. Fluids 16, 1668

(2004)Drake, R.P., Leibrandt, D.R., Harding, D.R., et al.: Phys. Plasmas 11,

2829 (2004)Fryxell, B., Olson, K., Ricker, P., et al.: ApJS, 131, 273 (2000)Haas, M.R., Erickson, E.F., Lord, S.D., et al.: ApJ 360, 257 (1990)Kifonidis, K., Plewa, T., Scheck, L., Janka, H.-Th., Muller, E.: A&A

453, 661 (2006)Kane, J., Arnett, W.D., Remington, B.A., et al.: ApJ 478, L75

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(1994)Miles, A.R., Braun, D.G., Edwards, M.J., et al.: Phys. Plasmas 11, 3631

(2004)Miles, A.R., Edwards, M.J., Blue, B., et al.: Phys. Plasmas 11, 5507

(2004)Robey, H.F., Kane, J.O., Remington, B.A., et al.: Phys. Plasmas 8, 2446

(2001)Weirs, G., Dwarkadas, V., Plewa, T., Tomkins, C., Marr-Lyon, Mark.:

Ap&SS 298, 341 (2005)

Springer


DOI 10.1007/s10509-006-9216-7


Models of Very-High-Energy Gamma-Ray Emission from the Jetsof Microquasars: Orbital Modulation

Markus Bottcher · Charles D. Dermer


Abstract The recent detection of very-high-energy (GeV –

TeV) γ -ray emission from the Galactic black-hole candidate

and microquasar LS 5039 has sparked renewed interest in jet

models for the high-energy emission in those objects. In this

work, we have focused on models in which the high-energy

emission results from synchrotron and Compton emission by

relativistic electrons in the jet (leptonic jet models). Particu-

lar attention has been paid to a possible orbital modulation

of the high-energy emission due to azimuthal asymmetries

caused by the presence of the companion star. Both orbital-

phase dependent γ γ absorption and Compton scattering of

optical/UV photons from the companion star may lead to

an orbital modulation of the gamma-ray emission. We make

specific predictions which should be testable with refined

data from HESS and the upcoming GLAST mission.


non-thermal . X-rays: binaries . Stars: winds, outflows

1. Introduction

Recent observations (Aharonian et al., 2005) of 250 GeV

γ -rays with the High Energy Stereoscopic System (HESS)

from the X-ray binary jet source LS 5039 establish that mi-

croquasars are a new class of γ -ray emitting sources. These

results confirm the earlier tentative identification of LS 5039

with the EGRET source 3EG J1824-1514 (Paredes et al.,

M. Bottcher ()Astrophysical Institute, Department of Physics and Astronomy,Ohio University, Athens, OH 45701, USA

C. D. DermerE. O. Hulburt Center for Space Research, Code 7653 NavalResearch Laboratory, Washington, D.C. 20375-5352

2000). In addition to LS 5039, the high-mass X-ray binary

LSI 61303 (V615 Cas) also has a possible γ -ray counterpart

in the MeV – GeV energy range (Gregory and Taylor, 1978;

Taylor et al., 1992; Kniffen et al., 1997). Microquasars now

join blazar AGNs as a firmly established class of very-high

energy γ -ray sources. The nonthermal continuum emission

of blazars is believed to be produced in a relativistic plasma

jet oriented at a small angle with respect to our line of sight.

Their radio through UV/X-ray emission is most likely due to

synchrotron emission by relativistic electrons in the jet, while

the high energy emission can be produced by Compton up-

scattering of lower-energy photons off relativistic electrons

(for a recent review, see, e.g. Bottcher, 2002), or through

hadronic processes (Mannheim and Biermann, 1992; Atoyan

and Dermer, 2001; Mucke et al., 2003).

Because of their apparent similarity with their supermas-

sive AGN cousins, it has been suggested that the same pro-

cesses may operate in the jets of Galactic microquasars,

which may thus also be promising sites of VHEγ -ray produc-

tion (e.g., Romero et al., 2003; Bosch-Ramon et al., 2005).

In the case of leptonic microquasar jet models, possible

sources of soft seed photons for Compton upscattering are

the synchrotron radiation produced in the jet by the same ul-

trarelativistic electron population (SSC = synchrotron self-

Compton; Aharonian and Atoyan, 1999), or external pho-

ton fields (Bosch-Ramon and Paredes, 2004; Bosch-Ramon

et al., 2005). In high-mass X-ray binaries like LS 5039, the 1–

10 keV luminosities from the accretion disk is typically much

lower than the characteristic bolometric luminosity of the

high-mass companions. Consequently, the dominant source

of external photons is the companion’s optical/UV photon

field.

In addition to providing an orbital-period dependent seed

photon field for Compton scattering, the intense radiation

field of the high-mass companion will also lead to γ γ

Springer


absorption of VHE γ -rays in the ∼100 GeV – TeV pho-

ton energy range if VHE photons are produced close to the

base of the jet.

We (Bottcher and Dermer, 2005; Dermer and Bottcher,

2006) have recently presented detailed analyses of the ex-

pected high-energy spectrum and its orbital modulation due

to the orbital-period dependence of both the soft photon field

and the γ γ absorption. Here, we briefly summarize and dis-

cuss those results.

2. Model description

We choose a generic model set-up in which the orbital plane

of the binary system defines the (x1, x2) plane. The jet, as-

sumed to be perpendicular to this plane, defines the x3 axis,

and is inclined with respect to our line of sight by an incli-

nation angle i . An azimuthal (phase) angle φ is defined such

that φ = 0 in the direction of the x2 axis. The line of sight

lies in the (x2, x3) plane. The γ -ray production site is located

at a height z0 along the jet. Model parameters of the sys-

tem have been motivated by recent observational results of

Casares et al. (2005) for LS 5039: Mass of the compact ob-

ject MX = 3.7+1.3−1.0 M⊙, orbital period P = 3.91 d, luminos-

ity of the companion star L∗ = 105.3 L⊙, effective tempera-

ture of the companion star Teff = 39, 000 K, inclination an-

gle i = 25, and an orbital separation of s ≈ 2.5 × 1012 cm.

3. Results

VHE γ -ray photons in the range ∼100 GeV – 1 TeV will be

efficiently absorbed by the intense photon field of the com-

panion star. The inset to Fig. 1 illustrates the shape of the

resulting absorption trough and its dependence on the orbital

phase. Here we assume that the intrinsic γ -ray spectrum is a

power-law with photon index αph = 2.5, and z0 = 1012 cm.

The various curves illustrate the orbital modulation of the

absorption trough, with the lowest (most heavily absorbed)

curve corresponding to φ = 0 and the highest (least ab-

sorbed) curve corresponding to φ = π . The modulation is

a combined consequence of two effects: for phase angles

closer to π , (a) the average distance of the star to any point

on the line of sight is longer and (b) the angle of incidence

θ is smaller, causing the threshold for γ γ pair production

to increase as ǫthr = 2/(ǫ∗ [1 − µ]). This leads to a decreas-

ing overall depth of the absorption trough, and a shift of the

minimum of the absorption trough towards higher photon

energies.

The main frame of Fig. 1 shows the dependence of the

absorption feature on the location z0 of the VHE γ -ray pro-

duction site. The γ γ opacity is plotted for two photon ener-

gies, E = 250 GeV, and E = 1 TeV at φ = 0. GeV – TeV

1011

1012

1013

z0 [cm]

10

10

10

100

101

τ γγ (

φ0 =

0)

E = 250 GeV

E = 1 TeV

101

102

103

104

E [GeV]

10

10

10

νF

ν

Fig. 1 γ γ opacity at 250 GeV and 1 TeV as a function of the distanceof the photon production region from the central compact object atphase φ = 0. The figure illustrates that (1) VHE photons producedwithin a few ×1012 cm (i.e., of the order of the orbital separation of thebinary system) would be subject to substantial γ γ absorption; (2) theminimum of the absorption trough (maximum of τγ γ as a function ofphoton energy) is shifting towards higher energies for larger distancesfrom the central source. Inset: Orbital modulation of the expected γ γ

absorption trough, assuming a power-law spectrum with photon indexαph = 2.5 and a photon production site at z0 = 1012 cm. The differentcurves represent the escaping photon spectrum at various orbital phases,from φ = 0 (lowest curve) to φ = π (highest curve) in steps of π/10

photons produced within z0 ∼ s from the compact object

will be heavily attenuated for this phase angle. For photons

produced at z0 ≫ s, γ γ attenuation becomes negligible. Our

results concerning the effects of γ γ attenuation can briefly

be summarized as follows:

(1) VHE γ -rays produced closer to the central engine than

z0 of the order of the binary separation s, would be

subject to very strong γ γ absorption due to the stellar

radiation field at orbital phases close to φ = 0.

(2) For VHE photon production sites at z0 s, the γ γ

opacity – and, thus, the VHE γ -ray flux – would be

strongly modulated on the orbital period of the binary

system. At orbital phases close to φ = π , the intrinsic

VHE γ -ray flux would still be virtually unabsorbed

even for z0 ∼ 1012 cm.

(3) The orbital modulation of the VHE γ -ray flux would be

characterized by a spectral hardening in the ∼300 GeV

– 1 TeV range during flux dips. At lower energies, the

spectrum softens with decreasing flux, while the oppo-

site trend would be observed at lower photon energies,

E 100 GeV.

Besides γ γ opacity effects, an orbital modulation of the

γ -ray flux is also expected from azimuthally asymmet-

ric Compton scattering of companion-star photons in the

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10-12

10-11

10-10

10-9

10-8

103

106

109

1012

νF

ν(e

rgs c

m-2

s-1

)

E(eV)

φ = π

φ = 0

φ = π/2

φ = 3π/4

φ = π/4

Average

Fig. 2 The νFν spectral energy distribution from Compton scatteringof starlight photons of a high-mas companion star in a microquasarjet. Standard parameters of the system, appropriate to LS 5039, havebeen chosen. In particular, the height of the emission region abovethe orbital plane is z0 = 2.5 × 1012 cm, the jet plasma outflow LorentzfactorŴ = 2, and the number index of the electron distribution is p = 3.Broken lines indicate the spectra at various orbital phases, while thesolid line shows the spectrum averaged over all phases

microquasar jet. This effect is illustrated in Fig. 2. Analogous

to the phase-dependent modulation of the incidence angle

for γ γ absorption, this geometrical effect would also yield

a more favorable angle for Compton scattering of starlight

photons into the γ -ray regime at phases φ ≈ 0.

This effect leads to an overall softening throughout the

GeV – TeV photon energy range with increasing flux be-

cause the Klein-Nishina cutoff becomes noticeable at lower

observed photon energies for scattering events happening

closer to head-on. This is in contrast to the trend caused by

γ γ absorption, where a higher flux will be accompanied by

a harder spectrum at photon energies ∼100 GeV.

Figures 3 and 4 illustrate the dependence of the orbital

modulation of the starlight Compton-scattering spectrum on

the height z0 of the emission region. Both figures indicate a

peculiarity in that the orbital modulation is particularly strong

under a specific orientation of the system, in which at phase

φ = π the companion star would appear directly behind the

emission region as seen by the observer. In that case, Comp-

ton scattering events producing γ -rays in the direction of the

observer, would have to happen almost tail-on – a geometry

which is highly unfavorable for the efficiency of Compton

scattering. Consequently, under this special orientation, the

starlight-Compton-scattering spectrum in the direction to-

wards the observer would be severely suppressed, and the

orbital modulation is maximized.

Acknowledgement This work was partially supported by NASAthrough XMM-Newton GO grant no. NNG 04GI50G, NASA INGE-

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

103

106

109

1012

νF

ν(e

rgs c

m-2

s-1

)

E(eV)

2.5x1012

cmxj = 10

12 cm

5x1012

cm

1013

cm

5x1012

cm

Fig. 3 The νFν spectral energy distribution from Compton scatteringof starlight photons for various values of the emission-region height x j

at phases φ = 0 (upper curves) and φ = π (lower curves), for standardparameters as used in Fig. 2

10-18

10-16

10-14

10-12

10-10

10-8

0 2 4 6 8 10 12

φ = π

xj (10

12 cm)

νF

ν(e

rgs c

m-2

s-1

)νF

ν (100 MeV)

φ = 0

Fig. 4 100 MeV γ -ray flux as a function of the height of the emissionregion x j . The strong dip illustrates the peculiarly strong orbital mod-ulation in the case that the star appears directly behind the emissionregion at φ = π . The vertical lines indicate the various values of x j forwhich the spectral energy distributions are plotted in Fig. 3

GRAL Theory grant no. NNG 05GK59G, and GLAST Science Inves-tigation no. DPR-S-1563-Y. The work of C. D. D. is supported by theOffice of Naval Research.

References

Aharonian, F., et al.: Sci. 309, 746 (2005)Aharonian, F., Aharonian, A.: MNRAS 302, 253 (1999)Atoyan, A., Dermer, C.D.: Phys. Rev. Lett. 87, 221102 (2001)Bottcher, M.: In: Goldwurm, A., Neumann. D.N., Van, J.T.T. (eds.) The

Gamma-Ray Universe, Proceedings of the XXII Moriond Astro-physics Meeting, p. 151 (2002)

Bottcher, M., Dermer, C.D.: A&A 634, L81 (2005)

Springer


Bosch-Ramon, V., Paredes, J.M.: A&A 417, 1075 (2004)Bosch-Ramon, V., Romero, G.E., Paredes, J.M.: A&A 429, 267 (2005a)Casares, J., Ribo, M., Ribas, I., Paredes, J.M., et al.: MNRAS 364, 899

(2005)Dermer, C.D., Bottcher, M.: A&A 643, in press (2006)Gregory, P.C., Taylor, A.R.: Nature 272, 704 (1978)Kniffen, D.A., et al.: ApJ 486, 126 (1997)

Mannheim, K., Biermann, P.L., A&A 253, L21 (1992)Mucke, A., et al.: Astropart. Phys. 18, 593 (2003)Paredes, J.M., Martı, J., Ribo, M., Massi, M.: Sci. 288, 2340 (2000)Romero, G.E., Torres, D.F., Kaufman Bernado, M.M., Mirabel, I.F.:

A&A 410, L1 (2003)Taylor, A.R., Kenny, H.T., Spencer, R.E., Tzioumis, A.: ApJ 395, 268

(1992)

Springer


DOI 10.1007/s10509-006-9241-6


Time-Dependent Synchrotron and Compton Spectrafrom Microquasar Jets

S. Gupta · M. Bottcher


Abstract Jet models for the high-energy emission of Galac-

tic X-ray binary sources have regained significant interest

with detailed spectral and timing studies of the X-ray emis-

sion from microquasars, the recent detection by the HESS

collaboration of very-high-energy γ -rays from the micro-

quasar LS 5039, and the earlier suggestion of jet models for

ultraluminous X-ray sources observed in many nearby galax-

ies. Here we study the synchrotron and Compton signatures

of time-dependent electron injection and acceleration, and

adiabatic and radiative cooling in the jets of Galactic micro-

quasars.


non-thermal . X-rays: binaries

1 Introduction

In many sources, the accretion process associated with high

energy emission from X-ray binaries (XRBs) is coupled with

the expulsion of collimated, mildly relativistic bipolar out-

flows (jets) most likely perpendicular to the accretion disk.

In the standard picture, the high-energy (X-ray – γ -ray)

spectra of X-ray binaries generally consist of a soft disk

blackbody with a typical temperature of kT ∼ 1 keV, and

a power-law at higher energies. Neutron-star and black-hole

X-ray binaries exhibit at least two main classes of spectral

states, generally referred to as the high/soft state (H/S), and

the low/hard state (L/H) (for a review see, e.g. Liang, 1998;

S. Gupta () · M. BottcherAstrophysical Institute, Department of Physics and Astronomy,Clippinger Hall 251B, Ohio University, Athens, OH 45701–2979,USAe-mail: [email protected]

McClintock and Remillard, 2004). Additionally, the Very

High (VHS) (Miyamoto et al., 1991) and Intermediate (IS)

(Mendez and van der Klis, 1997) states have been identi-

fied. In the L/H state, microquasars exhibit a continuously

generated, partially self absorbed compact jet, with its radio

luminosity showing strong, non-linear correlation with X-ray

luminosity. No radio emitting outflow is associated with the

H/S state, whereas discrete, often mutiple ejections attributed

to unstable disk radius, are associated with the VHS and IS.

Jet models of microquasars have recently attracted great

interest, especially after the detection of VHE γ -ray emission

from the high-mass X-ray binary and microquasar LS 5039,

in combination with the tentative identification of several

microquasars with unidentified EGRET sources. These de-

tections have confirmed the idea that microquasars are a dis-

tinctive class of high and very high energy γ -ray sources.

In a recent paper (Gupta et al., 2006), we presented a

detailed study of various plausible scenarios of electron in-

jection and acceleration into a relativistically moving emis-

sion region in a microquasar jet, and subsequent adiabatic

and radiative cooling, where we paid particular attention to

the X-ray spectral variability, as motivated above. Here we

briefly summarise the key results from our detailed parameter

study presented in (Gupta et al., 2006).

2 Model setup

The accretion flow onto the central compact object is eject-

ing a twin pair of jets, directed at an angle θ with respect

to the line of sight. Two intrinsically identical disturbances,

containing non-thermal plasma (blobs) originate from the

central source at the same time, traveling in opposite direc-

tions along the jet at a constant speed v j = β j c. Let d be

the distance to the source, and µ ≡ cos θ . The time at which

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any radio component is observed, is denoted by tobs. We as-

sume that over a limited range in distance x0 ≤ x ≤ x1, rela-

tivistic electrons are accelerated and injected in the emission

region.

The blob’s (transverse) radius, R⊥, scales with distance

from the central engine as R⊥ = R0⊥ (x/x0)a , i.e., a = 0 cor-

responds to perfect collimation, and a = 1 describes a coni-

cal jet. In the following, we will consider values of 0 ≤ a ≤ 1.

If the magnetic field is dominated by the parallel compo-

nent B‖, magnetic flux conservation leads to a magnetic-

field dependence on distance from the central black hole of

B(x) = B0 (R⊥/R0⊥)−2 = B0 (x/x0)−2a .

Each electron injected into the emission region at rela-

tivistic energies, will be subject to adiabatic and radiative

cooling, described by

−dγ

dt= 1

VB

dVB

dt

γ

3+ 4

3c σT

u

mec2γ 2 (1)

in the co-moving frame of the blob, where the first term

on the r.h.s. describes the adiabatic losses. The second

term describes synchrotron and Compton losses, with u =uB + urad, where uB = B2/8π is the magnetic-field energy

density, and urad is the seed photon energy density for Comp-

ton scattering in the Thomson regime. The term urad consists

of contributions from the X-ray emission from an optically

thick accretion disk, from the intrinsic synchrotron radiation,

and from external photons from the companion star.

The standard parameter choices for our “baseline” model,

summarized in Table 1, are broadly representative of GRS

1915+105 in the low/hard state, which also give equipartition

between the energy densities of the relativistic electrons and

the magnetic field in the ejecta.

Once a solution γi (ti ; γ, t) to Eq. (1) is found, the electron

distribution at any point in time (and thus at any point along

the jet) can be calculated through the expression

Ne(γ ; t) =∫ t

t0

dtid2 Ne(γi , ti )

dγi dti

∣

∣

∣

∣

dγi

dγ

∣

∣

∣

∣

. (2)

Radiation mechanisms included in our simulations are

synchrotron emission, Compton upscattering of synchrotron

photons, namely synchrotron self-Compton (SSC) emission,

and Compton upscattering of external photons. With the

time-dependent (and thus x-dependent) non-thermal electron

spectra in Eq. (2), we then use a δ-function approximation

(to estimate the νFν spectral output fsyǫ at a dimensionless

photon energy ǫ = hν/mec2 (in the observer’s frame). The

δ function approximation here indicates that electrons at any

particular energy γ emit only at a certain energy ǫ:

nsyn(ǫs, s) = uBσT c

3πǫcomov

∫ ∞

1

dγ γ 2 ne(γ )δ(ǫs − γ 2ǫB) (3)

giving,

f syǫ (ǫ, tobs) = D4

(c σT uB

6π d2

)

γ 3sy Ne(γsy, t), γsy ≡

√

ǫ

DǫB

(4)

Dermer et al. (1997), where D = [Ŵ j (1 − β jµ)]−1 is the

Doppler boosting factor, ǫB ≡ B/Bcr with Bcr = m2c3/he =4.414 × 1013 G, defined as the field at which the cyclotron

quantum number equals the rest mass energy of the electron,

and tobs is the observer time, so that tobs = t/D = t∗a,r/DŴ j .

Table 1 Parameter choices forour baseline model Parameter Symbol Value

Black-hole mass M 15 M⊙Distance d 3.75 × 1022 cm

Jet inclination angle θjet 70

Bulk Lorentz factor Ŵ j 2.5

Binary separation s∗ 1012 cm

Luminosity of companion star: L∗ 8 × 1037 ergs s−1

Surface temperature of the companion star T∗ 3 × 104 K

Initial blob radius R0 103 Rg

Jet collimation parameter a 0.3

Accretion Fraction m = M/MEdd 0.01

Accretion disk luminosity L D 1.9 × 1037 ergs s−1

Electron injection spectrum, low-energy cutoff γmin 10

Electron injection spectrum, high-energy cutoff γmax 104

Electron injection spectrum, spectral index q 2.4

Beginning of electron injection zone x0 103 Rg

End of electron injection zone x1 105 Rg

Magnetic field at x0 B0 5 × 103 G

Injection luminosity L inj 4.4 × 10−5 LEdd

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0.2 0.4 0.6 0.8 1.0

Fsoft

/Fsoft

max

10

10

10

10

Ha

rdn

ess R

atio

1000 G

2500 G

5000 G

7500 G

104

G

109

1012

1015

1018

1021

1024

ν[Hz]

107

108

109

1010

Flu

en

ce

[J

y H

z s

]

0.2 0.4 0.6 0.8 1.0

Fhard

/Fhard

max

10

10

100 H

ard

ne

ss R

atio

10 100

101

102

103

tobs

[s]

103

104

105

106

107

108

109

1010

νF

ν [Jy H

z]

a) b)

)d)c

3.5 keV

1 keV (x100)

Fig. 1 Effect of a changing magnetic field on (a) the time-integratedνFν (fluence) spectra, (b) the X-ray light curves at 1 keV (multipliedby 100 for clarity) and 3.5 keV, and (c,d) hardness-intensity diagrams(HIDs). In the HIDs, the soft flux is the integrated 0.1–2 keV flux, thehard flux is the 2–10 keV flux, and the hardness ratio is the ratio of thetwo. The abrupt shape of some of the HID tracks is an artifact of the δ

approximations used for some of our spectral calculations. The respec-tive magnetic fields are indicated in the legend; the other parametersare the baseline model values discussed in the text and listed in Table1. The vertical lines in panel (a) indicate the photon energies at whichthe light curves in panel (b) were extracted

For more details, see Gupta et al. (2006). For the time-

dependent νFν spectral output f ECǫ due to Compton up-

scattering of external photons from the star and the disk,

we use the Thomson approximation, represent the star as

a monochromatic point source, the disk as a thin annulus

at the radius Rmax where the differential energy output is

maximized, and use the optically thick, geometrically thin,

gas-pressure dominated accretion disk solution of Shakura

and Sunyaev (1973).

3 Results

A large number of simulations have been performed to study

the effects of the various model parameters on the resulting

broadband spectra, light curves, and X-ray hardness intensity

diagrams (HIDs). We started our parameter study with a base-

line model for which we used the standard model parameters

broadly representative of GRS 1915+105. Subsequently, we

investigated the departure from this standard set-up by vary-

ing (1) the initial magnetic field B0, (2) the luminosity of

the companion star L∗, (3) the injection electron spectral in-

dex q , (4) the low-energy cutoff γ1 of the electron injection

spectrum, (5) the high-energy cutoff γ2 of the electron injec-

tion spectrum, (6) the injection luminosity L inj, (7) and the

observing angle θobs and thus the Doppler boosting factor.

As an example, we show in Fig.1, the results for the case of

varying the initial magnetic field B0, where we focus on the

time-averaged photon spectra, light curves, and X-ray HIDs,

and explore the effect of variations of B0 on these aspects.

The detailed study of the whole set of parameters can be

found in (Gupta et al., 2006).

From this parameter study, one could draw the following

general conclusions:

A sudden increase of a light curve slope at a fixed observing

frequency, not accompanied by significant flaring activity

at other wavelengths, usually indicates the passing of a

new spectral component through the fixed observing fre-

quency range. Most notably, this diagnostic can be used to

investigate the presence of one or more external-Compton

component(s) in the X-ray/soft γ -ray regime. Clockwise spectral hysteresis in the hardness-intensity di-

agrams indicates the dominance of synchrotron emission

(in particular, before the end of the injection period in our

generic model setup). In this case, the frequency-dependent

light curve decay will be a useful diagnostic of the mag-

netic field strength in the jet. Counterclockwise spectral hysteresis in the hardness-

intensity diagrams indicates the dominance of Compton

emission (similar to the case of blazars, see, e.g., Bottcher

and Chiang, 2002).

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In our study, we found, quite often, a co-existence of clock-

wise and counterclockwise X-ray hysteresis loops, which

would provide a particularly powerful diagnostic, as it

would allow to probe the characteristic transition energy

between synchrotron and Compton emission, and its time

dependence.

Various spectral components (synchrotron, SSC, external-

Compton) could be easily distinguished if detailed snapshot

SEDs could be measured for microquasars, on the (often sub-

second) time scales of their X-ray variability. Unfortunately,

such detailed snapshot broadband spectra are currently not

available, and might not be available in the near future. There-

fore, we have exposed several other features pertinent to the

transition between different spectral components which will

be more easily observable in realistic observational data of

microquasars.

We conclude therefore that the X-ray variability as pre-

dicted by our model can be used as a diagnostic to gain

insight into the nature of the high energy emission in mi-

croquasar jets. In particular, a transition between clockwise

and counter-clockwise spectral hysteresis would allow not

only the distinction between different emission components,

but also parameters such as the magnetic field, the Doppler

boosting factor, and the characteristic electron injection /

acceleration time.

Acknowledgements The work of S.G. and M.B. was supportedby NASA through XMM-Newton GO grant no. NNG04GI50G andINTEGRAL theory grant NNG05GK59G.

References

Bottcher, M., Chiang, J.: ApJ 581, 127 (2002)Dermer, C.D., Sturner, S.J., Schlickeiser, R.: ApJS 109, 103 (1997)Gupta, S., Bottcher, M., Dermer, C.D.: astro-ph/0602439 (2006)Liang, E.P.: Phys. Rep. 302, 67 (1998)McClintock, J.E., Remillard, R.A.: astro-ph/0306213 (2004)Mendez, M., van der Klis, M.: ApJ 479, 926 (1997)Miyamoto, S., Kimura, K., Kitamoto, S., Dotani, T., Ebisawa, K.: ApJ

383, 784M (1991)Shakura, N.I., Sunyaev, R.A.: A&A 24, 337 (1973)Tavecchio, F., Maraschi, L., Ghisellini, G.: ApJ 509, 608 (1998)Mirabel, I.F., Rodrıguez, L.F.: Nature 371, 46 (1994)

Springer


DOI 10.1007/s10509-006-9289-3


New Experimental Platform for Studies of Turbulenceand Turbulent Mixing in Accelerating and Rotating Fluidsat High Reynolds Numbers

Sergei S. Orlov · Snezhana I. Abarzhi

Received: 25 July 2006 / Accepted: 8 December 2006C© Springer Science + Business Media B.V. 2007

Abstract We present a new experimental platform for stud-

ies of turbulence and turbulent mixing in accelerating and ro-

tating fluids. The technology is based on the ultra-high perfor-

mance optical holographic digital data storage. The state-of-

the-art electro-mechanical, electronic, and laser components

allow for realization of turbulent flows with high Reynolds

number (>107) in a relatively small form-factor, and quantifi-

cation of their properties with extremely high spatio-temporal

resolutions and high data acquisition rates. The technology

can be applied for investigation of a large variety of hydrody-

namic problems including the fundamental properties of non-

Kolmogorov turbulence and turbulent mixing in accelerating,

rotating and multiphase flows, magneto-hydrodynamics, and

laboratory astrophysics. Unique experimental and metrolog-

ical capabilities enable the studies of spatial and temporal

properties of the transports of momentum, angular momen-

tum, and energy and the identification of scalings, invari-

ants, and statistical properties of these complex turbulent

flows.

Keywords Turbulent mixing and turbulence .

Accelerating and rotating fluids . Multiphase and reactive

flows . Shocks . Supernova . Accretion disc

S. S. Orlov ()Stanford University, Stanford, CA, USAe-mail: [email protected]

S. I. AbarzhiThe University of Chicago, Chicago, IL, USAe-mail: [email protected]

1 Introduction

Turbulence and turbulent mixing in accelerating and rotat-

ing fluids play a key role in a wide variety of astrophysi-

cal phenomena. Stellar interiors and solar non-Boussinesq

convection, Rayleigh-Taylor (RT) and Richtmyer-Meshkov

instabilities in explosions of supernova type Ia and II, accre-

tion and proto-stellar disks, magneto-hydrodynamic and dy-

namo, formation of planets and stars are to list a few. In many

of these cases the acceleration is spatially non-uniform and

time-varying, the rotation is spatially varying, and the flow is

highly anisotropic and multiphase. Its statistical, spectral, and

scaling properties differ substantially from those of isotropic

Kolmogorov turbulence (Abarzhi et al., 2005; Baroud et al.,

2002; Frisch, 1995).

Theoretical description of the turbulent flows subjected to

acceleration and rotation remains one of the most challenging

problems in hydrodynamic theory, whose solution requires

innovative ideas and approaches. The computational treat-

ments of the problem are met with tremendous difficulties

as the numerical solutions appear to be very sensitive to the

initial and boundary conditions as well as to the influence of

small-scale structures on the turbulent dynamics and anoma-

lous character of energy transport (Calder et al., 2002). On

experimental side, flows with high rates of acceleration and

rotation and high Reynolds numbers (e. g.,>107) are not only

difficult to implement in a well-controlled laboratory envi-

ronment, but also very hard to quantify and measure with

sufficient accuracy and spatio-temporal resolution due to the

limitations of diagnostics and metrological tools currently

available (Adrian, 2005; Frisch, 1995).

Recent theoretical studies (Abarzhi et al., 2005) have sug-

gested that the rate of momentum loss can be a better indicator

of accelerated turbulent flows than the rate of energy dissi-

pation, which is the basic quantity of isotropic Kolmogorov

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turbulence (Frisch, 1995). Energy is complimentary to time,

momentum is complimentary to space, and spatial distribu-

tions of the flow quantities are important to monitor for a

reliable description of non-Kolmogorov turbulent processes,

with the analysis involving the statistical properties of the

transports of mass, momentum (angular momentum, if appli-

cable) and energy. These statistical characteristics are quite

difficult to obtain from the experimental diagnostics currently

used, as the most of the existing fluid dynamics experiments

perform measurements at one (or, a few) points and focus

on the temporal dependencies of the velocity fluctuations

and energy transport (Adrian, 2005; Frisch, 1995). To quan-

tify and better understand the non-Kolmogorov turbulence

and multi-phase turbulent mixing in accelerating and rotat-

ing fluids, new experimental approaches, capable of provid-

ing higher accuracy spatial and temporal measurements and

adequate statistics, are needed.

2 New technology for advanced diagnostics

The goal of our experiments in flow visualization is to per-

form the fully resolved measurements with high accuracy and

good statistics of the spatial and temporal distributions of the

turbulent flow quantities in a well controlled laboratory envi-

ronment. An advanced technology, which can be leveraged

to further the experimental capabilities, is the digital holo-

graphic data storage (HDS). Coufal et al. (2000) provides

a detailed review of the technology, its historical develop-

ments, and the state-of-the-art. One of the unique features

of the HDS is the capability to image and to capture with

high-resolution (106 pixels or more) and extremely low dis-

tortions the optical images at a very high transfer rate and

speed (up to 104 frames-per-second). In data images used

in HDS (see Fig. 1), each pixel is treated as a unique inde-

pendent data channel, and the spatial position of each pixel

is controlled with submicron accuracy throughout the entire

image, whereas the digitized pixels values are measured with

high (8 bits or higher) precision, providing extremely high

spatial resolution as well as signal-to-noise ratio over the

entire spatial extent of the imaged domain.

A state-of-the-art example of digital holographic data stor-

age technology is an experimental facility built at Stan-

ford University for the DARPA Holographic Data Storage

Systems consortium (Orlov et al., 2004). This platform in-

corporates the advanced mechanical, optical, imaging, and

programmable electronic components including the high-

precision high-speed air-bearing spindle device, pulsed fre-

quency doubled (λ = 532 nm) Nd:YAG lasers, dedicated

synchronization electronics for precision timing and signals

generation, 1000 frames-per-second 1 Mpixel digital cam-

era, high resolution optical imaging system, and advanced

automation.

The experimental and metrological capabilities of the

HDS technology can be employed to investigate the turbulent

flows (including the reactive, compressible, and multiphase

flows) subjected to accelerations and rotation. For the flow vi-

sualization, the Particle Imaging Velocimetry (PIV) approach

(Adrian, 2005) can be employed and significantly improved

by leveraging the high-resolution optical imaging capabili-

ties of the HDS (see Fig. 2). Spatial (∼1 µm over 10 cm area)

and temporal (<5 ns) resolutions of the HDS system allow

for mapping the flow velocity fields and capturing the fast

events with extreme accuracy and high data rate acquisition

(>1000 flow images per second) providing thus improved

statistical evaluation of the turbulent flow quantities (see Ta-

ble 1 for the characteristic values of the system parameters).

In order to provide simultaneous mapping of the velocity

field and density fluctuations in unsteady turbulent flows, the

PIV setup can be combined with an imaging interferometer

(compressible flows, Fig. 2) and PLIF diagnostics (for multi-

phase flows). The high rotation rate (up to 250 Hz) can result

Fig. 1 (Left) A high-resolution (1024 × 1024 pixels; 13.1 × 13.1 mm)holographic image captured at 1000 frames-per-second (Orlov et al.,2004); the octagonal shape and “voids” were imprinted during the signal

encoding; (Middle) same image after digital thresholding; (Right) en-larged fragment. Bright area looks like a complicated “maze” of whitepixels due to the spatial information coding used in HDS

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Table 1 Experimental and metrological capabilities of HDS platforms

Rotation rate Upto 15,000 RPM Stability: ±0.001%

Achievable Reynolds number >107

Acceleration

Radial Upto 5 × 105 m/s2 (50,000 g) At 15,000 RPM

Tangential (∼2 g) At 1000 RPM/s

Angular acceleration 0–1000 RPM/s Digitally controlled

Radial stability <10 nm Radial air-bearing

Optical spatial resolution ∼1 micron Over 10 cm field

Temporal resolution <5 ns Jitter <100 ps

Chamber size

Height Up to 30 cm

Diameter Up to 40 cm

CMOS camera resolutions 103×103 pixels or more 10 bit grayscale

Frame rate 103 (upto 104) fps ∼1 Gpixel bandwidth

Camera imaging data Up to 24 GB per experiment ∼2 × 104 of 1 Mpixel images

Interferometric density Sensitivity: n L/λ ∼ 0.1

fluctuation measurements ∼100 × 100 points per frame

Fig. 2 Schematic of the PIV setup for flow visualization combined with an imaging interferometer for the simultaneous, correlated, measurementsof the velocity field and the density fluctuations in compressible fluids

in the radial accelerations of up to 50,000 g for sustained

periods of time (tens of seconds), which, in combination

with high mechanical stability of the air-bearing suspension

(<10 nm spatial run-out), enables the high-accuracy exper-

imental studies of Rayleigh-Taylor instability. Compared to

linear accelerations obtained using, for instance, Linear Elec-

tric Motor (Dimonte et al., 2000), rotational arrangement al-

lows one to produce high accelerations in a relatively small

form-factor for a prolonged period of time and to study very

late stages of the mixing process. Our experimental approach

can provide the repeatability of the experimental conditions,

detailed diagnostics, and the ability either to control the sta-

bilizing/destabilizing factors or to measure their influences

with high accuracy. For instance, in experiments on RTI with

high acceleration, one of the mixing fluids can be a jelly-like

medium. As the value of acceleration induced by centrifu-

gal force exceeds the visco-elastic stress, the jelly acts as a

liquid-like medium (Meshkov et al., 1999). The initial per-

turbation in this case can be embossed mechanically on the

jelly surface using pre-designed plastic forms, whereas the

time-history of the rotation and acceleration can be controlled

via programmable electronic components, ensuring thus the

repeatability of the experimental conditions. In addition to

tracking of the interface and its dynamics, the simultaneous

PLIF and PIV measurements of the velocity fields in the bulk

of the mixing fluids will allow to quantify the basic statistics

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of the RT turbulent mixing (Abarzhi et al., 2005). The condi-

tions of high accelerations and the requirements for the high

spatial resolution (∼1 µm) both demand the optimization of

buoyancy and size of the imaging particle (for accurate flow

tracking) as well as the matching of refractive indexes of the

mixing fluids (McDougall, 1979) to minimize optical dis-

tortions in order to produce reliable high-spatial resolution

data.

The new diagnostics, high spatial and temporal resolu-

tions, as well as the capability to collect and store large imag-

ing data sets of tens of GBytes in a single experiment will

enable studies of many of the fluid dynamics phenomena

with accuracy and precision which has not been achieved

or demonstrated so far. As a future outlook, using precision

mechanical capabilities, in addition to RTI and multiphase

turbulent mixing, the magneto-hydrodynamics (MHD) and

turbulent dynamo experiments could be conducted under the

conditions of very stable (or time varying with precise con-

trol) and accurate (<± 0.001% variation) high-speed rota-

tion by utilizing conducting and magnetic liquids (e. g., liquid

sodium or liquid gallium) placed in a spherical or cylindrical

chamber.

3 Phenomenology and validation

Many of astrophysical objects (e.g., accretion disks, stellar

and planetary interiors) undergo turbulence, accelerated tur-

bulence mixing and rotation. Capturing scaling properties

of rotating and accelerating turbulent flows is important for

better understanding of the astrophysical phenomena. Recent

studies (Abarzhi et al., 2005) suggest that the basic invari-

ant properties of the accelerated turbulent mixing differ sub-

stantially from those of isotropic Kolmogorov turbulence.

In particular, the rate of momentum loss is the fundamen-

tal quantity of accelerated turbulent mixing, similarly to the

energy dissipation rate in isotropic Kolmogorov turbulence.

The invariance of the rate of momentum loss can be employed

further in studies of scaling and spectral properties. One can

show, for instance, that in accelerating turbulent flow, the

velocity scales as square root of length scale and the spec-

trum of kinetic energy (velocity fluctuations) is proportional

to ∼k−2, compared to the power 1/3 for velocity scale and

∼k−5/3 for velocity spectrum in classical Kolmogorov turbu-

lence. These quantitative differences are sufficiently large to

be distinguished in the high-precision experiments discussed

in the foregoing.

Another important application of our laboratory ex-

periments is the validation of hydrodynamic simulations.

Advanced astrophysical modeling platforms (such as the

FLASH code developed at the University of Chicago, see

Calder et al., 2002) produce numerical data, which are highly

resolved in space. These data are however rather difficult

to compare directly to existing experiments, which provide

mostly the temporal statistics at one (or just a few) spatial po-

sition(s). Higher accuracy spatial and temporal experimental

quantification of the model hydrodynamic problems (such

as multiphase turbulent mixing induced by Rayleigh-Taylor

instability, interface evolution in shock-driven Richtmyer-

Meshkov instability, or decaying turbulence in rotating flu-

ids) may significantly advance the code validation and, hence,

reliability of the hydrodynamic modeling of the complex tur-

bulent flows in astrophysical applications.

4 Summary

The unique capabilities developed in digital holographic stor-

age technology can be applied for advanced metrology and

high-resolution laboratory studies of a broad variety of hy-

drodynamic phenomena relevant to astrophysical problems.

High resolution measurements and quantitative characteri-

zations with good statistics are important for understanding

the basic invariant, scaling, spectral, and statistical proper-

ties of complex turbulent flows, including rotating, accel-

erating and multi-phase flows, and, particularly, turbulent

mixing induced by Rayleigh-Taylor instability. High resolu-

tion measurements of these phenomena performed in a well-

controlled laboratory environment can be applied for valida-

tion of hydrodynamic simulation platforms and for modeling

of astrophysical phenomena.

References

Abarzhi, S.I., Gorobets, A., Sreenivasan, K.R.: Phys. Fluids 17, 081705(2005)

Adrian, R.J.: Exp. Fluids 39, 159 (2005)Baroud, C.N., Plapp, B.B., She, Z.-S., Swinney, H.L.: Phys. Rev. Lett.

88, 114501 (2002)Calder, A.C., Fryxell, B., Plewa, T., Rosner, R., et al.: ApJS 143, 201

(2002)Coufal, H.J., Psaltis, D., Sincerbox, G.T. (eds.): Holographic Data Stor-

age. Springer-Verlag, Springer Series in Optical Sciences, Berlin(2000)

Dimonte, G., Schneider, M.: Phys. Fluids 12, 304 (2000)Frisch, U.: Turbulence, the Legacy of A. N. Kolmogorov. Cambridge

University Press, Cambridge, UK (1995)McDougall, T.J.: J. Fluid Mech. 93, 83 (1979)Meshkov, E.E., Nevmerzhitsky, N.V., Sotskov, E.A.: Comparative study

of evolution of periodical perturbations in liquid and solids us-ing jellies. In: Meshkov, E.E., Yanilkin, Yu., Zhmailo, V. (eds.),Proceedings of the International Workshop Phys. CompressibleTurbulent Mixing (1999)

Orlov, S.S., Phillips, W., Bjornson, E., et al.: Appl. Opt. 43, 4902 (2004)

Springer


DOI 10.1007/s10509-006-9288-4


Weibel Turbulence in Laboratory Experimentsand GRB/SN Shocks

Mikhail V. Medvedev


Abstract It has recently been realized that the Weibel insta-

bility plays a major role in the formation and dynamics of

astrophysical shocks of gamma-ray bursts and supernovae.

Thanks to technological advances in the recent years, exper-

imental studies of the Weibel instability are now possible

in laser-plasma interaction devices. We, thus, have a unique

opportunity to model and study astrophysical conditions in

laboratory experiments – a key goal of the Laboratory Astro-

physics program. Here we briefly review the theory of strong

non-magnetized collisionless GRB and SN shocks, empha-

sizing the crucial role of the Weibel instability and discuss the

properties of radiation emitted by (isotropic) electrons mov-

ing through the Weibel-generated magnetic fields, which is

referred to as the jitter radiation. We demonstrate that the jitter

radiation field is anisotropic with respect to the direction of

the Weibel current filaments and that its spectral and polariza-

tion characteristics are determined by microphysical plasma

parameters. We stress that the spectral analysis can be used

for accurate diagnostics of the plasma conditions in labora-

tory experiments and in astrophysical GRB and SN shocks.

Keywords Weibel instability . Plasma physics . Radiation

from plasma . Plasma diagnostics . Plasma astrophysics .

Collisionless shocks

1 Introduction

Internal and external shocks in gamma-ray bursters (GRBs),

internal shocks produced in jets of micro- and normal quasars

M. V. MedvedevDepartment of Physics and Astronomy, University of Kansas, KS66045, USAe-mail: [email protected]

and in active galactic nuclei jets, shocks in supernovae (SN)

remnants, merger shocks in galaxy clusters and large scale

structure (LSS), – all of them represent a single class of

strong collisionless shocks. The theoretical prediction that

the Weibel instability operating at the shock produces strong

magnetic fields (Medvedev and Loeb, 1999) has recently

been confirmed in a number of state-of-the-art numerical

sumulations (Silva et al., 2003; Frederiksen et al., 2004;

Nishikawa et al., 2003; Saito and Sakai, 2004; Kazimura

et al., 1998). It has also been predicted that radiation of rel-

ativisitc electrons from Weilbel-generated magnetic fields,

referred to as the “jitter radiation” has distinct spectral prop-

erties (Medvedev, 2000). The Weibel instability, its dynamics

and properties, can now be studied in laboratory experiments

od laser-plasma interactions (and even relevant to the fast ig-

nition concept). We can, thus, directly probe conditions of

distant astrophysical environments in the lab.

2 The Weibel mechanism

In general, shocks with the Mach number greater than three,

must be highly turbulent. The source and the mechanism of

the turbulence is thought to be of kinetic nature, in order to

prevent multi-stream motion of plasma particles. It has been

shown that the Weibel instability operates at the shock front

(Moiseev and Sagdeev, 1963; Medvedev and Loeb, 1999).

This instability is driven by the anisotropy of the particle

distribution function (PDF) associated with a large number

of particles reflected from the shock potential.

2.1 Linear regime of field growth and its saturation

The instability under consideration was first predicted by

Weibel (1959) for a non-relativistic plasma with an

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Fig. 1 Illustration of various stages of the Weibel instability. Colorcoding of particles: blue – the incoming particles from the IGM, red –the particles scattered from the shock

anisotropic distribution function. The simple physical in-

terpretation provided later by Fried (1959) treated the PDF

anisotropy more generally as a two-stream configuration of

cold plasma. Below we give a brief, qualitative description

of this two-stream magnetic instability.

Let us consider, for simplicity, the dynamics of one species

only (e.g., protons), whereas the other is assumed to provide

global charge neutrality.1 The electrons are assumed to move

along the z-axis with the velocities v = +zvz and v = −zvz ,

thus forming equal particle fluxes in opposite directions (so

that the net current is zero). Such a particle distribution occurs

naturally near the front of a shock (moving along z-direction),

where the “incoming” (in the shock frame) ambient gas par-

ticles meet the “outgoing” particles reflected from the shock

potential (loosely speaking, the low energy cosmic rays).

Thus, the particle velocities vz are of order the shock veloc-

ity, vz ∼ vsh. The counter-streaming particles may also have

some thermal spread. Since for high-Mach number shocks,

vth ≪ vsh in the upstream region, we may neglect the parallel

velocity spread in our consideration. The thermal spread in

the transverse direction cannot be neglected, however. We

parameterize the PDF anisotropy as follows:

A = ǫ‖ − ǫ⊥ǫtot

≃ M2 − 1

M2 + 1, (1)

where ǫ‖ ∝ v2z ≃ v2

sh is the energy of particle along z-

direction, ǫ⊥ ∝ (v2x + v2

y) ∝ v2thermal ≃ c2

s is the thermal en-

ergy in the transverse direction, ǫtot = ǫ‖ + ǫ⊥ is the total en-

ergy, cs is the sound speed upstream and the Mach number of

the shock is M = vsh/cs . Clearly, for strong shocks M ≫ 1,

the anisotropy parameter is close to unity, A ∼ 1. Next, ac-

cording to the linear stability analysis technique, we add

an infinitesimal magnetic field fluctuation, B = x Bx cos(ky).

The Lorentz force, e(v × B)/c, acts on the charged particles

1 In reality, the role of protons is more complicated. In particular, theycan play a crucial role in the electron acceleration in the downstreamregion. We do not consider such effects in this paper; they will be studiedelsewhere.

and deflects their trajectories, as is shown in Fig. 1. As a

result, the protons moving upward and those moving down-

ward will concentrate in spatially separated current filaments.

The magnetic field of these filaments appears to increase the

initial magnetic field fluctuation. The growth rate and the

wavenumber of the fastest growing mode (which, in fact,

sets the spatial correlation scale of the produced field) are

γB = A ωp,s(vz/c), kB = A ωp,s/c, (2)

where

ωp,s =(

4πe2ns

Ŵshockms

)1/2

≈ 1.32 × 103

(

ns

n p

m p

ms

)1/2

Ŵ−1/2shocks−1 (3)

is the plasma frequency defined for species s (electrons, pro-

tons, etc.), n p and m p are the number density and the mass of

the protons, respectively; Ŵshock is the Lorentz factor of a rel-

ativisitc shock (for non-relativistic shocks, Ŵshock ∼ 1). We

use cgs units throughout the paper, unless stated otherwise.

Thus, the instability is indeed driven by the PDF anisotropy

and should quench for the isotropic case. To put these facts in

the context of cosmological shocks, we give estimates of the

characteristic temporal and spatial scales. The order of mag-

nitude estimates of the magnetic field e-folding time and the

field correlation length at strong shocks (M ≫ 1) are readily

obtained as

τB ∼ 1/γB ≃ 2 × 10−1 v−18 n

−1/20 s, (4)

λB ∼ 2π/kB ≃ 108 n−1/20 cm (5)

for a typical ISM proton density of n ∼ 1 cm−3 and a typical

non-relativistic shock velocity v ∼ 108 cm s−1; as usual, we

denote n0 = n/(100 cm−3) and v8 = v/(108 cm s−1).

The Lorentz force deflection of particle orbits increases

as the magnetic field perturbation grows in amplitude. The

amplified magnetic field is random in the plane perpendicular

to the particle motion, since it is generated from a random

seed field. Thus, the Lorentz deflections result in a pitch angle

scattering, which makes the bulk of the PDF isotropic. If one

starts from a strong anisotropy, so that the thermal spread

is much smaller than the particle bulk velocity, most of the

particles will eventually isotropize and the thermal energy

associated with their random motions will be equal to their

initial directed kinetic energy. This final state will bring the

instability to saturation.

The saturation level of the magnetic field may readily be

estimated as follows. First of all, note that the instability is

due to the free streaming of particles. As the magnitude of

the magnetic field grows, the transverse deflection of particles

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gets stronger, and their free streaming across the field lines

is suppressed. The typical curvature scale for the deflections

is the Larmor radius,

ρL = v⊥B/ωc,s, (6)

where v⊥B is the particle velocity transverse to the direc-

tion of the local magnetic field and ωc,s = eB/msc is the cy-

clotron (Larmor) frequency of species s. On scales larger than

ρL , particles can only move along field lines. Hence, when

the growing magnetic fields become such that kBρL ∼ 1, the

particles are magnetically trapped and can no longer amplify

the field. Assuming an isotropic particle distribution at satu-

ration (v⊥B ∼ vsh), this condition can be re-written as

ǫB = B2/8π

msnsv2sh/2

≃ A2. (7)

For strong shocks (M ≫ 1, A ∼ 1), this corresponds to the

magnetic energy density close to equipartition with the ther-

mal energy of particles downstream the shock. Here again,

we evaluated the field in a non-relativistic shock.

Weibel instability has been modeled in numerical PIC 2D

and 3D simulations by our group Silva et al. (2003) as well

as by several other research groups (Frederiksen et al., 2004;

Nishikawa et al., 2003; Saito and Sakai, 2004; Kazimura

et al., 1998). We examined the instability, which occurs in

a collision of two inter-penetrating unmagnetized electron-

positron clouds with zero net charge. This is the simplest

model for the formation region of a shock front, as well as a

classic scenario unstable to electromagnetic and/or electro-

static plasma instabilities.

The relativistic electromagnetic 3D PIC code OSIRIS

(Fonseca et al., 2002) was used. The simulations were per-

formed in a simulation cube of size 256 × 256 × 100 grid

points, ten grid points correspond to one plasma skin depth

c/ωp,e. We had more than 108 particles in the simulation

box. Periodic boundary conditions were imposed. The initial

state is spatially homogeneous with two identical groups of

particles moving with some velocity ±vz . The particles in

both groups have a small thermal velocity vth ≃ vz/6. The

system has no net charge and no net current, and initially the

electric and magnetic fields are set to zero.

The results of the simulations are shown in Figs. 2 and 3.

Figure 2 shows the temporal evolution of the magnetic

equipartition parameter ǫB . In Fig. 3 the three-dimensional

structure of the magnetic fields and currents are shown at

two different times: (a) during linear regime, at t ≃ 13ω−1p,e,

and (b) just after the saturation, at t ≃ 20ω−1p,e. The left

panels show the structure of magnetic field lines and the

right panels show the number density of particles (blue

are moving downward, red are moving upward). We see

that during the linear stage of the instability (ωp,et 15)

Fig. 2 The temporal evolution of the magnetic field energy densitynormalized by the initial kinetic energy of the particles

Fig. 3 The 3D structure of the magnetic fields and currents fromthe simulations at two different times: (a) during linear regime, att ≃ 13ω−1

p,e, and (b) just after the saturation, at t ≃ 20ω−1p,e. The left

panels show the structure of magnetic field lines and the right panelsshow the number density of particles (blue are moving downward, redare moving upward). The units of axes x1, x2, x3 are c/ωpe

there is exponential generation of a magnetic field, which

predominantly lies in the plane of the shock (x − y-plane),

i.e., perpendicular to the direction of motion of the plasma

clouds. The produced magnetic field is highly inhomoge-

neous, with the characteristic correlation scale comparable

to the plasma skin depth length c/ωp,e. It is also seen that the

magnetic field generation is associated with the separation

of the particle streams in spatially distinct regions and the

formation of straight current filaments.

Saturation of the instability occurs at time t ∼ 15ω−1p,e,

which is indicated by the peak of ǫB in Fig. 2. At this moment,

most of the particles are randomized over the pitch angle by

the Lorentz deflections. Thus the PDF anisotropy, which is

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the free energy source for the instability is removed. At the

time of saturation, the magnetic field energy density reaches

its maximum ∼20%.

2.2 Nonlinear regime: Long-term field evolution

At longer times, one sees the substantial decrease in ǫB ,

during the initial stage of the nonlinear regime in which

current filaments begin to interact with each other, forcing

like currents to approach each other and merge. During this

phase, initially randomly oriented filaments cross each other

to form a more organized, large-scale quasi-regular pattern,

hence much current and B-field is annihilated. At later times,

t 35ω−1p,e, the filament coalescence continues, as is indi-

cated by the increase of the filament sizes. Note that the

spatial distribution of currents is now quite regular, so that

filaments with opposite polarity no longer cross each other,

but simply interchange, staying always far away. The total

magnetic field energy is ǫB ∼ 0.25% and does not change

any more. Note also that the residual magnetic field is highly

inhomogeneous, seen as a collection of magnetic field fila-

ments or “bubbles”. The amplitude of the field in the bubbles

is close to equipartition. Therefore, the overall decrease of

the magnetic field energy is mostly associated with the de-

creasing filling factor of the field.

The filament coalescence is a hierarchical and self-similar

process (Medvedev et al., 2005). Initially the system consists

of filaments of diameter D0, carrying current I0 and sepa-

rated by distance d0 ≃ 2D0, which are randomly located in

space. This configuration is unstable because opposite cur-

rents repel each other, whereas like currents are attracted to

each other and tend to coalesce and form larger current fila-

ments. At the k-th merger level, i.e., after k pairwise mergers:

Ik = 2k I0, Dk = 2k/2 D0, dk ∼ Dk/2. Expressing k interms

of tite t , for the non-relativistic and relativistic filaments

respectively, we obtain for the magnetic field correlation

length

λB(t) = D02t/(2τ0,N R ), λB(t) ≃ ct. (8)

Here, the typical non-relativistic time-scale is determined by

Equation (8). The coalescence time may be written as

τ0,N R ∼= η (c/vsh)√ǫB ωp,s

∼ 105 ω−1p,s, (9)

Here we assumed the typical values: ǫB ∼ 10−3 and η ∼ 1.

Also, the parameter η accounts for the fact that the field corre-

lation scale at the onset of non-linear evolution (t ∼ 15ω−1p,s)

is somewhat larger than λs . We include this uncertainty via

the parameter η > 1 as D0 ∼ η (c/ωp,s). The transition from

the non-relativistic to relativistic coalescence regime occurs

at the time

t∗ = 2 log2(c/v0m) τ0,N R ∼ 10τ0,N R . (10)

We now compare our theoretical predictions with the re-

sults of particle-in-cell numerical simulations, performed

using code OSIRIS. We have performed 2D simula-

tions (1280 × 1280 cells, 128.0 × 128.0 (c/ωp,e)2, 9 parti-

cles/(cell × species), 4 species) of the collision of electrically

neutral clouds (electron-positron – e−e+, and electron-proton

– e− p) moving in the z direction, across the xy simulation

plane, with parameters similar to those in Silva et al. (2003).

In order to save on computation time and trace a substantial

period of field evolution, we set the particles bulk veloc-

ity to vsh ≃ v ≃ 0.6c. Relativistic effects do not play any

significant role because the corresponding Lorentz factor is

γsh ∼ 1.17. The change of the field correlation length with

time is clearly seen in these runs (not shown). The growth

of this length is substantially slower and the magnetic field

filling factor is respectively larger in the electron-proton run.

The temporal evolution of λB as measured in the simulations

is shown in Fig. 4.

Both a non-power-law non-relativistic regime (until t ≈10 − 20/ωp,e) and a power-law regime are clearly seen. The

power-law fits yield λB(t) ∝ tα with α ≈ 0.8. Note also that

the second power-law segment with the same index is present

at t 100/ωp,e in e− p run, indicating proton filament coales-

cences. A similar behavior was also observed in 3D simula-

tions, but the significantly larger simulation planes presented

here allow for improved statistics. At late times t 100/ωp,e,

the evolution of λB rolls off and is slower when the number

of filaments in the simulation box becomes relatively small.

We also note that in some respect, the field scale growth is

analogous to the inverse cascade in two-dimensional magne-

tohydrodynamic (MHD) turbulence. The crucial difference

is, however, the entirely kinetic nature of the process; at such

Fig. 4 Temporal evolution of the field correlation length in electron-positron (solid line) and electron-proton (dashed line) runs. In the range7/ωp,e < t < 50/ωp,e, λB ∝ tα with α ≃ 0.8

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small scales ∼c/ωp the MHD approximation is completely

inapplicable.

Numerous runs were performed to isolate the effect of the

finite box size and make sure that the results are not affected.

We found that the periodic boundary conditions affect the dy-

namics when the filament sizes are comparable or, at most,

a factor of three-four smaller than the box size. Initially, the

filaments are created on the skin-depth scale, c/ωpe, which is

about 1/25 of the box size. Hence, the linear instability, satu-

ration and early nonlinear dynamics are accurately simulated.

3 Theory of jitter radiation in 3D

The angle-averaged spectral power emitted by a relativistic

particle moving through small-scale random magnetic fields,

under the assumption that the deflection angle is negligible

and the particle trajectory is a straight line, has been de-

rived elsewhere (Rybicki and Lightman, 1979; Landau and

Lifshitz, 1971; Medvedev, 2000) and it reads

dW

dω= e2ω

2πc3

∫ ∞

ω/2γ 2

|wω′ |2ω′2

(

1 − ω

ω′γ 2+ ω2

2ω′2γ 4

)

dω′.

(11)

Here γ is the Lorentz factor of a radiating particle and

wω′ =∫

weiω′t dt is the Fourier component of the transverse

particle’s acceleration due to the Lorentz force. We need to

express the temporal Fourier component of the Lorentz ac-

celeration, w = (e/γmc)v × B, taken along the particle tra-

jectory in terms of the Fourier component of the field in the

spatial and temporal domains Medvedev (2006). In the static

case, i.e., when the magnetic field is independent of time, the

ensemble-averages acceleration spectrum reads:

〈|wω′ |2〉 = (2πV )−1

∫

|wk|2δ(ω′ + k · v) dk, (12)

|wk|2 = (ev/γmc)2(δαβ − v−2vαvβ) V Kαβ(k), (13)

Kαβ(r, t) = T −1V −1∫

Bα(r′, t ′)Bβ(r′ + r, t ′ + t) dr′dt is

the second-order correlation tensor of the magnetic field.

3.1 The magnetic field spectrum

We adopt the following geometry: a shock is located in the

x-y-plane and is propagating along z-direction. As it has ini-

tially been demonstrated by Medvedev and Loeb (1999) and

later confirmed via 3D PIC simulations (Silva et al., 2003;

Nishikawa et al., 2003; Frederiksen et al., 2004), the magnetic

field at relativistic shocks is described by a random vector

field in the shock plane, i.e., the x – y plane. As the shock is

propagating through a medium, the produced field is trans-

ported downstream (in the shock frame) whereas new field

is continuously generated at the shock front. Thus, the field

is also random in the parallel direction, i.e., the z-direction.

Thus, Weibel turbulence at the shocks is highly anisotropic.

Both the theoretical considerations and realistic 3D simula-

tions of relativistic shocks indicate that the dynamics of of

the Weibel magnetic fields in the shock plane and along the

normal to it are decoupled. Hence, the Fourier spectra of the

field in the x − y plane and in z direction are independent.

Thus, for the Weibel fields at shocks, the correlation tensor

has the form

Kαβ(k) = C(δαβ − nαnβ) fz(k‖) fxy(k⊥), (14)

where n is the unit vector normal to the shock front, C is the

normalization constant proportional to 〈B2〉, fz and fxy are

the magnetic field spectra along n and in the shock plane,

respectively, k⊥ = (k2x + k2

y)1/2 and k‖ = kz , and finally, the

tensor (δαβ − nαnβ) is symmetric and its product with n is

zero, implying orthogonality of n and B.

Numerical simulations (Frederiksen et al., 2004) also indi-

cate that the field transverse spectrum, fxy , is well described

by a broken power-law with the break scale comparable to

the skin depth, c/ωp, where ωp = (4πe2n/Ŵm)1/2 is the rel-

ativistic plasma frequency and Ŵ is the shock Lorentz factor.

We expect that the spectrum fz , has similar properties. There-

fore, we use the following models:

fz(k‖) =k

2α1

‖(

κ2‖ + k2

‖)β1

, fxy(k⊥) = k2α2

⊥(

κ2⊥ + k2

⊥)β2

, (15)

where κ‖ and κ⊥ are parameters (being, in general, a function

of the distance from the front, Medvedev et al., 2005) deter-

mining the location of the peaks in the spectra,α1, α2, β1, β2

are power-law exponents below and above a spectral

peak (β1 > α2 + 1/2 and β2 > α2 + 1, for convergence at

high-k).

3.2 Radiation spectra from a shock viewed at different

angles

We now evaluate Equations (12, 13). The scalar product of

the two tensors is

(δαβ − vαvβ/v2)(δαβ − nαnβ)

= 1 + (nαvα)2/v2 = 1 + cos2 , (16)

where we used that δαα = 3. Here is the angle between the

normal to the shock and the particle velocity (in an observer’s

frame), which is approximately the direction toward an ob-

server, that is v‖k for an ultra-relativistic particle (because of

relativistic beaming, the emitted radiation is localized within

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Fig. 5 The log − log plots of |wk|2 vs k (thin lines) and of dW/dω vsω (thick lines), for three viewing angles = 0, π/10, π/2. The axesunits are arbitrary. In this calculation we used fz = fxy withα = 2, β =20, κ = 10, v = 1. The exponent ζ = ζ (α, β) is model dependent. Wealso chose γ = 1 in order to align the peaks of |wk|2 and dW/dω. Notethat the actual peaks are at values k, ω lower than 10 by a factor two orthree. Note also that the spectrum dW/dω levels off at oblique angles atfrequencies much smaller than κvγ 2 sin, whereas |wk|2 indeed startsto flatten at k ∼ κv sin

a narrow cone of angle ∼1/γ ). Equation (12) becomes

〈|wω′ |2〉 = C

2π(1 + cos2 )

×∫

fz(k‖) fxy(k⊥)δ(ω′ + k · v) dk‖d2k⊥. (17)

Equations (11, 17) fully determine the spectrum of jitter ra-

diation from a GRB shock. We now consider special cases.

A spectrum from a shock viewed at an arbitrary angle,

0 ≤ ≤ π/2, is illustrated in Fig. 5, which represents full

numerical solutions of Equations (11), (15), (17) for three

different viewing angles. In calculation of dW/dω, the emit-

ting electrons were assumed monoenergetic, for simplicity.

An important fact to note is that the jitter radiation spectrum

varies with the viewing angle. When a shock velocity is along

the line of sight, the low-energy spectrum is hard Fν ∝ ν1,

harder than the “synchrotron line of death” (Fν ∝ ν1/3). As

the viewing angle increases, the spectrum softens, and when

the shock velocity is orthogonal to the line of sight, it becomes

Fν ∝ ν0. Another interesting feature is that at oblique angles,

the spectrum does not soften simultaneously at all frequen-

cies. Instead, there appears a smooth spectral break, which

position depends on . The spectrum approaches ∼ν0 below

the break and is harder above it.

The jitter spectra can deliver much information on the

structure of the Weibel magnetic fields. As one can see, the

spectrum depends on the spatial spectra of the magnetic fields

modeled by Equations (15). In particular, when viewing an-

gles are 0 and 90 degrees, the contributions of the parallel

and transverse magnetic field spectra are decoupled. For in-

stance, for θ = 0, the peak of the jitter radiation spectrum and

its high-energy asymptotic slope are uniquely determined by

the parallel correlation length κ‖ and the large-k magnetic

field spectrum slope ks with s = 2α1 − 2β1. Similarly, the

transverse jitter spectrum (at θ = π/2) allows one to deduce

these parameters for the transverse magnetic field spectrum,

fxy . At intermediate angles, one can determine the relative

orientation of the current (and magnetic) filaments in the tar-

get and the radiation detector.

Acknowledgments This work has been supported by DOE grant DE-FG02-04ER54790.

References

Frederiksen, J.T., Hededal, C.B., Haugbølle, T., Nordlund, A.: ApJ 608,L13 (2004)

Fonseca, R.A., et al.: Lecture Notes in Computer Science 2329, III-342,Springer-Verlag, Heidelberg (2002)

Fried, B.D.: Phys. Fluids 2, 337 (1959)Kazimura, Y., Sakai, J.I., Neubert, T., Bulanov, S.V.: ApJ 498, L183

(1998)Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon

Press, Oxford (1971)Medvedev, M.V., Loeb, A.: ApJ 526, 697 (1999)Medvedev, M.V.: ApJ 540, 704 (2000)Medvedev, M.V., Fiore, M., Fonseca, R.A., Silva, L.O., Mori, W.B.:

ApJ 618, L75 (2005)Medvedev, M.V.: ApJ 637, 869 (2006)Moiseev, S.S., Sagdeev, R.Z.: J. Nucl. Energy C 5, 43 (1963)Nishikawa, K.-I., Hardee, P., Richardson, G., Preece, R., Sol, H.,

Fishman, G.J.: ApJ 595, 555 (2003)Rybicki, G.B., Lightman, A.P.: Radiative Processes in Astrophysics.

Wiley, New York (1979)Saito, S., Sakai, J.I.: ApJ 604, L133 (2004)Silva, L.O., Fonseca, R.A., Tonge, J.W., Dawson, J.M., Mori, W.B.,

Medvedev, M.V.: ApJ 596, L121 (2003)Weibel, E.S.: Phys. Rev. Lett. 2, 83 (1959)

Springer


DOI 10.1007/s10509-006-9290-x


Diagnostics of the Non-Linear Richtmyer-MeshkovInstability

M. Herrmann · S. I. Abarzhi

Received: 27 July 2006 / Accepted: 13 December 2006C© Springer Science + Business Media B.V. 2007

Abstract We study analytically and numerically the evolu-

tion of the two-dimensional coherent structure of bubbles and

spikes in the Richtmyer-Meshkov instability (RMI) for flu-

ids with a finite density ratio. New diagnostics and scalings

are suggested for accurate quantification of RMI dynamics.

New similarity features of the late-time instability evolution

are observed. The results obtained can serve as benchmarks

for high energy density laboratory experiments.

Keywords Richtmyer-Meshkov . Supernovae .

Diagnostics . Similarity . Multi-scale dynamics

Introduction

The Richtmyer-Meshkov instability develops when a shock

wave refracts through the interface between two fluids with

different values of the acoustic impedance (Richtmyer, 1960;

Meshkov, 1969). The instability results in a growth of the in-

terface perturbations and produces with time extensive inter-

facial mixing of the fluids (Dimonte, 2000). The shock-driven

turbulent mixing plays an important role in many astrophys-

ical phenomena. In particular, in supernova type II, RMI is

considered as a plausible mechanism of the extensive mixing

of the inner and outer layers of the progenitor star (Chevalier,

1992). Laboratory experiments and simulations suggest the

following evolution of RMI. Initially, the light fluid accel-

erates “implusively” the heavy fluid, with the acceleration

value determined by the shock-interface interaction. With

time, a coherent structure of bubbles and spikes appear: the

M. Herrmann () · S. I. AbarzhiCenter for Turbulence Research, Stanford University, Stanford,CA, USAe-mails: [email protected]; [email protected]

light (heavy) fluid penetrates the heavy (light) fluid in bub-

bles (spikes). Small-scale structures are produced by shear,

and roll-up of vortices causes a mushroom-typed shape of the

spike. Eventually a mixing zone develops, and in the chaotic

regime the spikes and bubbles decelerate as power-law with

time (Dimonte, 2000).

The dynamics of RMI is governed by two, in general in-

dependent, length scales: the spatial period of the structure

λ, set by the initial perturbation or by the mode of fastest

growth, and the amplitude h, which is the bubble (spike) po-

sition (Abarzhi and Herrmann, 2003; Abarzhi et al., 2003).

To quantify the evolution of RMI, experiments and simu-

lations were focused on diagnostics of the length scale h,

readily available for measurements. The observation results

have been interpreted with several empirical models, which

presumed a single-scale character of the interface dynamics,

governed solely by the spatial period λ, and extensively used

adjustable parameters for data comparison. Despite the ef-

forts made, the primary issue for observations still remains

”How to quantify these flows reliably?” Here we suggest

new diagnostics and scaling, which allow for a more accu-

rate description of RMI dynamics, and discuss new similarity

features exhibited at its late stages.

Diagnostics and evolution of the nonlinear unstable

interface

To study RMI dynamics, we find numerical and analytical

solutions for the system of compressible two-dimensional

Navier-Stokes equations with the initial and boundary con-

ditions at the fluid interface. The density of the heavy (light)

fluid is ρh(l), and the Atwood number is A = (ρh − ρl)/(ρh +ρl). The normal component of velocity and pressure are

continuous at the fluid interface. The flow has no mass

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sources, and the initial co-sinusoidal perturbation has small

amplitude.

In our numerical simulations, we solve the compress-

ible Navier-Stokes equations using a finite volume hy-

brid capturing-tracking scheme (Smiljanovski et al., 1997;

Schmidt and Klein, 2003). The contact discontinuity at the

interface is tracked by a level set scalar, and the boundary

conditions at the interface are used to reconstruct the exact

states on each side of the interface, so that the interface re-

mains an immiscible discontinuity with out unphysical fluxes

across (Abarzhi and Herrmann, 2003). Several test runs are

performed for code validation. Figures 1 and 2 show a com-

parison of the amplitude evolution obtained in our simula-

tions with the experiments of Jones and Jacobs (1997) and

with the linear theory of Wouchuk (2001), and the agreement

with the experiments and the theory is excellent.

Figure 2 shows that the growth-rate of the amplitude ex-

hibits oscillations. These oscillations are caused by the rever-

berations of sound waves, and, predicted first by the linear

0

5

10

15

20

0 1 2 3 4 5 6 7

Am

pli

tude

[mm

]

Time [ms]

Fig. 1 Simulated (solid line) and measured (circles) (Jones and Jacobs,1997) amplitude versus time for A = 0.663 and Ma = 1.1. The set ofexperimental data is a combination of single shot measurements of aseries of different experiments

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8v

v v

Fig. 2 Simulated (solid line) and linear theory (dashed line) (Wouchuk,2001) growth-rate a in the linear regime of RMI for A = 0.9 andMa = 1.2. The numerical and theoretical solutions coincide in thelinear regime and start to deviate in the weakly nonlinear regime fortv∞/λ > 0.3

theory of Wouchuk (2001), they are accurately reproduced

in our simulations. The amplitude of the oscillations is very

small compared to the speed of sound, so the oscillations do

not induce significant pressure fluctuations. In Fig. 1 the ve-

locity oscillations are also present and can be derived from

our numerical solution as the time-derivative of the ampli-

tude. The experiments of Jones and Jacobs (1997) do not

capture the oscillations, as the data set in Fig. 1 is a combi-

nation of single shot measurements of a series of different

experiments.

To quantify the evolution of RMI, we define first the length

scale and the time scale of the flow. The length scale is the

period of the coherent structure λ. The proper choice of the

time-scale is a non-trivial issue. To perform a comparative

study of various stages of RMI (linear, weakly and highly-

nonlinear), most of exisiting observations use the time-scale

set by the initial growth-rate v0, as λ/v0 (Cheng et al., 2000;

Robey et al., 2003; Glendinning et al., 2003; Miles et al.,

2004; Jacobs and Krivets, 2005). However, as is seen from

Fig. 2, the value of the growth-rate oscillates and the ampli-

tude of the oscillations is ∼10% − 20% of v0 . Hence, for

more accurate quantification of RMI evolution, we choose

the time-scale set by the velocity v∞, at which an ideally pla-

nar interface would move after shock passage. The velocity

v∞ can be obtained straightforwardly from 1D calculations

(Meshkov, 1969).

For studies of the nonlinear RMI, four different Atwood

numbers are considered in our simulations with A = 0.55,

0.663, 0.78, and 0.9. In all runs, the dynamic viscosity

µ is the same in the heavy and light fluids, with Re =ρlv∞λ/µref = 13042 (A = 0.55), Re = 11572 (A = 0.663),

Re = 9700 (A = 0.78), and Re = 6968 (A = 0.9). Periodic

boundary conditions are used in the transverse direction,

and the effect of viscosity on the large-scale dynamics is

negligible. The initial perturbation has spatial period λ =2π/k = 3.75 cm and amplitude a0 = 0.064λ and is located

at z(x, t = 0) = a0 cos(kx) inside a [−40.667λ, 1.333λ] ×[−0.5λ, 0.5λ] box resolved by 5376 × 128 equidistant carte-

sian grid cells. The initial shock is weak, Ma = 1.2, and

propagates from the light to the heavy fluid. Our simulations

stop as the outlet reflected shock hits the interface. Still, as

the domain size is large, they run much longer compared to

other observations (Cheng et al., 2000; Robey et al., 2003;

Glendinning et al., 2003; Miles et al., 2004; Jacobs and Kriv-

ets, 2005).

To quantify the nonlinear evolution of RMI, we use two di-

agnostic parameters, the bubble velocity and curvature. The

bubble velocity is a traditional diagnostic parameter. In many

existing observations, the velocity of the bubble front in RMI

is determined relative to a ”middle line” – half of the distance

between the tips of the bubble and spike, i.e. half the ampli-

tude. To calculate the bubble velocity more accurately, we ac-

count for the fact that RMI develops relative to a background

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motion with a constant velocity v∞. Therefore, in the lab-

oratory frame of reference the bubble velocity is (v + v∞),

whereas in the frame of references moving with velocity v∞,

the bubble velocity is v.

We compare our numerical solutions with the results of the

nonlinear theory of Abarzhi et al. (2003), which suggests the

following evolution of the bubble front in RMI. After a short

stage of the shock-interface interaction, the bubble curvature

ζ and velocity v change linearly with time t ; then, in the

weakly non-linear regime, the curvature reaches an extreme

value, dependent on the initial conditions and the Atwood

number; asymptotically, the bubble flattens, ζ → 0, and de-

celerates, v → 0. For fluids with similar densities, A ≪ 1,

the bubbles move faster than those with contrasting densi-

ties, A → 1. The flattening of the bubble front is a distinct

feature of RMI universal for all A, which indicates that the

nonlinear evolution of RMI is governed by two length scales,

the amplitude h and the wavelength λ, and has therefore a

multi-scale character (Abarzhi et al., 2003).

The evolution of the bubble front is shown in Fig. 3, repre-

senting the phase diagram of the bubble velocity v(t) versus

bubble curvature ζ (t) with time t being a parameter. Ini-

tially the bubble exhibits an abrupt acceleration caused by

the shock- interface interaction (Richtmyer, 1960; Meshkov,

1969). Then the bubble starts to decelerate, while the abso-

lute value of the bubble curvature increases and reaches an

extreme value. We emphasize that the bubble velocity and

curvature reach their extreme values at two distinct moments

of time. As the instability evolves, the bubble continues to

decelerate and its curvature approaches zero, as found by the

theory of Abarzhi et al. (2003). Our results are in qualita-

tive agreement with the experiments of Jacobs and Krivets

(2005), who observed that in the nonlinear regime the front of

the RM bubble flattens at the tip and its velocity decreases,

whereas for fluids with similar densities the bubbles move

faster than those for fluids with highly contrasting densities.

Our simulation results indicate that it may be hard to es-

timate accurately the time-dependence of the asymptotic ve-

locity of the bubble front in compressible RMI. Figure 4

presents the log-log plot of the bubble velocity versus time

in the nonlinear regime of instability for A = 0.55. This den-

sity ratio is close to that in the experiments of Jacobs and

Krivets (2005), while the time considered is longer. We see

that the evolution of the bubble velocity is accompanied by

oscillations. These oscillations appear in the linear regime

of the instability and are induced by the revereberations of

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5

v v

v v v v

v v

Fig. 3 Bubble velocity v as function of absolute bubble curvature |ζ | for A = 0.55, 0.663, 0.78, and 0.9 (top left to bottom right); white squaremarks the asymptotic solution given by our non-local theory, black box marks the solution predicted by drag model

Springer


A = 0.55

0.004

0.005

0.006

0.007

0.008

0.01

10 20

v v

v

Fig. 4 Log-log plot of the bubble velocity v(t) in the nonlinear regimefor A = 0.55

sound waves (see Fig. 2). In the nonlinear regime of RMI,

the bubble velocity decreases, whereas the amplitude of the

velocity oscilllations is damped only slightly, and an estimate

of the time-dependence of the bubble velocity from the simu-

lation data is thus a challenging problem. We emphasize that

in most of exisiting experiments, the data sampling do not

have high temporal resolution and do not capture the oscil-

lations. Therefore, the accuracy and resolution of available

experimental data may be insufficient to make a quantitative

statement on the asymptotic time-dependencies of the non-

linear RMI. Further improvement of the interface diagnostics

in experiments is required, in particular, in the temporal res-

olution and data rate acquisition.

Our simulations confirm the results of the theory of

Abarzhi et al. (2003) and do not confirm the solution given

by drag models (Alon et al., 1995; Oron et al., 2001), which

presume a single-scale character of the interface evolution

and predict that for all values of the Atwood number, the RM

bubble remains curved asymptotically with ζD = −π/3λ. As

we see from Fig. 3, the shape of the bubble front in RMI is

not determined solely by the spatial period λ and is sensi-

tive to the length scale h which plays the role of the integral

scale for energy dissipation in small-scale structures. For

fluids with contrasting densities, A = 0.663, 0.78, and 0.9,

the decay in the curvature value is obvious, Fig. 3. In the

case of A = 0.55 the flattening process is slower. At a time

when the reflected shock hits the interface and our simula-

tions stop, the absolute value of the bubble curvature is still

finite |ζ | = 0.516/λ. Yet this is already about 50% smaller

than |ζD| (Oron et al., 2001). Figure 3 illustrates that the non-

linear dynamics in RMI is a multi-scale process, governed

by two macroscopic scales: the spatial period of the structure

λ and the amplitude h, which is the bubble displacement. In

the nonlinear regime of RMI the velocity of the bubble front

v = (dh/dt) and its curvature ζ mutually depend on one

another: dh/dt = v∞ f (|ζλ|), where f is an algebraic func-

tion. The processes of deceleration, d2h/dt2, and flattening,

d(ζλ)/dt , are therefore inter-related, and exhibit new fea-

tures of universality and similarity, see Figure 3, which can

be used in future models of RM turbulent mixing.

Summary

We report analytical and numerical solutions describing the

dynamics of the two-dimensional coherent structure of bub-

bles and spikes in the Richtmyer-Meshkov instability for flu-

ids with a finite density ratio. Good agreement between the

analytical and numerical solutions is achieved. To quantify

accurately the interface evolution in the observations, new di-

agnostics and scalings are suggested. The velocity, at which

the interface would move if it would be ideally planar, is used

to set the flow time-scale as well as the reference point for

the bubble (spike) position. Our data sampling has high tem-

poral resolution and captures the velocity oscillations caused

by sound waves. The bubble velocity and curvature are both

monitored. The obtained results indicate that the evolution of

the Richtmyer-Meshkov instability is a multi-scale process,

governed by two length scales, the spatial period and the am-

plitude of the coherent structure, and exhibits new similarity

features in the late-time evolution. Our results can serve as

benchmarsk for high energy density laboratory experiments

(Robey et al., 2003; Glendinning et al., 2003; Miles et al.,

2004).

References

Abarzhi, S.I., Herrmann, M.: New type of the interface evolution inthe Richtmyer-Meshkov instability. In: Annual Research Briefs-2003, Center for Turbulence Research, pp. 173–183. Stanford, CA(2003)

Abarzhi, S.I., Nishihara, K., Glimm, J.: Rayleigh-Taylor andRichtmyer-Meshkov instabilities for fluids with a finite densityratio. Phys. Lett. A 317, 470–476 (2003)

Alon, U., Hecht, J., Offer, D., Shvarts, D.: Power laws and similarityof Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts at alldensity ratios. Phys. Rev. Lett. 74, 534–537 (1995)

Cheng, B.L., Glimm, J., Sharp, D.H.: Density dependence of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts. Phys. Lett. A268(4–6), 366–374 (2000)

Chevalier, R.A.: A model for the radio brightness of the supernovaremnant 1987a. Nature 355(6361), 617–618 (1992)

Dimonte, G.: Spanwise homogeneous buoyancy-drag model forRayleigh-Taylor mixing and experimental evaluation. Phys. Plas-mas 7, 2255–2269 (2000)

Glendinning, S.G., Bolstad, J., Braun, D.G., Edwards, M.J., Hsing,W.W., Lasinski, B.F., Louis, H., Miles, A., Moreno, J., Peyser, T.A.,Remington, B.A., Robey, H.F., Turano, E.J., Verdon, C.P., Zhou,Y.: Effect of shock proximity on Richtmyer-Meshkov growth.Phys. Plasmas 10(5), 1931–1936 (2003)

Jacobs, J.W., Krivets, V.V.: Experiments on the late-time develop-ment of single-mode Richtmyer-Meshkov instability. Phys. Fluids17(034105), 1–10 (2005)

Jones, M.A., Jacobs, J.W.: A membraneless experiment for the studyof Richtmyer-Meshkov instability of a shock-accelerated gas in-terface. Phys. Fluids 9, 3078–3085 (1997)

Meshkov, E.: Sov. Fluid Dyn. 4, 101 (1969)Miles, A.R., Edwards, M.J., Blue, B., Hansen, J.F., Robey, H.F., Drake,

R.P., Kuranz, C., Leibrandt, D.R.: The effect of a short-wavelengthmode on the evolution of a long-wavelength perturbation drivenby a strong blast wave. Phys. Plasmas 11(12), 5507–5519 (2004)

Springer


Oron, D., Alon, U., Offer, D., Shvarts, D.: Dimensionality dependenceof the Rayleigh-Taylor and Richtmyer-Meshkov instability late-time scaling laws. Phys. Plasmas 8, 2883–2889 (2001)

Richtmyer, R.: Taylor instability in shock acceleration of compressiblefluids. Commun. Pure Appl. Math. 13(2), 297 (1960)

Robey, H.F., Zhou, Y., Buckingham, A.C., Keiter, P., Remington, B.A.,Drake, R.P.: The time scale for the transition to turbulence in a highReynolds number, accelerated flow. Phys. Plasmas 10(3), 614–622(2003)

Schmidt, H., Klein, R.: A generalized level-set/in-cell-reconstructionapproach for accelerating turbulent premixed flames. Combust.Theory Modelling 7, 243–267 (2003)

Smiljanovski, V., Moser, V., Klein, R.: A capturing-tracking hybridscheme for deflagration discontinuities. Combust. Theory Mod-elling 1, 183–215 (1997)

Wouchuk, J.: Growth rate of the linear Richtmyer-Meshkov instabil-ity when a shock is reflected. Phys. Rev. E 63(56303), 1–13(2001)

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DOI 10.1007/s10509-006-9184-y


Density Measurements of Shock Compressed Matter Using ShortPulse Laser Diagnostics

M. Koenig · A. Ravasio · A. Benuzzi-Mounaix ·

B. Loupias · N. Ozaki · M. Borghesi · C. Cecchetti ·

D. Batani · R. Dezulian · S. Lepape · P. Patel ·

H. S. Park · D. Hicks · A. Mckinnon · T. Boehly ·

A. Schiavi · E. Henry · M. Notley · R. Clark ·

S. Bandyopadhyay

Received: 21 April 2006 / Accepted: 31 May 2006C© Springer Science + Business Media B.V. 2006

Abstract In this paper, experimental results on X-ray and

proton radiography of shock compressed matter are pre-

sented. It has been performed at the Rutherford Appleton

Laboratory (RAL) using three long pulse beams to generate

a shock wave in a multi-layer foil and a short pulse beam

to create either an X-ray or protons source for a transverse

radiography. Depending on the probe material (aluminium

M. Koenig ()· A. Ravasio· A. Benuzzi-Mounaix· B. Loupias·N. OzakiLaboratoire pour l’Utilisation des Lasers Intenses,CNRS-CEA-Universite Paris VI-Ecole Polytechnique, 91128Palaiseau, France

M. Borghesi· C. CecchettiDepartment of Physics and Astronomy, The Queen’s University ofBelfast, Belfast, BT7 1NN, UK

D. Batani· R. DezulianDipartimento di Fisica ‘G. Occhialini’, Universita diMilano-Bicocca and INFM, Piazza Della Scienze 3, 20126Milano, Italy

S. Lepape· P. Patel· H. S. Park· D. Hicks· A. MckinnonLawrence Livermore National Laboratory, Livermore,CA 94550, USA

T. BoehlyLaboratory for Laser Energetics, University of Rochester, USA

A. SchiaviDepartment of Energetics, University of Rome “La Sapienza”,Rome, Italy

E. HenryDepartement de conception et realisation des experiences(DECRE), CEA-DIF, BP 12, 91680 Bruyeres-le-Chatel, France

M. Notley· R. Clark· S. BandyopadhyayCentral Laser Facility, Rutherford Appleton Laboratory, Chilton,Oxfordshire, OX11 0QX, UK

or carbon foam) a Molybdenum Kα source or a proton beam

are used. Density data of the shocked aluminium, in the mul-

timagabar regime are presented.

Keywords Shocks . Laser plasmas . Radiography

The knowledge of Equation Of State (EOS) and related pa-

rameters of dense matter is important in several fields of

physics. For instance, in astrophysics the star evolution is

mainly governed by the thermodynamic properties of mat-

ter. EOS is also fundamental for the knowledge of internal

structure of giant or telluric planets (Stevenson, 1981). In-

ertial Confinement Fusion (ICF) success depends directly

from the understanding of shell pellet implosion and the final

core compression. Both of these processes implies a precise

knowledge of the microballoon material (Koenig et al., 1998)

and the fuel (deuterium) EOS at very high pressures (Collins

et al., 1998; Knudson et al., 2001) (>100 Gpa).

Since several years, a large effort has been done in laser

driven shock wave experiment to develop relevant diagnos-

tics to perform high precision EOS measurements (Cauble et

al., 1997; Koenig et al., 1995) (Benuzzi-Mounaix et al., 2002;

Hicks et al., 2005). Shock-wave-EOS experiments require

that two parameters, usually the shock and fluid velocities,

be measured to infer the thermodynamic properties of the

material. While a few experiments have used X-ray radiog-

raphy on low-Z materials to determine both velocities(Cauble

et al., 1997; Collins et al., 1998), most of them rely on the

shock velocities measurement via optical interferometry in

transparent media or by observation of shock breakout times

on steps of known thickness in optically opaque materials

(Hicks et al., 2005; Koenig et al., 1999; Koenig et al., 1995).

The latter technique result in indirect EOS determinations

through a method known as impedance matching. In opaque

high/mean Z materials, it is not possible to get information

Springer


about the shock characteristics inside the sample and then

to measure fluid velocities directly unless the free surface

velocity can be determined (Benuzzi-Mounaix et al., 2002).

Therefore, impedance matching is the only possibility to in-

fer EOS data. Moreover, in measurements based on velocity

determination, it is impossible to have a good precision on

such an important physical quantity as density. Indeed this

crucial parameter, as seen in the recent deuterium various

experiments (Collins et al., 1998; Knudson et al., 2001), due

to error amplification going through Rankine-Hugoniot rela-

tions remains uncertain . The development of direct probing

techniques to obtain information on another shock parame-

ter, such as density, would allow precise absolute EOS de-

terminations and would represent a real breakthrough in the

field.

Such attempt has been made on plastic several years ago

(Hammel et al., 1993) and more recently (Hicks et al., 2004;

Boehly et al., 2005) with “conventional” (long pulse laser

beam) X-ray sources. Moreover, density measurements have

been conducted using high explosives to drive a shock in a

metal and a high energy proton beam (800 MeV) produced

by a conventional accelerator to probe it (Holtkamp et al.,

2003). These experiments, dedicated to the study of spal-

lation, i.e., in a pressure regime much lower than the one

we are interested in, provided a density measurement due

to protons collisional stopping power. In a recent experi-

ment, high-energy protons, produced by the interaction of

an intense laser pulse with a thin solid target, in a point

projection imaging scheme, were used to characterize in

situ the spatial and temporal evolution of a laser-driven

shock propagating through a low-Z material. The particu-

lar properties of laser produced protons beam (small source,

high degree of collimation, short duration) make them of

great interest for radiographic applications (Borghesi et al.,

2001). However due to high level scattering of the pro-

ton beam, inferring density implies development of specific

propagation code for the proton in the shock compressed

material.

In this paper we present density measurement of shocked

aluminium using hard X-ray radiography (17.5 keV) and

shock propagation in a carbon foam probed by a proton beam.

This experiment was performed at the RAL (Rutherford

Appleton Laboratory) in the TAW target area. Three long

pulse beam converted at 2ω with a 2 ns pulse duration gen-

erated a shock wave into a multilayer target (Fig. 1) de-

livering a total maximum energy on target E2ω ≈ 300 J. To

generate the shock wave, we used Random Phase Plates

in order to eliminate large scale spatial intensity modula-

tions and obtain a Gaussian type profile in the focal spot

(FWHM ≈ 300µm), corresponding to a maximum laser in-

tensity IL≤81013 W/cm2.

A schematic view of the experimental set-up and the tar-

get design is shown in Figure 1. The three laser beams, were

Fig. 1 Experimental set-up and diagnostics

focused on an ablator-pusher foil to generate a shock into a

sliver which was aluminium or carbon foam for the X-ray

or proton source respectively. The pusher design was opti-

mized using 1D radiative hydrodynamic simulations accord-

ing to the laser characteristics. We had a three layer target

(10µm CH-10µm Al-10µm CH), the last CH layer acting

as a “witness” plate. Two main sets of diagnostics were im-

plemented: on the rear side of the target, a streak camera

(“Self Emission”) collected photons emitted by the target

giving the shock mean shock velocity of the rear side CH

layer. On the transverse axis, we had specific detectors for

the probe beams either Image Plates (IP) for the hard X-ray

source or Radiochromic Films (RCF) for protons. Magnifi-

cation was adjusted according to the detector resolution (80

microns for IP or few microns for the RCF) and was 40 and

13 respectively.

X-ray radiography

The relativistic electrons generated by the interaction of an

intense short laser pulse induce Kα radiation. The energy of

this radiation can be adapted to the material to be probed by

choosing the appropriate backlit target material. In our case,

we used a molybdenum foil that generated a 17.5 keV Kα

line. According to previous electron transport experiments,

the Kα X-ray source has a minimum size of the order of

50µm (Martinolli et al., 2006; Stephens et al., 2004) which

is too big for the desired spatial resolution. This is the reason

why we used a scheme developed recently (Park et al., 2006)

which consists to use the foil thickness (5µm in our case) as

the way to limit the source size (Fig. 1). In order to check the

final resolution, we did a test with a 100 lpi tungsten mesh.

From the data, we could determine a 20µm resolution along

the shock propagation (foil thickness), 60µm in the other di-

rection due the Kα source minimum size. We then performed

several shots compressing the aluminum with the long pulse

beams and varying the probe beam delay, between 5 and 10 ns

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Fig. 2 X-ray radiograph ofcompressed aluminum

after the beginning of the shock drive beams. This ensures

that the shock has propagate a suffisant length into the sliver

in order to be able to deduce the density (Fig. 2). To deter-

mine the density, in fact, we must consider that the observed

signal on the detector is a line integral of the local absorp-

tion. The use of RPP generated a symmetrical shock along

the propagation axis making the situation treatable with typ-

ical Abel inversion techniques. Taking azimuthal symmetry

into account, the areal density in the radiography z direction

can be written as:

ρz = 2

a∫

y

ρ(r )rdr√

r2 − y2+ ρ0(L − 2

√

a2 − y2),

where a is the y coordinate of the shock front and L the

transverse thickness of the target. The X-ray intensity I, after

propagation into the sliver is a function of the areal density,

I = I0 exp(−µmρz). The measured data transmission F(y)

is then given by

F(y) = − 1

ρ0µm

ln

(

I

I0

)

− L = 2

∫ a

y

β(r )rdr√

r2 − y2,

where β(r ) = ρ

ρ0− 1 is inferred from Abel inversion. From

the data shown in Fig. 2, we deduced the shock compressed

density (Fig. 3). The density is maximum in the center, the

compression ratio ρ/ρ0=2.2 in good agreement with ex-

pected value given, for example, by simulations. The error

bars, taking into account the total incident spectrum uncer-

tainty, Abel inversion evaluation, are at least of the order of

±10%.

Fig. 3 Deduced density profile of shock compressed aluminum. Dotsare the Abel inverted data, plain curve is a fit to those data

Proton radiography

Due to high level of scattering of protons generated by an

ultra-intense laser pulse in a high Z solid target as pointed

out recently (LePape et al., 2006), we deliberately tried to

probe a low density material such as carbon foam (CRF).

According to simple calculations (geometrical+scattering

Fig. 4 Resolution test for the proton beam through CRF foam and a1000 lpi grid. A line out is taken where protons goes through the foam

Fig. 5 Line out of Fig. 4 on the foam+grid region

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Fig. 6 Proton radiograph ofcompressed carbon foam taken7 ns (a) & 11 ns (b) after themain pulses

angles given by the SRIM code), the expected resolution

for the proton beam going through a 500µm wide 0.1 g/cm3

CRF foam sliver must be typically 25µm. This was probed

experimentally having a 1000 lpi grid mesh behind the CRF

sliver shooting only the short pulse beams, i.e., just with the

proton beam (Fig. 4). As we can observe in the zone where

there is no foam, the grid is very contrasted and well de-

fined. When the protons propagate through the CRF sliver,

there is a loss of contrast and signal due to scattering and

slight absorption. However by taking a lineout in the foam

region (Fig. 4), we obtain the modulations (Fig. 5) due to the

grid spacing (25µm). As expected the deduced resolution is

<15µm, not as good as for direct transmission through the

grid (with no foam) but high enough to give high contrast

for the shock compressed case. As for the hard x-ray, we did

several shots delaying the probe beam with respect to the

main long pulses. Figure 6 shows a clear shock front in the

middle of the foam sliver (a) and just at the end of the CRF

target (b) for the 7 ns and 11 ns cases respectively. One can

also remark (Fig. 6b) the position of the pusher behind the

shock front. The shock between the two these shots traveled

through 120µm of foam, giving an estimate of the velocity

Us ≈ 30 km/s which is in good agreement with expected val-

ues from hydrodynamic simulations. To deduce the density

from the experimental results, it cannot be done as for the

x-ray case because we are not in a pure absorption situation,

moreover not in cold matter either; Therefore one has to treat

the proton propagation in a 2D density and temperature pro-

files. To this end we developed a specific Monte-Carlo code

to deal with the proton propagation in warm dense matter

coupled to density and temperature profiles given by 2D hy-

drodynamic simulations (DUED code (Atzeni, 1986)). We

then need to iterate between the MC code and density pro-

file to retrieve the experimental data. This final stage will be

completed in the next few months so final density from the

compressed foam will be determined.

Conclusions

In this paper, we present new diagnostics for direct density

measurements using hard X-ray or proton beam source gen-

erated by ultra intense short lasers pulses. Hard X ray radiog-

raphy (E> 10 keV) allowed, for the first time, to infer shock

characteristics inside the high-Z dense target impossible to

obtain with standard thermal x-ray sources. Concerning the

proton radiography, we have shown the possibility to diag-

nose shock propagation in a low density medium (carbon

foam) with high spatial resolution (<25µm). We are on the

way to infer the compressed material density by coupling

a Monte-Carlo code, for the proton propagation in a warm

dense plasma, with 2D hydrodynamic simulations. Higher

proton energy beams, using for example the newt PW laser

at LULI (Pico, 2000) will probably allow to detect shock front

in standard density materials such as plastic or diamond.

Acknowledgements The authors would like to thank the RAL peoplewho contributed to the success of this work. This experiment has beensupported by the Laserlab EU program FP6 contract RII3-CT-2003-506350.

References

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Henry, E., Tomasini, M., Marchet, B., Hall, T.A., Boustie, M.,Resseguier, T.D., Hallouin, M., Guyot, F., Andrault, D., Charpin,T.: Phys. Plasmas 9, 2466–469 (2001)

Boehly, T.R., Vianello, E., Miller, J.E., Meyerhofer, D.D., Hicks, D.G.,Eggert, J.H., Hansen, J.F., Celliers, P.M., Collins, G.W.: Bull. Am.Phys. Soc. 50, 73 (2005)

Borghesi, M., Schiavi, A., Campbell, D.H., Haines, M.G., Willi, O.,MacKinnon, A.J., Gizzi, L.A., Galimberti, M., Clarke, R. J., Ruhl,H.: Plasma Phys. Control. Fusion 43, 267–276 (2001)

Cauble, R., Silva, L. B.D., Perry, T.S., Bach, D.R., Budil, K.S., Celliers,P., Collins, G.W., Ng, A., Barbee, T.W., Hammel, B.A., Holmes,

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N.C., Kilkenny, J.D., Wallace, R.J., Chiu, G., Woolsey, N.C.: Phys.Plasmas 4 (1997)

Collins, G., Silva, L.D., Celliers, P., Gold, D., Foord, M., Wallace,R., Ng, A., Weber, S., Budil, K., Cauble, R.: Science 281, 1178(1998)

Hammel, B.A., Griswold, D., Landen, L., Perry, T.S., B, A.R., Mille,P.L., Peyser, T.A., Kilkenny, J.D.: Phys. Fluids B 5, 2259–2264(1993)

Hicks, D., Hansen, F., Bradley, D., Celliers, P., Eggert, J., Collins,G., Boehly, T., Meyerhofer, D.: Bull. Am. Phys. Soc. 49, 234(2004)

Hicks, D.G., Boehly, T.R., Celliers, P.M., Eggert, J.H., Vianello, E.,Meyerhofer, D.D., Collins, G.W.: Phys. Plasmas 12, 082702 (2005)

Holtkamp, D., Clark, D., Ferm, E., Gallegos, R., Hammon, D., Hemsing,W., Hogan, G., Holmes, V. King, N.: in: Furnish, M.D., Gupta,Y.M., Forbes, J. W. (eds.), A survey of high explosive-induceddamage and spall in selected metals using proton radiography,AIP Press, Portland, Or, USA, pp. 477–482 (2003)

Knudson, M.D., Hanson, D.L., Bailey, J.E., Hall, C.A., Asay, J. R.,Anderson, W. W., Phys. Rev. Lett. 8722 (2001)

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Koenig, M., Benuzzi, A., Philippe, F., Batani, D., Hall, T., Grandjouan,N., Nazarov, W.: Phys. Plasmas 6, 3296–3301 (1999)

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LePape, S., Koenig, M., Vinci, T., Martinolli, E., Bennuzzi-Mounaix,A., Hicks, D., Patel, P., Mackinnon, A., Romagnani, L., Borghesi,M., Kar, S., Boehly, T.: HEDP (2006)

Martinolli, E., Koenig, M., Baton, S.D., Santos, J.J., Amiranoff, F.,Batani, D., Perelli-Cippo, E., Scianitti, F., Gremillet, L., Melizzi,R., Decoster, A., Rousseaux, C., Hall, T.A., Key, M.H., Snavely, R.,MacKinnon, A., Freeman, R.R., King, J.A., Stephens, R., Neely,D., Clark, R.J.: Phys. Rev. E, to be published (2006)

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Stevenson, D.J.: Science 214, 611–619 (1981)

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DOI 10.1007/s10509-006-9285-7


Hydrogen and Helium at High Densityand Astrophysical Implications

G. Chabrier · D. Saumon · C. Winisdoerffer


Abstract We briefly summarize the present status of theoret-

ical and experimental investigations aimed at describing the

thermodynamic properties of hydrogen and helium at high

density. We confront various theoretical models to presently

available experiments and we consider the astrophysical im-

plications for giant planet interiors.

Keywords Plasma physics . Equation of state

1 Introduction

An accurate determination of the thermodynamic properties

of matter under extreme conditions of temperature and den-

sity, in particular hydrogen and helium, is required for a cor-

rect description of the mechanical and thermal properties of

various cool and dense astrophysical bodies, including giant

planets, brown dwarfs and low-mass stars (i.e., stars less mas-

sive than the Sun). These objects are composed dominantly

of ion-electron plasmas, where ions are strongly correlated

and electrons are strongly or partially degenerate. Pressure

ionization of both hydrogen and helium occurs along the

internal profile of these bodies, which bears major conse-

quences for their structural, cooling and magnetic properties.

Furthermore, complex phenomena such as phase transition

or phase separation may take place in their interior, present-

ing challenging problems. The characterization of the struc-

G. Chabrier () · C. WinisdoerfferEcole Normale Superieure de Lyon, C.R.A.L.(UMR 5574CNRS), Lyon, Francee-mail: [email protected]

D. SaumonLos Alamos National Laboratory, Mail Stop P365, Los Alamos,NM 87545, USA

ture and cooling of these astrophysical bodies thus requires

the knowledge of the equation of state (EOS) and the trans-

port properties of such dense plasmas, including a correct

description of pressure ionization and of the mixture’s ther-

modynamic stability. In this short contribution, we focus on

the range of astrophysical objects characteristic of the so-

called substellar objects, i.e. objects with masses below the

hydrogen-burning limit (m ≤ 0.075 M⊙). This encompasses

brown dwarfs and gaseous planets.

2 The equation of state of dense fluid hydrogen

and helium

2.1 Hydrogen: pressure dissociation and ionization

The description of hydrogen pressure-ionization, or met-

allization, has remained a challenging problem since the

pioneering work of Wigner and Huntington (1935). Much

experimental work has been devoted to this problem, but

no definitive result has been reached yet. Several high-

pressure shock wave experiments have been conducted in

order to probe the EOS of deuterium in the regime of pres-

sure ionization. Gas gun shock compression experiments

are generally limited to pressures below 1 Mbar (Nellis

et al., 1983), probing only the domain of molecular hydro-

gen. New techniques include laser-driven shock-wave exper-

iments (Collins et al., 1998, 2001; Mostovych et al., 2000;

Boehly et al., 2004), pulse-power compression experiments

(Knudson et al., 2004) and convergent spherical shock wave

experiments (Belov et al., 2002; Boriskov et al., 2003) that

can achieve pressures up to 5 Mbar in fluid deuterium at

high temperature, exploring for the first time the regime of

pressure-dissociation and ionization. These recent experi-

ments give different results at P1 Mbar, however, and this

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controversy needs to be settled unambiguously before a ro-

bust comparison between experiment and theory can be made

in the very domain of hydrogen pressure ionization.

On the theoretical front, a lot of efforts has been de-

voted to describing the pressure ionization of hydrogen. The

EOS commonly used for modeling Jovian planet interiors is

the Saumon-Chabrier-Van Horn (SCVH) EOS (Saumon and

Chabrier, 1991, 1992; Saumon et al., 1995) which includes

a detailed description of the partial ionization regime. This

EOS reproduces the Hugoniot data of Nellis et al. (1983) but

yields temperatures about 30% higher than the gas reshock

data, indicating insufficient D2 dissociation (Holmes et al.,

1995). A slightly revised version (Saumon et al., 2000) re-

covers the gas gun reshock temperature data as well as

the laser-driven shock wave results (Collins et al., 1998),

with a maximum compression factor of ρ/ρ0 ≃ 6, where

ρ0 = 0.17 g cm−3 is the initial density of liquid deuterium at

20 K. On the other hand, the earlier SESAME EOS (Kerley,

1972), based on a similar formalism, predicts a smaller com-

pression factor, with ρ/ρ0 ≃ 4, in general agreement with

all the other recent shock wave experiments. Ab initio ap-

proaches for the description of dense hydrogen include path

integral Monte Carlo (PIMC) (Militzer and Ceperley, 2000;

Militzer et al., 2001; Bezkrovniy et al., 2004) and Quantum

Molecular Dynamics (QMD) simulations. The latter com-

bine molecular dynamics (MD) and Density Functional The-

ory (DFT) to take into account the quantum nature of the

electrons (Lenosky et al., 2000; Bagnier et al., 2001; Des-

jarlais, 2003; Bonev et al., 2004). The relevance of earlier

MD-DFT calculations was questioned on the basis that these

simulations were unable to reproduce data from gas-gun ex-

periments (Lenosky et al., 2000). This problem has been

solved with more accurate simulations (Bagnier et al., 2001;

Desjarlais, 2003; Bonev et al., 2004). Although an ab ini-

tio approach is more satisfactory than the phenomenological

approach based on effective potentials, in practice these sim-

ulations also rely on approximations, such as the handling of

the so-called sign problem for the antisymmetrization of the

fermion wave functions, or the calculation of the electronic

density functional itself (in particular the exchange and corre-

lation effects), or the use of effective pseudo-potentials of re-

stricted validity, not mentioning finite size effects. Moreover,

these simulations are too computationally intensive for the

calculation of an EOS covering several orders of magnitude

in density and temperature, as necessary for the description

of the structure and evolution of astrophysical bodies.

Figures 4 and 5 of (Saumon and Guillot, 2004) and

Figure 1 of (Chabrier et al., 2006) compare experimental and

theoretical Hugoniots in the P-ρ and P-T planes. Whereas

the SCVH EOS achieves a maximum compression similar to

the laser-driven data, most of the other models predict maxi-

mum compression factors in the P-ρ plane in agreement with

the more recent data. The MD-DFT results, however, pre-

dict temperatures for the second shock significantly larger

than the experimental results (Holmes et al., 1995). Even

though the experimental double-shock temperature may be

underestimated due to unquantified thermal conduction into

the window upon shock reflection, and thus may represent a

lower limit on the reshock temperatures, the disagreement in

the T-V plane is significant. As mentioned above, the degree

of molecular dissociation has a significant influence on the

thermodynamic properties of the fluid and insufficient dis-

sociation in the simulations may result in overestimates of

the temperature. It has been suggested that the LDA/GGA

approximations used in MD-DFT underestimate the disso-

ciation energy of D2 (Stadele and Martin, 2000). A correct

calculation would thus lead to even less dissociation. The fact

that compression along the experimental Hugoniot remains

small thus suggests compensating effects in the case of hydro-

gen. More recent, improved QMD simulations (Bonev et al.,

2004), however, seem to partly solve this discrepancy and to

produce reshock temperatures in better agreement with the

experimental results. Peak compression in the modern MD-

DFT simulations occurs in the ∼0.2–0.5 Mbar range around

a dissociation fraction of ∼50%.

The differences in the behaviour of hydrogen at high den-

sity and temperature illustrated by these various results bear

important consequences for the structure and evolution of

our Jovian planets. These differences must be correctly un-

derstood before the description of hydrogen pressure disso-

ciation and ionization stands on firm grounds. As noted by

Boriskov et al. (2005), all the recent experiments agree quite

well in terms of the shock speed us versus the particle velocity

up, almost within their respective error bars. When solving

the Rankine-Hugoniot equations, error bars and differences

in (us , u p) are amplified in a P-ρ diagram by a factor of

(ρ/ρ0 − 1). These are challenging experiments as the differ-

ences seen in the P-ρ diagram arise from differences in us and

up of less than 3%. Ongoing high-pressure isentropic com-

pression experiments (Knudson, this volume) are promising

techniques to help address this challenge.

2.2 Helium: equation of state and phase diagram

A model EOS for helium at high density, covering the

regime of pressure ionization, has been developped recently

by Winisdoerffer and Chabrier (WC05) (Winisdoerffer and

Chabrier, 2005). This EOS, based on effective interaction po-

tentials between He, He+, He++ and e− species, reproduces

adequately experimental Hugoniot and sound speed mea-

surements up to ∼1Mbar. Figure 12 of WC05 illustrates the

expected phase diagram of fluid helium at high temperatures

and densities, for conditions characteristic of the interior of

substellar objects.

In this model, pressure ionization is predicted to occur

directly from He to He++ for T 105K. Because of the

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uncertainties in the treatment of the interactions at high den-

sity, however, the predicted ionization density ranges from

a few to ∼10 g cm−3, i.e. P ∼ 9–20 Mbar, depending on

the temperature. This is significantly larger than the ρ ≈ 1 g

cm−3 density above which available measurements of elec-

trical conductivity of helium (Ternovoi et al., 2001; Fortov

et al., 2003) predict that the plasma is substantially ionized.

These measurements, however, conflict with more recent

conductivity measurements (Cellier, this volume) and MD-

DFT calculations (Mazevet, this volume). It must be kept in

mind that the reported measurements are model dependent

and that conductivity determinations imply some underlying

EOS model.

3 Plasma phase transition. Hydrogen-helium

phase separation

The pressure ionization and metallization of hydrogen have

been predicted to occur through a first order phase transi-

tion, the so-called plasma phase transition (PPT) (Wigner

and Huntington, 1935; Saumon and Chabrier, 1992; Norman

and Starostin, 1968; Ebeling and Richert, 1985; Saumon and

Chabrier, 1989; Kitamura and Ichimaru, 1998). Nearly all of

these PPT calculations are based on chemical EOS models.

Such models are based on a model Helmholtz free energy

that includes contributions from (1) neutral particles (atoms

and molecules), (2) a fully ionized plasma, and (3) usually a

coupling between the two. It is well-known that realistic fully

ionized plasma models become thermodynamically unstable

at low temperatures and moderate densities. This is analogous

to the behavior of expanded metals at T = 0 that display a re-

gion where d P/dρ < 0 and even P < 0 (Pines and Nozieres,

1958). This behavior of the fully ionized plasma model is for-

mally a first order phase transition and reflects the formation

of bound states in the real system. In other words, the chem-

ical models have a first order phase transition built in from

the onset, and this phase transition coincides, not surpris-

ingly, with the regime of pressure ionization. This represents

a common flaw in this type of models and it follows that

their prediction of a PPT in hydrogen is not credible. On the

other hand, recent ab initio simulations find a sharp (6 ±2%) volume discontinuity at constant pressure (Bonev et al.,

2004; Scandalo, 2003) or d P/dT < 0 at constant volume

(Magro et al., 1996), a feature consistent with the existence

of a first order phase transition. At the same time, the pair

correlation function exhibits a drastic change from a molecu-

lar to an atomic state with a metallic character (finite density

of electronic states at the Fermi level). These transitions are

found to occur in the ∼0.5–1.25 Mbar and ∼1500–3000 K

temperature range. While these results are suggestive, a sys-

tematic exploration of this part of the phase diagram remains

to be done. Note that a first order structural transition for H2

at T = 0 is predicted to occur at a pressure P 4.0 Mbar,

from DFT calculations based on exact exchange calculations

(Stadele and Martin, 2000). There is so far no published ex-

perimental evidence for the PPT but it cannot yet be ruled out.

Given the difficulty of modeling this region of the phase dia-

gram of hydrogen, only experiments can ultimately establish

whether a PPT exists or not.

The existence of a phase separation between hydrogen and

helium under conditions characteristic of Jupiter and Saturn

interiors was first suggested by Smoluchowski (1967) and

Salpeter (1973) and the first detailed calculations were done

by Stevenson and Salpeter (1977). A phase separation is a

first order transition which implies a concentration and thus

a density discontinuity below a critical temperature between

two phases in equilibrium, as given by the Gibbs phase rule:

µIi = µI I

i ⇒ x Ii = x I I

i e− GkT , (1)

where µi , xi denote respectively the equilibrium chemical

potential and number concentration of the species i in phase

I and II and G is the excess (non-ideal) mixing enthalpy

between the two phases. Under the influence of a planet’s

gravity field, a density discontinuity yields an extra source

of gravitational energy as the dense phase dropplets (namely

helium-rich ones in the present context) sink towards the

center of the planet. This, in turn, translates into a delayed

cooling time and thus a longer time scale (age) t to reach a

given (observed) luminosity L:

t ≈ E

L≈

Mgρ

ρR

L, (2)

where M is the mass fraction experiencing phase separa-

tion, ρ is the density difference between the two phases, g

is the planet gravity and R is the planet’s radius.

In Saturn’s case, such an additional source of energy is

required to explain the observed luminosity at the correct age,

i.e. the age of the solar sytem, ∼4.5 × 109 yr (Fortney and

Hubbard, 2004). There are few studies of the H/He phase di-

agrams. Stevenson’s (1982) calculations are based on the so-

called binary ionic mixture (BIM) model, where the electrons

are considered as a rigid background. These calculations

were extended by Guillot and Chabrier (unpublished) by in-

cluding the electron polarizability. Although yielding differ-

ent values for the critical temperature, these calculations yield

similar qualitative results, i.e. a critical temperature increas-

ing with decreasing pressure. Both calculations, however,

assume that phase separation takes place in the fully ion-

ized part of the planet interior, i.e. in the H+/He++ domain.

Electronic structure calculations for the T = 0 H/He solid

alloy, with no assumption on the degree of ionization of the

plasma, were first conducted by Kleipeis et al. (1991). Finite

temperature results were obtained by applying an estimated

entropy correction. The following, finite temperature calcu-

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lations were performed by Pfaffenzeller et al. (1995) within

the framework of the MD-DFT. A striking result of these

latter calculations is the prediction of an increasing critical

temperature with increasing pressure, a result qualitatively

different from the fully ionized calculations, and which has

major consequences for jovian planet evolution. Although the

most detailed calculations of the H/He phase separation di-

agram so far, these calculations, however, suffer from some

limitations. First of all, these calculations were performed

in the canonical ensemble and a thermostat must thus be

used to calculate the forces acting on the ions when solving

Newton’s equation of motion. The effects due to the ther-

mostat remain unquantified. Although expected to be small,

the possibility that these effects yield uncertainties compa-

rable to the calculated excess (non-ideal) enthalpy, itself a

very small (1%) contribution to the total mixing enthalpy,

can not be excluded. Second, the excess enthalpy in Pfaffen-

zeller et al. (1995) is calculated directly from the difference

between the final and the initial states, yielding an incorrect

evaluation of the contraction work thus of the pression along

the changes of equilibrium states (see Winisdoerffer et al.,

2004). No accurate H/He phase diagram is thus available

yet. A new MD-DFT method, allowing the calculation of

the excess chemical potential of a fraction of helium atoms

immersed in a hydrogen fluid in the microcanonical ensem-

ble has been derived recently (Winisdoerffer et al., 2004).

The microcanonical ensemble is the most natural ensem-

ble for MD calculations, preventing the use of thermostats.

Derivation of the complete phase diagram with an accurate

functional for the electrons, however, is prohibitively time

consuming. Comparisons for a few test cases between these

calculations and calculations conducted in a different ensem-

ble, however, might enable us to quantify the effects due to

external thermostats.

The problem of the phase diagram of dense H/He mixtures

is perhaps the most important remaining problem regarding

the interior structure and evolution of Jupiter and Saturn and

needs to be addressed both experimentally and theoretically.

It is also one of the few such problems that is well con-

strained by astrophysical observations. Indeed, some of the

aforementioned H/He critical point or critical line determina-

tions can be excluded from astrophysical constraints arising

from Jupiter and Saturn cooling history (Guillot et al., 1995).

4 Implications for Jupiter and Saturn interiors

A detailed study of the influence of the EOS of hydrogen and,

to a lesser extent, of helium on the structure and evolution of

Jupiter and Saturn has been conducted recently (Saumon and

Guillot, 2004). Fortunately, some shock wave experiments

overlap the adiabats of Jupiter’s and Saturn. As demonstrated

by Saumon and Guillot (2004), the small (≤5%) difference on

the (P, ρ) relation along the adiabat between the SCVH and

SESAME EOS, representative of the two sets of experimen-

tal results, is large enough to affect appreciably the interior

structure of the models. A slightly modified version of the

SESAME EOS, which recovers the H2 entropy at low temper-

ature and density, yields Jupiter models with a very small core

mass, Mcore ∼ 1 M⊕ (M⊕ is the mass of the Earth) and a mass

MZ ∼ 33 M⊕ of heavy elements (Z > 2) mixed in the H/He

envelope. The SCVH EOS yields models with Mcore ∼0–6

M⊕ and MZ ∼ 15–26 M⊕. Models of Saturn are less sensi-

tive to the EOS differences, since only ∼70% of its mass lies

at P > 1 Mbar, compared to 91% for Jupiter. Models com-

puted with the SCVH and the modified SESAME EOS have

Mcore = 10–21 M⊕ and MZ = 20–27 M⊕ and 16–29 M⊕,

respectively. As shown by Saumon and Guillot (2004), the

temperature along the adiabat is quite sensitive to the choice

of the EOS. This affects the thermal energy content of the

planet and thus its cooling rate and evolution. Equations of

state which are adjusted to fit the deuterium reshock temper-

ature measurements (Ross, 1998) lead to models that take

∼3 Gyr for Jupiter to cool to its present state. Even when

considering uncertainties in the models, or considering the

possibility of a H/He phase separation, such a short cooling

age is unlikely to be reconciled with the age of the solar sys-

tem. This astrophysical constraint suggests that the reshock

temperature data are too low.

5 Conclusions

In this brief review, we have examined the present status of

the description of the thermodynamic properties of dense

hydrogen and helium. The description of the pressure ion-

ization and more generally the EOS of these elements at

high density determine the mechanical and thermal proper-

ties, thus the structure and the evolution of substellar objects,

brown dwarfs and jovian planets. Modern high-pressure ex-

periments and/or observations remain for now too uncertain

to enable us to discriminate between most EOS models in

planet interiors. Upcoming experiments like pre-compressed

targets or isentropic compression experiments, however, will

lead eventually to a better determination of these EOS and,

eventually, of the characterization of phase separation and

phase transition in the interior of these objects.

Acknowledgments The work of DS was supported in part by theUnited States Department of Energy under contract W-7405-ENG-36.

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DOI 10.1007/s10509-006-9231-8


Graded-Density Reservoirs for Accessing High Stress LowTemperature Material States

Raymond F. Smith · K. Thomas Lorenz · Darwin Ho · Bruce A. Remington ·

Alex Hamza · John Rogers · Stephen Pollaine · Seokwoo Jeon · Yun-Suk Nam ·

J. Kilkenny


Abstract In recently developed laser-driven shockless com-

pression experiments an ablatively driven shock in a primary

target is transformed into a ramp compression wave in a sec-

ondary target via unloading followed by stagnation across an

intermediate vacuum gap. Current limitations on the achiev-

able peak longitudinal stresses are limited by the ability of

shaping the temporal profile of the ramp compression pulse.

We report on new techniques using graded density reser-

voirs for shaping the loading profile and extending these

techniques to high peak pressures.

Keywords Quasi-isentropic compression . High

compression rates

Understanding high stress-low temperature compressive

states is relevant to the study of planetary interiors

(Remington et al., 2005). Traditionally multi-Mbar longi-

tudinal stress (Px) states have been accessed by launching

a near instantaneous compressive shock into the material.

Shock waves are associated with large jumps in temperature

which greatly increases the thermal contribution to Px, and

can cause melting of the material under study. In aluminum,

shock stresses above 1.4 Mbar produce temperatures above

4000 K and melting of the sample (Chijioke et al., 2005).

R. F. Smith () · K. T. Lorenz · D. Ho · B. A. Remington ·A. Hamza · S. PollaineLawrence Livermore National Laboratory, P.O. Box 808,CA 94550e-mail: [email protected]

J. Rogers · S. Jeon · Y.-S. NamUniversity of Illinois at Urbana-Champaign

J. KilkennyGeneral Atomics, P.O. Box 85608, CA 92186–5608

Recent laser-driven shockless compression techniques

(Smith et al., 2006; Edwards et al., 2004; Lorenz et al.,

2006; Swift and Johnson, 2005) have demonstrated quasi-

isentropic compression (ICE) in an aluminum sample to peak

stresses over 1 Mbar (Smith et al., 2006) and at estimated

temperatures of 500 K. The isentrope generally lies to the

compressive side of the Hugoniot in pressure-volume (P-V)

space. Since the Al melt temperature is measured to increase

with compression, isentropic loading from room tempera-

ture to multi-Mbar stresses will stay below the melt line. The

smaller amount of internal energy imparted into the material

within the ICE platform allows for greater compression for

comparable Px on shock experiments. The technique has been

demonstrated with several drivers such as the magnetic pulse

loading of the Sandia Z-machine (Asay, 1999; Reisman et al.,

2001; Hall et al., 2001), pillow impactors in gas guns facilities

(Chhabildas and Barker, 1997; Asay, 1997) and the chemi-

cal energy of high explosives (Barnes et al., 1974; Tasker

et al., 2004). The time scales for these experimental plat-

forms range from 100’s of ns to several microseconds. In

the case of the gas-gun driven ICE platform the mm thick

impactor is constructed using a graded density layered-plate

approach that initially produces a series of small steps in

the loading, which subsequently transition to smooth com-

pression as a result of wave interactions in the layer plates

(Chhabildas and Barker, 1997).

In laser-driven ICE experiments the loading time is over

tens of nanoseconds. Laser-driven ramp compression exper-

iments have recently been used to measure material strength

(Lorenz et al., 2005) and the kinetics of polymorphic phase

transformations (Smith et al., 2006). Currently the high-

est pressure achieved on laser-driven ICE targets is 2 Mbar

(Lorenz et al., 2006). With current laser ICE target designs

the ramp compression rise time scales inversely with peak

Px, which for high Px results in hydrodynamic steepening

Springer


of the ramp compression wave into a shock over short dis-

tances. For a given target thickness this places a limit on the

maximum Px that can be applied to a sample while still en-

suring shockless compression. Within this paper we describe

new techniques which incorporate graded densities into the

standard laser ICE target design for increasing the rise time

within laser-driven ramp compression experiments. The de-

velopments of these techniques are important for realizing

the potential of shockless compression to a peak Px over

10 Mbar on the National Ignition Facility (NIF) (Remington

et al., 2005; Edwards et al., 2004).

The target design for the laser-driven shockless compres-

sion consists of a low-Z reservoir foil followed by a vacuum

gap and the target to be shocklessly compressed (as shown

in Fig. 1(a)). In the experiments described here one beam

of the Janus laser at 527 nm delivered a maximum of 380 J

in a 4 ns square pulse onto the front surface of the reser-

voir material. A kinoform phase plate (KPP), inserted into

the beamline to spatially smooth and shape the laser focal

spot, generated a ∼1 mm square planar (I/I ∼ 5%) region

at the focal plane which contained an estimated 80% of the

total drive energy, giving a maximum on-target intensity of

∼5 × 1012 W/cm2. The focused laser beam launches an ab-

latively driven shock through the reservoir. Reservoir materi-

als typically consist of a plastic foil (<300µm thick) doped

with a higher Z material (e.g. Br) in order to absorb x-rays

generated in the laser corona. After breakout from the rear

surface shock heating and momentum cause the reservoir

material to dissociate and unload across a vacuum gap. Tran-

sit across the vacuum gap causes the mass density gradients

along the target axis to relax as a function of distance from

the original reservoir/vacuum gap interface. The unloading

reservoir material monotonically loads up against the sample

and the imparted momentum launches a ramp longitudinal

stress wave through the material. The temporal profile of

the compression wave may be shaped to a limited extent by

varying the size of the vacuum gap, the density of the reser-

voir or by controlling the laser energy (Lorenz et al., 2006).

In the experiments described here we apply the ramp com-

pression wave to an Al/LiF target. As the ramp stress wave

reaches the back surface of the Al, the sample begins to ac-

celerate into a 500µm thick LiF window material. The time

history of the Al/LiF interface acceleration is recorded with

a line imaging velocity interferometer (VISAR) with two

channels set at different sensitivities (Celliers et al., 2004)

(Fig. 1(b)). The output of the VISAR is recorded by a fast

optical streak camera. Fringe movement is linearly propor-

tional to the velocity of the Al/LiF interface. This allows for

accurate measurement of the interface velocity (after taking

into account the refractive index of the window (Wise and

Chhabildas, 1986)) as a function of time. As Al and LiF are

well impedence matched the VISAR effectively records the

Al particle velocity history. This information allows for the

Fig. 1 (a) Schematic of Laser ICE target. (b) The output of theVISAR as recorded with a streak camera gives a temporal history of Alparticle veocity and a spatial record of the applied longitudinal stresswave. Fringe movement is linearly proportional to the Al/LiF interfacevelocity

compression source to be determined via a back-integration

technique where the time dependent particle velocity at the

rear surface is used as an input (Hayes, 2001; Hayes et al.,

2003, 2004). Due to increase of sound speed with increasing

Px the ramp compression wave will eventually steepen up into

a shock within the Al sample. This would result in a near in-

stantaneous jump in entropy and off-isentropic compression

(and possible target melting) would ensue. The maximum Al

thickness is therefore designed to be less than the calculated

shock-up thickness.

For all reported Laser ICE experiments the laser pulse

duration is designed to be less than the shock transit time

through the reservoir. Therefore by the time the shock reaches

the reservoir-vacuum interface the initial steady shock has

transformed into a blast wave which contains no information

about the temporal history of the laser drive. For a fixed

target design higher ramp pressures may be achieved in the

sample by increasing the input laser energy. This results in

increased peak stresses in the blast wave exiting the reservoir

and increased velocities of the material unloading across the

vacuum gap. Increased peak stresses are then launched into

the sample material but over increasingly shorter timescales.

Using this technique peak longitudinal stresses of 0.1 and

2 Mbar have ramp compression rise times of ∼30 ns (Smith

et al., 2006) and 5 ns (Lorenz et al., 2006), respectively.

The dynamics of the unloading solid reservoir produces a

shaped ramp compression profile which tends to shock up

Springer


at low Px. There is a need therefore to develop techniques

for shaping the time history of the compression wave which

will firstly increase the loading up time for a given peak Px

and secondly soften the gradient in the ramp wave such that

eventual steeping into a shock will tend to occur at the top of

the loading profile. The net effect in both cases, for a given

peak Px, is to increase the shock up distance and hence for a

fixed target thickness higher levels of shockless compression

may be obtained.

In the experiments reported here we use two different tech-

niques for applying a density gradient onto the gap side of

the reservoir material. The expectation is that the density gra-

dient slows the rate at which momentum is imparted into the

target and by customizing this gradient we can ultimately

customize the shape of the pressure profile. The first reser-

voir design used consists of a 120µm thick 1% Br/CH laser

ablator with 60µm of SU8 (CH6O4N5) photopolymer glued

onto the vacuum side. Recently developed phase contrast

lithographic techniques (Jeon et al., 2004) are used to pro-

duce 3D nanostructures which reduce in size with increasing

depth within an 60µm thick SU8 photopolymer (see Fig.

2(a) insert). The resultant density gradient is characterized

with transmission x-ray radiography, with a spatial resolu-

tion of 0.3µm, to go from full density to 19% density over

60µm (Fig. 2(a)). Using the target conditions described in

Fig. 1(a) data is taken for solid density and graded density

SU8. The time history of the ramp pressure profiles are shown

in Fig. 2(b). It is observed that for the same peak stress the

rise time of the ramp compression wave is ∼30% longer for

the graded density reservoir case. Importantly the slope of

the ramp compression wave associated with the graded den-

sity reservoir is noticeable reduced in the stress range over

which shocks typically develop within solid density reser-

voirs. Approximately 30% more laser energy was required

in the graded density target to reach in the same peak stresses

as for the solid density reservoir. The peak longitudinal stress

is related to the total amount of momentum imparted by the

impacting reservoir material into the Al sample. The higher

amount of laser energy is therefore needed for the graded

density material to match the momentum associated with the

solid density reservoir. Also shown are the calculated pres-

sure profiles from the LASNEX plasma physics code (Zim-

merman and Kruer, 1975) which as an approximation used a

linear density gradient from full density to 19% density over

the 60µm SU8 layer. The simulations show some disagree-

ment in the gradient of the pressure profiles but qualitatively

agree with the experimental observations of increased com-

pression rise time with the use of a density gradient. Fur-

ther improvements are expected in the compression rise time

through customization of the density gradient profile.

Another approach for incorporating an effective integrated

density gradient into a solid density reservoir is by direct

micro-machining three dimensional features into the gap

Fig. 2 (a) Novel phase contrast lithographic techniques (Jeon et al.,2004) produce 3D nanostructures which reduce in size with increasingdepth within an 60µm thick SU8 photopolymer (see insert). The resul-tant density gradient is characterized with x-ray radiography to go fromfull density to 19% density over 60µm. (b) Using the target conditionsdescribed in Fig. 1(a) the time history of the ramp pressure profile withand without a density gradient in the SU8 is shown. Also shown are thecalculated pressure profiles from the LASNEX plasma physics code(Zimmerman and Kruer, 1975). The pressure and time axes have beennormalized to make comparisons easier

side of the reservoir material. In these experiments, a saw-

tooth feature was diamond turned into one side of a 225µm

Polyimide [C22H10N2O5] foil. This feature was machined

in one dimension and was characterized with high resolution

imaging to have a period of 10µm and a depth of 8µm (insert

of Fig. 3). Longitudinal stress profiles measured from identi-

cal target and irradiation conditions for polyimide reservoirs

with and without the sawtooth feature are shown in Fig. 3.

The target without the micro-machined feature exhibits a

smooth ramp up to a peak pressure of ∼0.19 Mbar. The rise

time of the graded density reservoir has increased over the

solid density case. The target with the density gradient shows

a more structured rise with two mini-plateaus which is due to

local softening followed by steepening of the ramp gradients

when compared to the solid density case. Locally there is ex-

pected to be a lot of turbulence as the stress wave breaks out

of the sawtooth reservoir but experimentally this is observed

to be annealed out at the distance of the vacuum gap. Future

experiments will concentrate on varying the structure of the

Springer


Fig. 3 An average graded density is produced within a 225µm thickpolyimide foil by machining of a sawtooth feature into the vacuum gapside. A scanning electron microscope (SEM) image (insert) reveals thediamond turned feature to have a depth of 8µm and a period of 10µm.The resultant ramp compression profiles with and without the sawtoothfeature are shown

machined feature in order to extend the rise time and smooth

out the gradients within the ramp profile.

Increasing the rise time for a given peak stress in

laser driven quasi-isentropic compression experiments is

necessary to drive these technique into the multi-Mbar stress

regime where low temperature compressive material states

relevant to planetary interiors may be accessed. Using two

separate techniques to introduce a graded density in the gap

side of the reservoir it has been shown that the rise time of the

compression wave in laser-driven ICE is increased. Further

improvements are expected by customizing the shape of the

density gradient which will facilitate shockless compression

experiments on NIF to peak pressures over 10 Mbar.

Acknowledgements This work was performed under the auspices ofthe U.S. Dept. of Energy by the University of California, LawrenceLivermore National Laboratory under contract No. W-7405-Eng-48.

References

Asay, J.R.: Int. J. Impact Engng. 20, 27 (1997)

Asay, J.R.: Isentropic compression experiments on the Z Accelerator.In: Furnish, M.D., Chhabildas, L.C., Hixson, R.S. (eds) ShockCompression of Condensed Matter–1999, AIP Conf. Proceedings,NY

Barnes, J.F., Blewett, P.J., McQueen, R.G., Meyer, K.A., Venable, D.:J. Appl. Phys. 45, 727 (1974)

Celliers, P.M., Bradley, D.K., Collins, G.W., Hicks, D.G., Boehly, T.R.,Armstrong, W.J.: Rev. Sci. Instrum. 75, 4916 (2004)

Chhabildas, L.C., Barker, L.M.: Sandia Report SAND86–1888 (1986)Chijioke, A.D., Nellis, W.J., Silvera, I.F.: J. App. Phys. 98, 073526

(2005)Edwards, M.J., Lorenz, K.T., Remington, B.A., Pollaine, S., Colvin,

J., Braun, D., Lasinski, B. F., Reisman, D., McNaney, J.M.,Greenough, J.A., Wallace, R., Louis, H., Kalantar, D.: Phys. Rev.Lett. 92(7), 075002/1–4 (2004)

Hall, C.A., Asay, J.R., Knudson, M.D., Stygar, W.A., Spielman, R.,Pointon, T.D., Reisman, D.B., Toor, A., Cauble, R. C.: Rev. Sci.Instrum. 72, 3587 (2001)

Hayes, D.B., Backward Integration of Equations of Motion to Correctfor Free Surface Perturbations, Sandia Report SAND2001–1440(2001)

Hayes, D.B., Hall, C.A., Asay, J.R., Knudson, M.D.: J. Appl. Phys.94(4), (2003)

Hayes, D.B., Hall, C.A., Asay, J.R., Knudson, M.D.: J. Appl. Phys.96(10), (2004)

Jeon, S., Park, J.-U., Cirelli, R., Yang, S., Heitzman, C.E., Braun,P.V., Kenis, P.J.A., Rogers, J.A.: PNAS 101(34) 12428–12433(2004)

Lorenz, K.T., Edwards, M.J., Glendinning, S.G., Jankowski, A.F.,McNaney, J., Pollaine, S.M., Remington, B.A.: Phys. Plasmas 12,056309 (2005)

Lorenz, K.T., Edwards, M.J., Jankowski, A.F., Pollaine, S.M., Smith,R.F., Remington, B.A.: Submitted to J. High Energy Den. Phys.(2006)

Reisman, D.B., Toor, A., Cauble, R.C., Hall, C.A., Asay, J.R., Knudson,M.D., Furnish, M.D.: J. Appl. Phys. 89(3), 1625–1633 (2001)

Remington, B.A., Cavallo, R.M., Edwards, M.J., Ho, D.D.-M., Lorenz,K.T., Lorenzana, H.E., Lasinski, B.F., McNaney, J.M., Pollaine,S.M., Smith, R.F.: Astrophys. Space Sci. 298(1–2) (2005)

Smith, R.F., et al.: Submitted to Phys. Rev. Letts. (2006)Smith, R.F., et al.: Submitted to Phys. Rev. Letts (2006)Swift, D.C., Johnson, R.P.: Phys. Rev. E 71, 066401 (2005)Tasker, D.G., Goforth, J.H., Oona, H., Fowler, C.M., King, J.C., Herrera,

D., Torres, D.: Shock Compression of condensed Matter- AIP Con-ference Proceedings, 706, pp. 1239–1242 (2004)

Wise, J.L., Chhabildas, L.C.: pp. 441–454, Plenum, New York (1986)Zimmerman, G.B., Kruer, W.L.: Comments on Plasma Physics for Con-

trolled Thermonuclear Fusion 2, 51–61 (1975)

Springer


DOI 10.1007/s10509-006-9185-x


He Conductivity in Cool White Dwarf Atmospheres

S. Mazevet · M. Challacombe · P. M. Kowalski ·

D. Saumon


Abstract We investigate the conductivity of warm dense

helium under conditions found in the atmospheres of cool

white dwarfs using ab initio simulations. The calculations

performed consist of quantum molecular dynamics simula-

tions where the electronic wavefunction at each time step

is obtained using density functional theory, while the ion

trajectories are calculated using the resulting quantum me-

chanical forces. We use both conventional DFT (PW91) and

hybrid (PBE0) functionals to calculate the conductivities that

provide an estimate of the ionization fraction. While the cal-

culations are in good agreement with the measurements for

the equation of state, a significant discrepancy exists with the

recently measured conductivity.

Keywords Atomic processes . Dense matter . Equation of

state . Plasmas . Radiation mechanisms:general

Introduction

The possibility of using white dwarfs for dating stellar pop-

ulations has generated a renewed interest in modeling their

cooling rate (Fontaine et al., 2001). The coolest white dwarfs

are the most important for this task but are rather poorly un-

S. Mazevet () · M. ChallacombeTheoretical Division, Los Alamos National Laboratory, LosAlamos NM 87545, USA

D. SaumonTheoretical Division, Los Alamos National Laboratory, LosAlamos NM 87545, USA; Applied Physics Division, Los AlamosNational Laboratory, Los Alamos, NM 87545, USA

P. M. KowalskiApplied Physics Division, Los Alamos National Laboratory, LosAlamos, NM 87545, USA

derstood. Modeling their outer layers requires the calculation

of the physical properties of He and H-He mixtures at high

densities (few g/cm3) and low temperatures (T < 10000 K)

where conventional methods to calculate opacities usually

fail (Iglesias et al., 2002). For very cool white dwarfs, the

opacity of the outer layers controls the cooling rate of the

star. In pure He atmospheres, the opacity is dominated by

the free-free absorption process which depends directly on

the ionization fraction. At the conditions of interest here, the

ionization fraction is very low and uncertain as witnessed by

the broad range of predictions for the density at which helium

pressure ionizes (from 0.3 to 14 g/cm3 (Young et al., 1981;

Fontaine et al., 1977)). To eliminate this uncertainty, we per-

formed He conductivity calculations at conditions relevant

to white dwarfs atmospheres using ab initio simulations.

We performed quantum molecular dynamics (QMD)

simulations where the electrons receive a full quantum me-

chanical treatment using finite temperature density functional

theory (DFT) Mermin (1965), while the ions are propagated

in time classically using the resulting quantum mechanical

forces. We estimate the ionization fraction by performing

electrical conductivity calculations within linear response

theory (Mazevet et al., 2004). We find that the ab initio simu-

lations are in good agreement with the experimental data for

the equation of state. In contrast, the value of conductivity

recently measured at densities of about 1 g/cm3 as well as its

density dependence can not be reproduced by the method.

Equation of state

For the calculations presented here, we used the VASP ab ini-

tio simulation code (Kresse and Hafner, 1996). The simula-

tions were ran for typically 2 ps with time steps ranging from

2 fs for the lowest densities, to 0.5 fs at the highest, and using

Springer


54 and 128 atoms in the simulation cell. We used a Projected

Augmented Wave (PAW) pseudo-potential in the Perdew-

Wang 91 (PW91) (Perdew and Wang, 1992) parametrization

of the Generalized Gradient approximation (GGA) (Blochl,

1994). We paid particular attention to the convergence of the

various properties of interest as a function of the plane wave

energy cutoff. This issue was recently raised for the case of

hydrogen (Desjarlais, 2003). For helium, we find that it is

necessary to use a cutoff energy of 800 eV in order to ob-

tain an accurate description of the experimental Hugoniot. In

contrast to the hydrogen findings, we find that the real space

projection used in the QMD calculations does not introduce

significant error for the pressure or the internal energy.

In Fig. 1, we compare the QMD Hugoniot with the ex-

perimental data (Nellis, 1984) and with the results obtained

from a free energy minimization method based on a chemical

model we have developed. The Rankine-Hugoniot Equation

(10)

(U0 − U1) + 1

2(V0 − V1)(P0 + P1) = 0, (1)

describes the shock adiabat through a relation between the

initial and final volume, internal energy, and pressure, respec-

tively, (V0,U0, P0) and (V1,U1, P1). As described in (Kress

et al., 2001), the DFT-MD P and U for a fixed V1 are least-

squares fitted to a quadratic function in T to calculate the

Hugoniot point for a given V1 and a set of T1’s. P1 and T1

are determined by substituting these functions and solving

Equation (1).

While the high pressure experimental data are very sparse,

it is reassuring to find an overall good agreement with

the QMD calculations as these four experimental points

are extensively used to adjust a large majority of helium

EOS at high pressures. For the principal Hugoniot calcu-

lation, the initial condition used is P0 = 0GPa, and ρ0 =0.1245 g/cm3, in agreement with the experimental data. The

reference energy point for the QMD calculations was taken

as U0 = 0 eV/atom and was obtained by calculating the in-

ternal energy of an isolated atom in a box of 12 A. For the

principal Hugoniot, the agreement at the highest pressure

measured is rather good and within the experimental error

bars. For the reshocked points, we used the QMD Hugoniot

point obtained at the highest pressure as an initial condi-

tion,ρ1 = 0.4 g/cm3,U1 = 1.7 eV/atom and P1 = 14.7 GPa.

The highest QMD reshocked point calculated also agrees,

within the error bars, with the experimental measurement.

The QMD calculations indicate, however, a slightly softer

reshock Hugoniot, in agreement with the EOS obtained us-

ing the chemical model.

The chemical EOS is inspired from the He EOS of Saumon

et al. (Saumon et al., 1995) which is widely used in the as-

trophysics community. While both EOS are calibrated to the

0.3 0.4 0.5 0.6 0.7 0.8

ρ (g/cm3)

0

10

20

30

40

50

60

P (

GP

a)

Exp. principal Hugoniot

Exp. second shock

QMD principal Hugoniot

QMD second shock

Chem. principal Hugoniot

Chem. second shock

Fig. 1 Helium principal and second shock Hugoniot as a function ofdensity

first and second-shock gas gun data shown in Fig.1, they

differ in several significant ways. The chemical EOS pre-

sented here was developed specifically for the calculation of

opacities of He in the warm dense matter regime. Because

opacities are dominated by processes involving trace species

(especially free electrons), particular attention was devoted

to the numerical accuracy of the chemical equilibrium. The

trace species He+2 has been added as it affects the ionization

at low temperatures and high densities. On the other hand,

only the He-He interactions are included and the free elec-

trons are treated classically. The latter approximation is jus-

tified as long as the degree of ionization remains small, but

Fermi-Dirac statistics need to be introduced once pressure

ionization becomes significant (at densities above 2 g/cm3 in

this model). The agreement between the QMD and chemical

EOS is excellent below 1 g/cm3.

Electrical properties

We now turn to the principal motivation of the current study

which is the calculation of the electrical properties of he-

lium at high pressures and for conditions relevant to white

dwarf atmospheres. From the QMD trajectories produced,

we calculate the electrical properties on a set of configura-

tions using the Kubo-Greenwood formulation of the optical

conductivity where the real part is given as(Harrison, 1970;

Callaway, 1974).

σ (ω)R = 2πe2

3ω

1

∑

k

W (k)∑

n,m,α

(

f kn − f k

m

)

×∣

∣

⟨

ψkn

∣

∣∇α

∣

∣ψkm

⟩∣

∣

2δ(

Ekm − Ek

n − ω)

, (2)

In Equation (2), ω is the frequency, e is the elec-

tronic charge, ψkn and Ek

n are the electronic eigenstates and

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0111.0

density (g/cm3)

0

2

4

6

8

10

12

14

16

18

20G

ap (

eV)

T=0K fccMD T=0.52eVMD T=1.5eVMD T=3eV

Fig. 2 GGA gap variation as a function of density and for severaltemperatures

eigenvalues for the electronic band n at a given k-point in

the Brillouin zone, W (k) is the k-point weight in the Bril-

louin zone using the Monkhorst-Pack scheme and f kn is the

Fermi distribution function. ∇α is the velocity operator along

each direction (α = x, y, z) and the volume of the simu-

lation cell. Finally, the Kubo-Greenwood formulation is ap-

plied using the all-electron PAW potential, which does not

require the correction term related to the non-locality of the

pseudo-potential that would be needed if, for example, an

ultrasoft pseudo-potential was used. Calculations were per-

formed at the Ŵ point. Additional details on this type of cal-

culation can be found in (Mazevet, 2004).

At normal conditions, helium is an insulator with a rather

large band gap of around 20 eV. As can be seen from

Equation (2), the conductivity, and hence the ionization frac-

tion, increases as the population of the conduction band be-

comes large. This occurs when either the band gap diminishes

or if the temperature increases or both. Figure 2 shows the

variation of the band gap as a function of density and tempera-

ture as obtained using DFT. For a regular Fcc lattice at T = 0,

the method predicts that the He band gap closes at a density

of 13.5 g/cm3 in agreement with previous LMTO calcula-

tions [3]. At low temperatures, 0.5 eV (1 eV = 11604 K), the

band gap obtained from the simulations is almost constant in

the density range investigated here. At finite temperature, the

band gap shown in Fig. 2 is obtained by averaging the energy

difference of the orbitals on either side of the Fermi energy

along the trajectory. This is equivalent to the HOMO-LUMO

gap which is defined, at zero temperature, as the energy dif-

ference between the highest occupied orbitals and the lowest

unoccupied one. We show, in Fig. 2, the band gap obtained

at finite temperature as the latter is the quantity entering in

Equation (2). At higher temperatures, we further see that the

band gap diminishes as a function of both density and tem-

perature with a closure of the band gap predicted at around

0 1 2 3 4

density (g/cm3)

2.8

3

3.2

3.4

3.6

Hybri

d G

ap c

orr

ecti

on (

eV)

T=0.52eVT=3eV

Fig. 3 Variation of the hybrid band gap correction as a function ofdensity and temperature

10 g/cm3. At the two highest temperatures, the band gap re-

duces faster than when an Fcc structure is used for the ions

as a result of increased disorder.

At low densities, the gap tends to a value of about 15 eV.

For the isolated atom, calculation of the first excited state

gives a value of 15.8 eV using either the PW91 (Perdew and

Wang, 1992) or PBE (Perdew et al., 1996) functionals. The

experimental value for the position of He first excited state is

19.82 eV. The DFT method, which is a ground state approach,

is well known to systematicaly underestimate band gaps as

well as excited state energies. In effect, the excited state en-

ergies used in Equation (2), Ekm , and obtained from the diag-

onalization of the Kohn-Sham Hamiltonian are quasi excited

states only and neglect, for example, the electronhole interac-

tion which appears when an electron is promoted to a higher

level. For semi-conductors and insulators, the eigenenergies

need to be corrected using either time dependent DFT or

Greens function approaches such as GW to allow for an ac-

curate calculation of the electrical properties using Equation

(2) (Onida et al., 2002). These methods are rather expensive

computationally and can not, at present, be applied to dense

plasmas where the simulation cell contains from fifty to a

few hundreds atoms.

To estimate the band gap correction needed, we performed

DFT calculations using a hybrid functional, PBE0, with the

chemistry code MondoSCF, a program suite for 0(N) SCF

theory and ab initio MD (Challacombe et al., 1996). Hybrid

functionals combine the orbital-dependent Hartree-Fock ex-

change and an explicit density functional, in the present case

the GGA PBE functional (Perdew et al., 1996). Among other

properties, hybrid functionals allow for an improved calcu-

lation of the band gap and at an accuracy comparable to GW

calculations (Martin, 2004). Figure 3 shows the band gap

correction obtained using hybrid calculations that were per-

formed on a single configuration randomly selected from the

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trajectory of interest. We further note that the hybrid calcu-

lations are performed at zero temperature. The temperature

dependence indicated in Fig. 3 results from the use of ionic

configurations obtained from a trajectory corresponding to

the thermodynamics conditions indicated. We see in Fig. 3

that the correction is approximately constant in both density

and temperature at about 3 eV. We notice a small increase

of the correction of about 0.7 eV as the density increases be-

tween 0.1 and 4 g/cm3. While this variation could be physical,

we further note that it is within the uncertanties of the calcu-

lation performed here. The MondoSCF electronic structure

code used to perform the hybrid DFT calculations uses Gaus-

sian basis sets. In extended systems, care must be taken when

the density is varied as numerical linear dependencies arise

when the basis contains orbitals that are too diffuse. We find

that the situation is exacerbated for helium which requires

rather diffuse orbitals to converge the band gap.

To account for the result obtained using the hybrid func-

tional, we apply a uniform E = 3 eV correction to the

eigenenergies above the Fermi energy when calculating the

conductivities using Equation (2). We note that the occupa-

tion number f kn appearing in Equation (2) also needs to reflect

the correction applied to the eigenenergies. To first order, the

matrix elements appearing in Equation (2) do not need to be

corrected (Hedin, 1965). We show, in Fig. 4, the variation of

the DC conductivity along the 2000 K, 6000 K as well as the

17406 K (1.5 eV) isotherms. The DC conductivity is obtained

by taking the zero frequency limit of Equation (2). For all the

conditions studied here, the optical conductivity has a Drude

like form for photon energies below 10eV. This confirms that

the opacity is dominated by the free-free contribution which

in turn is directly related to the electrical conductivity and the

ionization fraction. For the three isotherms shown in Fig. 4,

we find that pressure ionization remains moderate (weak) at

these densities. While the DC conductivity varies by about 2

orders of magnitude as the density is increased from 0.5 to

6 g/cm3, the fluid stays mostly neutral along the two lowest

isotherms and up to the highest densities investigated. The

temperature dependence of the DC conductivity is, in con-

trast, more drastic with an increase over ten orders of mag-

nitude as the temperature is raised from 2000 K to 17406 K.

Figure 4 also shows that the DC conductivity is reduced by a

factor of about three when the hybrid band gap correction is

introduced. We further note that at a temperature of 2000K,

the calculated conductivity reaches the limit of accuracy of

the method and are given as an order of magnitude estimate

only. As such, we do not apply the hybrid correction at this

particular temperature.

Finally, Fig. 4, shows a direct comparison between the

calculated conductivities and recent experimental measure-

ments performed by (Fortov et al., 2003). The measurements

strongly suggest that helium starts becoming a conductor at

densities around 1 g/cm3. It is further important to notice that

0 1 2 3 4 5 6

density (g/cm3)

10-12

10-10

10-8

10-6

10-4

10-2

σD

C[1

06(Ω

m)-1

]

T=2000KT=6000KT=17406Kexp. [ ]

T=30240K

Fig. 4 Comparison between the calculated and measured. He conduc-tivities. Dotted lines: GGA calculations. (Solid lines) using the hybridcorrection

while the conductivities are measured for temperatures be-

tween 15000 and 30000 K. there is no correlation between

the temperature and the increase in conductivity shown in

Fig. 4. In effect, the rise in the experimental conductivity

around 1 g/cm3 was attributed to the effect of pressure ion-

ization and to a rapidly closing band gap (Fortov et al., 2003).

This result is in sharp contradiction with the QMD calcula-

tions which indicate conductivity values lower by about a

factor of three, a weak density dependence, and a wide band

gap persisting to densities well above 1 g/cm3. The hybrid

calculations which improve on the conductivity obtained us-

ing DFT properties further exacerbate the comparison with

the experimental data by leading to lower conductivity values

in this density range.

Overall, while the experimental conductivities are of the

same magnitude as those calculated, we can not reconcile the

ab initio results with the experimental measurements. While

the calculations performed here provide a solid benchmark

for physical models to describe He at conditions found in

white dwarf atmospheres, the significant disagreement with

the currently available conductivity data calls for additional

measurements to bring new light to this discrepancy. Mea-

surements of the He reflectivity under similar conditions are

currently performed at the OMEGA laser facility and pre-

liminary results are reported in these proceedings (Cellier,

Private communication).

Conclusion

Using Quantum Molecular Dynamics, we calculate the dy-

namical and electrical properties of helium in a regime

relevant to white dwarf atmospheres. We find very good

agreement with the experimental principal and second-shock

Hugoniot below 1 g/cm3. As the experimental measurements

at high pressures are sparse, the current study first provides

a useful benchmark for EOS modeling above 1 g/cm3. The

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conductivity calculated using QMD and linear response the-

ory is, however, in significant disagreement with recent ex-

perimental measurements.

Acknowledgements Work supported under the auspices of the U.S.Department of Energy at Los Alamos National Laboratory under Con-tract W-7405-ENG-36.

References

Blochl, P.E.: Phys. Rev. B 50, 17953 (1994); Kresse, G., Joubert, J.:Phys. Rev. B 59, 1758 (1999)

Cellier, P., Loubeyre, P.: Private CommunicationChallacombe, M., Tymczak, C.J., Nemeth, K., Weber, V., Gan, C.K.,

Schwegler, E., Henkelman, G., Niklasson, A.: Los Alamos Na-tional Laboratory, LA-CC-04-086

Callaway, J.: Quantum theory of the solid state. Academic Press NewYork (1974)

Desjarlais, M.P.: Phys. Rev. B 68, 064204 (2003)Fontaine, G., Brassard, P., Bergeron, P.: PASP 113, 409 (2001)Fontaine, G., Graboske, H.C., Van Horn, H.M.: ApJS 35 (1977)Fortov, V. et al.: JETP 97, 259 (2003)

Harrison, W.A.: Solid state theory. Mc Graw-Hill (1970)Hedin, L.: Phys. Rev. 139, A796 (1965)Iglesias, C.A., Rogers, F.J., Saumon, D.: Astrophysical Journal Letter,

569, L111 (2002)Kresse, G., Hafner, J.: Phys. Rev. B 47, RC558 (1993); Kresse, G.,

Furthmuller, J.: Comput. Mat. Sci. 6, 15–50 (1996); Kresse, G.,Furthmuller, J.: Phys. Rev. B 54, 111

Kress, J.D., Mazevet, S., Collins, L.A., Wood, W.W.: Phys. Rev. B 63,024203 (2001)

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temperature hydrodynamic phenomena (Academic Press, NewYork, 1966)

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DOI 10.1007/s10509-006-9224-7


The Structure of Jupiter, Saturn, and Exoplanets: Key Questionsfor High-Pressure Experiments

Jonathan J. Fortney


Abstract We give an overview of our current understanding

of the structure of gas giant planets, from Jupiter and Saturn to

extrasolar giant planets. We focus on addressing what high-

pressure laboratory experiments on hydrogen and helium can

help to elucidate about the structure of these planets.

Keywords Planetary systems . Jupiter . Saturn

1 Introduction

In order to understand the formation of giant planets, and

hence, the formation of planetary systems, we must be able

to determine the interior structure and composition of giant

planets. Jupiter and Saturn, our solar system’s gas giants,

combine to make up 92% of the planetary mass of our solar

system. Interestingly, knowledge of only a few key quan-

tities allows us to gain important insight into their interior

structure. The equation of state of hydrogen, together with

measurements of the mass and radius of Jupiter and Saturn

is sufficient to show that these planets are hydrogen-helium

rich objects with a composition similar to that of the Sun

(Demarcus, 1958). Furthermore, estimates of the transport

coefficients of dense metallic hydrogen and the observation

that Jupiter emits more infrared radiation than it absorbs from

the Sun (Low, 1966), is sufficient to show that gas giant planet

interiors are warm, fluid, and convective, not cold and solid

(Hubbard, 1968). It has also been clear for some time that the

composition of Jupiter and Saturn is not exactly like that of

the Sun – these planets are enhanced in “heavy elements”

J. J. FortneySpace Science and Astrobiology Division, NASA Ames ResearchCenter, MS 245-3, Moffett Field, CA 94035e-mail: [email protected]

(atoms more massive than helium), compared to the Sun

(Podolak and Cameron, 1974). An understanding of how

these planets attained these heavy elements, and their rel-

ative ratios, can give us a wealth of information on planetary

formation and the state of the solar nebula.Looking beyond Jupiter and Saturn, we now have 200 ex-

trasolar giant planets (EGPs) that have been found to orbit

other stars. A subclass of these planets are the “hot Jupiters”

that orbit their parent stars at around 0.05 AU. To date, ten

planets (with masses from 0.36 to 1.5 MJ) have been seen to

transit their parents stars. All of these objects are hot Jupiters,

with orbital periods of only a few days (see Charbonneau

et al., 2006). These transiting planets are important because

we can measure their masses and radii, thereby allowing

us access to information on their interior structure (Guillot,

2005). While our understanding of the interiors of these plan-

ets will never be as detailed as that for Jupiter and Saturn,

we will eventually have a very large sample of these tran-

siting objects at various masses, compositions, and orbital

distances, which will allow for an understanding of the mass-

radius relation for giant planets under a variety of conditions.By far the most important physical input into giant planet

structural models is the equation of state (EOS) of hydrogen.

The decade of pressure that is most important for under-

standing the interiors of giant planets is 1–10 Mbar (100–

1000 Gpa) (Saumon and Guillot, 2004). In the past decade

experiments have been able to probe into the lower end of

this pressure range (Weir et al., 1996; Collins et al., 1998;

Knudson et al., 2001; Boriskov et al., 2005). In this paper,

instead of focusing on equation of state physics we will focus

on key questions for understanding the structure and compo-

sition of giant planets. As we discuss giant planet interiors

we will investigate how high pressure laboratory experiments

have and will continue to allow us to better answer these

questions.

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2 Key questions

2.1 Are planetary atmospheric abundances representative

of the entire H/He envelope?

This question is directly related to whether hydrogen’s

molecular-to-metallic transition is continuous or first-order.

Whether or not hydrogen’s transition to a metal in the fluid

state is first order has always been an open issue. The impor-

tance of this question to giant planets cannot be overstated.

If the transition is first order (a so-called “plasma phase tran-

sition”, or PPT) then there will be an impenetrable barrier to

convection within the planet and there must also be several

discontinuities at this transition. One is a discontinuity in en-

tropy (Stevenson and Salpeter, 1977; Saumon and Chabrier,

1992). In the 1970s, W. B. Hubbard discussed that, for a

fully convective and adiabatic giant planet, a measurement

of the specific entropy in the convective atmosphere would

essentially allow us to understand the run of temperature vs.

pressure for the entire planet, as all regions would share this

specific entropy (see Hubbard, 1973). However, if a PPT ex-

ists, this will not be true (Chabrier et al., 1992).

Another discontinuity at the PPT would be in chemical

composition, due to the Gibbs phase rule. Modern struc-

tural models of Jupiter and Saturn aim to constrain the bulk

abundance and distribution of heavy elements in the inte-

riors of these planets. We would like to understand what

fraction of the heavy elements are distributed throughout the

H/He envelope, and what fraction are in a central core. See

Guillot (1999) and Saumon and Guillot (2004, 2005) for

recent computations of the interior structure of Jupiter and

Saturn. The main constraints on these models are planetary

mass, radius, rotation period, and gravity field. Additional

constraints would be most welcome. One potentially impor-

tant constraint would be atmospheric abundances derived

from entry probes or spectra. If it could be clearly shown

that the molecular-to-metallic transition is indeed continu-

ous, then mixing ratios of chemical species in the atmosphere

should be representative of the entire H/He envelope, as the

entire envelope should be well-mixed due to efficient con-

vection. This could constrain the amount of heavy elements

in the H/He envelope and allow for a much more precise de-

termination of the core mass and bulk heavy element abun-

dance. For Jupiter, the Galileo Entry Probe has measured the

abundances of the important species methane and ammo-

nia (Atreya et al., 2003). However, the abundance of water,

presumably the most abundant species after helium, is still

highly uncertain.

Perhaps the clearest indication of the physical state of hy-

drogen in the molecular-to-metallic transition region (∼l–5

Mbar) would be a measurement of the hydrogen’s conduc-

tivity. To date, Weir et al. (1996) and Nellis et al. (1999) have

measured the conductivity of hydrogen using a reverberation

shock technique up to 1.8 Mbar (180 Gpa). They found a

four order of magnitude increase in conductivity from 0.93

to 1.4 Mbar that plateaued between 1.4 and 1.8 Mbar at a

conductivity consistent with that of the minimum conductiv-

ity of a metal. These measurements appear to indicate that

hydrogen’s transition to a metallic state is indeed continuous

(at least at their measured temperature of 2600 K). However,

the measured conductivity is still over an order of magnitude

less than that expected for a fully ionized hydrogen plasma

(Hubbard et al., 2002), so these measurements cannot be

considered a definitive refutation of a PPT. Another open

question is how the presence of neutral atomic helium (10%

by number in a solar composition mixture) may affect this

transition.

2.2 Heavy Elements: How much and where are they?

The pressure-density relation of hydrogen is the single most

important input in giant planet structural models. All things

being equal, the more compressible hydrogen is, the smaller

a planet will be at a given mass and composition. This has a

direct bearing on model-derived constraints on the amount of

heavy elements within a planet’s interior. Saumon and Guillot

(2004) computed detailed interior models for Jupiter and

Saturn that were consistent with all available observational

constraints. They found that Jupiter models that used EOSs

consistent with the 6-fold limiting deuterium compression

data of Collins et al. (1998) lead to core sizes of 0–10 M⊕,

with total heavy element abundances (envelope plus core) of

10–25 M⊕. Models computed using EOSs consistent with

the harder 4.3-fold limiting compression of Knudson et al.

(2001, 2004) and Boriskov et al. (2005) led to smaller cores

sizes (0–3 M⊕) but larger heavy elements abundances (25–35

M⊕). Since other experiments have not been able to replicate

the soft Collins et al. (1998) data, and the data of Knudson

et al. and Boriskov et al. agree quite well while using different

experimental setups, these harder EOS data sets are currently

viewed by many as the most reliable. (For recent reviews, see

Nellis, 2005, 2006). Tests of the hydrogen or deuterium EOS

off of the single-shock Hugoniot, perhaps at pressures of up

to a few Mbar, but temperatures below 104 K, would be most

valuable. For helium, our second most important constituent,

new EOS data are sorely needed. No helium EOS data have

been published since Nellis et al. (1984), and this data set

only reached a maximum pressure of 560 kbar (56 Gpa).

In Fig. 1 we show schematic interior structures of

Jupiter and Saturn. We show pressures and temperatures

at three locations: the visible atmosphere (1 bar), near the

molecular-to-metallic transition of hydrogen (2 Mbar), and at

the top of the heavy element core of each planet. Atmospheric

elemental abundances, as determined by the Galileo Entry

Probe for Jupiter and by spectroscopy for Saturn, are shown

within a grey box (Atreya et al., 2003). These abundances

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Fig. 1 Schematic interiorstructure of Jupiter and Saturn.Pressures and temperature aremarked at 1 bar (100 kpa, visibleatmosphere), 2 Mbar (200 Gpa,near the molecular-to-metallictransition of hydrogen), and atthe top of the heavy elementcore. Temperatures areespecially uncertain, and aretaken from Guillot (2005).Approximate atmosphericabundances for “metals”(relative to solar) are shownwithin the grey box, in themolecular H2 region. Possiblecore masses, in M⊕ (labeled as“ME ”) are shown as well(Saumon and Guillot, 2004)

should at least be representative of the entire molecular H2

region. If a PPT does not exist, these abundances should

be representative of the entire H/He envelope. In both plan-

ets, the molecular H2 region is depleted in helium relative

to protosolar abundances (von Zahn et al., 1998; Conrath

and Gautier, 2000) indicating sedimentation of helium into

metallic H layers. Recent evolutionary models for Saturn in-

dicate this helium may rain through the metallic H region

and form a layer on top of the core (Fortney and Hubbard,

2003).

2.3 What are the temperatures in the deep interiors of

Jupiter and Saturn?

While the interior pressure-density relation sets the structure

of the planet, it is the pressure-temperature relation that de-

termines the thermal evolution. The temperature of the deep

interior sets the heat content of the planet. The higher the

temperatures in the planet’s interior, the longer it will take to

cool to a given luminosity. This has been investigated recently

by Saumon and Guillot (2004) for Jupiter. They computed

evolution models of Jupiter using several different hydrogen

EOSs that span the range of data obtained from LLNL laser

(Collins et al., 1998) and Sandia Z (Knudson et al., 2004)

data. These different EOSs predict temperatures than can

differ by as much as 30% at 1 Mbar. They find that Jupiter

models cool to the planet’s known luminosity in ∼3 to 5.5

Gyr using these various EOSs. This 2.5 Gyr uncertainty is

rather significant.

The atmospheres of Jupiter and Saturn are both depleted

in helium relative to protosolar composition (Atreya et al.,

2003). This observation, together with theoretical work indi-

cating that helium has a limited solubility in metallic hydro-

gen at planetary interior temperatures of ∼104 K (Stevenson,

1975; Hubbard and Dewitt, 1985; Pfaffenzeller et al., 1995),

indicates helium is phase separating from hydrogen and be-

ing lost to deeper layers in each planet. The evolution of

Saturn, and perhaps Jupiter, must be able to accommodate

the substantial additional energy source due to differentiation

within the planet. This “helium rain”, if present, has been

shown to be the dominant energy source for several-Gyr-

old giant planets (Stevenson and Salpeter, 1977; Fortney and

Hubbard, 2003, 2004). In order to understand to what degree

helium phase separation has progressed in Jupiter and Sat-

urn, and how far down into the planet the helium has rained

to, we must understand the deep interior temperature of these

planets.

To date, temperature measurements have been published

by Holmes et al. (1995) and Collins et al. (2001). These exper-

iments were performed using gas gun and laser apparatuses,

respectively. Both found temperatures generally lower than

most calculated hydrogen EOSs, which if indeed correct,

would lead to shorter cooling timescales for giant planets.

This faster cooling would more easily accommodate the ad-

ditional energy source due to helium rain. Additional data,

especially at the high pressures and “cool” temperatures of

planetary interest (off of the single-shock Hugoniot) would

be of great interest.

2.4 Do all giant planets possess heavy element

enrichments?

If we are to understand giant planets as a class of astronomical

objects, we must understand how similar other giant planets

are to Jupiter and Saturn. The mass-radius relation of exo-

planets allows us, in principle, to understand if these planets

have heavy element enrichments that are similar to Jupiter

and Saturn. Figure 2 shows the mass and radius of Jupiter,

Saturn, and the 10 known transiting hot Jupiters. It is interest-

ing to note while Jupiter and Saturn differ in mass by a factor

of 3.3, their radii only differ by 18%. However, while the hot

Jupiters differ in mass by a similar factor (of 4) they differ in

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Fig. 2 Radius and mass of Jupiter, Saturn, and the 10 known transitinghot Jupiters, as of April 2006. See Charbonneau et al. (2006) and refer-ences therein. One RJ is 71492 km, Jupiter’s equatorial radius at P = lbar. Curves of constant density (in g cm−3) are over-plotted with a dottedline. Data are taken from Charbonneau et al. (2006) and McCulloughet al. (2006)

radius by a factor of 2. This large spread is presumably due

to large difference in the interior heavy element abundances

of these planets (Fortney et al., 2006; Guillot, 2005; Guillot

et al., 2006). Giant planets under intense stellar irradiation

cool and contract more slowly that those far from their parent

stars, so radii larger than 1 RJ are expected (Guillot et al.,

1996).

Planet HD 149026b, with a radius of only 0.73 RJ, must

be on the order of 2/3 heavy elements by mass to explain

its small radius (Sato et al., 2005; Fortney et al., 2006).

Its parent star has a metallicity 2.3 × that of the Sun, so

this may point to a connection between stellar and planetary

abundances. However, the determination of planetary core

sizes appears to be complicated by the need for an addi-

tional interior energy source (yet to be definitely identified)

for planet HD 209458b, and perhaps also OGLE-Tr-l0b (Bo-

denheimer et al., 2001; Guillot and Showman, 2002; Winn

and Holman, 2005). These planets have radii that are too large

to be explained by conventional cooling/contraction models

(Chabrier et al., 2004; Laughlin et al., 2005). Therefore, the

spread in Fig. 2 is likely due to a combination of differing

magnitudes of this interior energy source and heavy element

abundances, which adds significant complications to this pic-

ture. Guillot et al. (2006) have recently proposed a correlation

between the heavy element abundances in transiting planets

and the metallicity of the planets’ host stars, assuming an ad-

ditional energy source that scales linearly with the incident

stellar flux absorbed by the planets.

In Fig. 3 we show a first look at comparative interior

structure of the core-dominated planet HD 149026b, Saturn,

and Neptune. The figure shows the current interior density

distribution as a function of normalized radius for two HD

149026b models from Fortney et al. (2006) compared to inte-

Fig. 3 Interior density as a function of normalized radius for two pos-sible models for HD 149026b compared with Neptune and Saturn. Allplanet models have been normalized to the radius at which P = l bar. TheNeptune profile is from Podolak et al. (1995) and the Saturn profile isfrom Guillot (1999). The Saturn and Neptune models have a two-layercore of ice overlying rock. The two profiles of HD 149026b assumea metallicity of 3 times solar in the H/He envelope and a core madeentirely of either ice or rock

rior models of Saturn (Guillot, 1999) and Neptune (Podolak

et al., 1995). The Saturn and Neptune models both have two-

layer cores of rock overlain by ice. The ratio of ice to rock in

these cores is based more on cosmogonical arguments than

on physical evidence. The interior structure of HD 149026b

may be a hybrid of the ice giants and gas giants. Uranus and

Neptune are ∼90% heavy elements, while Saturn is ∼25%

and Jupiter 10% (Saumon and Guillot, 2004). Although

HD 149026b is more massive than Saturn, it has a bulk mass

fraction of heavy elements (50–80%) more similar to that of

the solar system’s ice giants. Clearly, the field of exoplanets

is allowing us to study and understand planets unlike any we

have in our solar system.

3 The future

The path towards a better understanding of the structure of

giant planets seems clear. Along with additional laboratory

work at high irradiance laser, Z-pinch, and other facilities,

space missions will also allow us better insight into giant

planets. For Saturn, NASA’s Cassini spacecraft will allow

us to place better constraints on Saturn’s gravity field. For

Jupiter, NASA’s Juno mission, still scheduled to launch in

2010, will map the planet’s gravity field at high precision

and to high order, and will derive the abundances of water

vapor and ammonia in the planet’s atmosphere below

their respective cloud layers. For extrasolar planets, the

European COROT and NASA Kepler missions will allow

us to detect potentially hundreds of additional transiting

planets. The scientific gain from all of these missions is

directly dependent on our understanding of hydrogen and

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helium at high pressure. Experiments in the future should

focus on the following issues:

Is the fluid molecular-to-metallic transition of hydrogen a

continuous transition? Does the presence of a 10% mixture

of helium effect this transition? What are the EOSs of hydrogen and helium along the in-

ternal adiabats of Jupiter and Saturn? What is the temperature of hydrogen along the relatively

“cool” adiabats of giant planets?

Acknowledgements JJF acknowledges the support of an NASA Post-doctoral Program (NPP) fellowship and a travel grant from the HEDLAconference organizers.

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Charbonneau, D., Brown, T.M., Burrows, A., Laughlin, G.: astro-ph/0603376, Protostars and Planets V, in press (2006)

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Collins, G.W., da Silva, L.B., Celliers, P., Gold, D.M., Foord, M.E.,Wallace, R.J., Ng, A., Weber, S.V., Budil, K.S., Cauble, R.: Science21, 1178 (1998)

Conrath, B.J., Gautier, D.: Icarus 144, 124 (2000)Demarcus, W.C.: AJ 63, 2 (1958)Fortney, J.J., Hubbard, W.B.: Icarus 164, 228 (2003)Fortney, J.J., Hubbard, W.B.: ApJ 608, 1039 (2004)Fortney, J.J., Saumon, D., Marley, M.S., Lodders, K., Freedman, R.S.:

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DOI 10.1007/s10509-007-9385-z


Modeling Planetary Interiors in Laser Based Experiments UsingShockless Compression

J. Hawreliak · J. Colvin · J. Eggert · D. H. Kalantar ·

H. E. Lorenzana · S. Pollaine · K. Rosolankova ·

B. A. Remington · J. Stolken · J. S. Wark

Received: 8 May 2006 / Accepted: 22 January 2007C© Springer Science + Business Media B.V. 2007

Abstract X-ray diffraction is a widely used technique for

measuring the crystal structure of a compressed material.

Recently, short pulse x-ray sources have been used to mea-

sure the crystal structure in-situ while a sample is being dy-

namically loaded. To reach the ultra high pressures that are

unattainable in static experiments at temperatures lower than

using shock techniques, shockless quasi-isentropic compres-

sion is required. Shockless compression has been demon-

strated as a successful means of accessing high pressures.

The National Ignition Facility (NIF), which will begin doing

high pressure material science in 2010, it should be possible

to reach over 2 TPa quasi-isentropically. This paper outlines

how x-ray diffraction could be used to study the crystal struc-

ture in laser driven, shocklessly compressed targets the same

way it has been used in shock compressed samples. A sim-

ulation of a shockless laser driven iron is used to generate

simulated diffraction signals, and recent experimental results

are presented.

Keywords Laser shock . Shockless compression .

Diffraction . Dynamic diffraction . Simulation

1 Introduction

Models of planetary interiors rely on accurate laboratory

measurements of equations of state (Guillot, 2005). High

pressure measurements are generally limited by the experi-

mental apparatus. Diamond anvil experiments can be used to

study statically compressed samples up to only a few hundred

J. Hawreliak () · J. Colvin · J. Eggert · D. H. Kalantar ·H. E. Lorenzana · S. Pollaine · K. Rosolankova ·B. A. Remington · J. Stolken · J. S. WarkLawrence Livermore National Laboratory, Livermore, CA, USAe-mail: [email protected]

GPa (Clark et al., 1995). Dynamic techniques are needed to

reach pressures exceeding 4 TPa that exist at the cores of

the gas giants (Guillot, 1999). These pressure should be ob-

tainable in the next generation of high energy density exper-

imental facilities such as the National Ignition Facility (NIF)

(Remington et al., 2005).

This paper discusses how the crystal structure can be mea-

sured in dynamically compressed samples using in-situ x-ray

diffraction (Kalantar et al., 2003a,b; Loveridge et al., 2001).

First, a brief discussion of shock and shockless compression

is presented as a means of obtaining high pressures. Then we

describe how the x-ray diffraction techniques would differ

between the shock and shockless drive. Finally a compar-

ison is made between data from a recent experiment and

simulation.

2 Dynamic compression techniques

Shock compression has been used over the last half century to

provide a wealth of information about materials at high pres-

sures (Meyers, 1994). Traditionally experiments have been

done using a gas gun to launch a projectile at a stationary tar-

get, or using high explosives (HE) to apply a sharp pressure

pulse (Asay, 1997). More recently, shock wave research has

extended to large laser and z-pinch facilities. Even though

the technology to create a shock is different in each method,

the sample will always transform to a single temperature

and pressure described by the shock Hugoniot, the curve in

phase space that represents the shock response of a mate-

rial. To study samples under the ultra high pressure condi-

tions relevant to planetary interiors in the laboratory, dynamic

compression techniques which do not produce a shock are

required. These techniques are labeled as “quasi-isentropic”

because they produce high pressures off the shock Hugoniot

Springer


in materials while maintaining low temperatures using a se-

quence of small shocks.

Quasi-isentropic compression has been performed using

pillow impactors on gas guns (Asay, 1997), temporal pulse

shaping on z-pinch facilities (Hall, 2000), HE expanding

across a vacuum gap (Barnes et al., 1974) and a two compo-

nent target on laser facilities (Edwards et al., 2004; Lorenz

et al., 2005). In a similar fashion to the HE case, the laser

target uses a plastic reservoir which is shocked with a high-

powered laser and the plasma blow off from the ensuing blast

wave crosses a vacuum gap and pressurizes the sample. The

parameters of the target can be tailored to suit a particular

compression regime. Unlike shock compression, the sample

does not end up on a single point on the shock Hugoniot, but

samples a continuum of states along the compression ‘quasi-

isentrope’. Where measurements made at different times look

at different pressure and temperatures.

An example of a laser driven shockless compression target

is shown in Fig. 1, and simulated pressure, temperature, and

density profiles are shown in Fig. 2 for an on target drive laser

intensity of 1 × 1013 W/cm2. This shows that over 5 to 10 ns

period the pressure increases in the sample until it reaches a

peak pressure, which is maintained for approximately 10 ns.

The beryllium layer serves as a heat shield to prevent the

sample from melting on initial contact with the plasma, but

ultimately a thermal wave causes the sample to heat up, set-

ting up a large temperature gradient in the sample. The long

time-scales and spatial gradients in the isentropically com-

pressed case change the way the sample can be diagnosed

using in-situ x-ray diffraction.

3 X-ray diffraction from dynamically compressed

materials

With the development of x-ray diffraction of dynamic sys-

tems, (Kalantar et al., 2003; Loveridge et al., 2001; Kalantar

8 µm Iron Crystal4 µm Berylium

Heat Shield250 µm Plastic Backing

250 µm Gap

250 µm Plastic

Reservoir

Laser Drive Beams

X-ray

Source

Diffracted X-rays

To Film Detector

Fig. 1 A schematic diagram of a shockless compression target. Thedrive beams are incident on the plastic reservoir, which causes the blowoff from the back of the plastic to cross the gap and pile up on the ironsample. A thin layer of beryllium is used to prevent thermal contactbetween the iron sample and the plasma

0

0.2

0.4

0 20 40 60

PressureCompressionTemperature

300

350

400)0

01

/a

PG(

erusser

P-

1(n

oisserp

mo

Cρ

o/ρ

)

Time (ns)

)K(

erut

are

pm e

T

X-ray preheat

Reservoir driven

Thermal conduction wave

Thermal expansion

Fig. 2 A plot of the simulated temperature, pressure and compressionat the center of the 8 µm iron foil from a 1D LASNEX simulation.The simulation shows slight preheating of the sample from the x-raysgenerated at the drive surface of the reservoir, causing a slight expansionbefore the pressure wave arrives at the center of the sample. The threelines indicate the time at which the simulated diffraction images areshown in Fig. 5

et al., 1999; Gupta et al., 1999) it is possible to make a deter-

mination of crystal structure in-situ during the compression

process. These in-situ measurements are critical to under-

standing how materials react under presssure as the crys-

tallographic structure is fundamental in determining mate-

rial properties. A change in crystal structure, known as a

phase change, is usually accompanied by a dramatic change

in bulk properties, i.e. volume collapse, change in electrical

conductivity, change in magnetism, etc. We will discuss mod-

ifications to the existing wide angle x-ray diffraction tech-

niques used on shock compression experiments (Kalantar

et al., 2003) to make crystallographic measurements on

quasi-isentropically compressed samples.

Figure 3 shows how the x-ray probes will differ between

the two techniques. The gray shaded region denotes the time

and the finite penetration depth of the x-ray probes on the

crystal. In the shock case, shown in Fig. 3a, there is a dis-

continuity at the shock front between the uncompressed and

shock compressed material. For x-ray diffraction this discon-

tinuity means we can record a diffraction signal from both the

unshocked sample and the shock compressed sample with a

single pulse of x-rays which is time synchronous with the

shock front reaching the material. The measurement of the

unshocked signal is critical to making quantitative measure-

ments because it provides a method for calibrating the exper-

imental geometry of the detector and the initial crystal state.

In the isentropically compressed case, the compression

happens over a longer timescale, as illustrated in Fig. 3b.

This means that it will require two separate x-ray pulses.

One x-ray pulse is required to record a zero pressure signal.

A second x-ray pulse is required to interrogate the sample at

pressure. Unlike the shock case, where the shock front pro-

vides a discontinuity between compressed material and the

zero pressure material , the isentropically compressed targets

will have a continuous compression with large gradients into

the sample.

Springer


Fig. 3 Schematic diagram showing a time-distance plot of representa-tive lattice planes in iron that is compressed by (a) shock loading and (b)isentropic compression. The shaded region represents the x-ray probetime and depth in the crystal. Both the uncompressed and compressedlattice are probed by a single x-ray pulse in the shocked case, where

as two separate pulses are required to probe in the isentropically com-pressed case. d0 is the initial plane spacing, ds is the plane spacing inthe shocked sample and ∇d is the direction of increasing plane spacingin the ramp compressed sample

4 Diffraction from iron [100] crystals

The properties of iron under high pressure and temperature

have always been of interest owing to iron being a major

component of the earth’s inner core and its technological rel-

evance to modern society. The α − ǫ phase transition at 13

GPa is of particular importance to the shock physics com-

munity as it was first proposed by Bancroft et al. (1956)

to explain the discrepancy between Walsh’s measurements

(Walsh, 1954) and Bridgman’s original work up to 10 GPa

(Bridgman, 1964). The crystallographic nature of the ǫ phase

was not determined until later with static compression mea-

surements using x-ray diffraction by Jamison and Lawson

(1962). It was not until recent in-situ x-ray diffraction ex-

periments of laser shocked samples that this was confirmed

(Kalantar et al., 2005). With the recent success of in-situ x-

ray diffraction in determining the crystal structure of shocked

iron, we will use iron as the example to discuss the x-ray

diffraction technique for isentropically compressed samples.

We employ a simulation to show how the gradients in the

sample will affect the width of the diffraction lines that will

be recorded on the film.

Figure 4 shows how the compression of the iron varies in

the sample at different times in the simulation. During the

pressure ramp, 30 ns after the drive beams are incident on

the reservoir surface, the density gradient is large going from

ambient density to a compression of nearly 15%. Since the

x-rays can penetrate the entire sample, the diffracted peak

will contain information from the full range of compressions

at this time, and it will be significantly broadened. Close to the

peak of the pressure pulse, 33.5 ns, maximum compression is

reached at the front surface, but there is still a large gradient

in compression which will broaden the diffracted signal and

make it difficult to extract detailed information.

Figure 5 illustrates line broadening effects due to the strain

gradient in the iron sample. Figure 5a shows a line out from

an experimental image from iron sample shock compressed

with a laser using an iron K-shell backlighter (Kalantar et al.,

0

0.1

0.2

0µm 4µm 8µm

30.0ns33.5ns41.5 ns

-1(

noisser

pm

oC

ρο/ρ

)

Position from iron surface

Fig. 4 This plot shows the compression at different times through thesample. Both 30 and 33.5 ns show steep gradients in the compression.At 41.5 ns the compression is more uniform across the sample

2005; Hawreliak et al., 2006). As stated before in the shock

case we can probe several states of the material with a single

pulse due to the penetration depth of the x-rays. In this case

we get signal from pristine material (deepest in the sample),

a compressed BCC lattice (intermediate phase) and the trans-

formed HCP phase (closest to the surface). For comparison,

Fig. 5b and c show calculated signals at 3 times for iron and

copper backlighters respectively. Two different examples are

show for each time. The solid line includes some instrument

broadening of the diffraction peaks. Figure 5b shows that the

diffraction signals recorded at 30 and 33.5 ns will be broaden

significantly by the gradient in compression of the sample.

The diffraction signal at 41.5 ns forms a tighter peak than

the diffraction peak at the other two times, which can be

attributed to the smaller gradient in the compression.

The broadening in the diffraction signal due to the strain

gradient can be reduced by the choice of backlighter. When

using K -shell laser generated x-rays from an iron backlighter

at 6 keV the 1/e depth is about 7 µm for the (002) plane,

generating diffraction signal from nearly the entire sample.

Alternatively the K -shell radiation from a laser based copper

backlighter at 8 keV the depth is reduced to less then 2 µm.

Springer


00 200 400 600

Pixel

)stin

U.

brA(

gniretta c

S

a)

b)c)

2 2.2 2.4

30 ns - Cu BL w Broadening33.5 ns - Cu BL w Broadening41.5 ns - Cu BL w Broadening

Scattering Vector [001]

Unshocked Material

(002) BCC

Compressed

New Phase

HCP

2 2.2 2.4

30 ns - Fe BL w Broadening33.5 ns - Fe BL w Broadening41.5 ns - Fe BL w Broadening

Scattering Vector [001]

Unshocked Material

recorded using

early x-ray pulse

Compressed Material

recorded using

late x-ray pulse

Unshocked Material

recorded using

early x-ray pulse

Compressed Material

recorded using

late x-ray pulse

Fig. 5 A comparison of line outs from (a) raw experimental x-raydiffraction data from a shocked, single crystal sample and simulateddiffraction using (b) an iron backlighter and (c) a copper backlighter.

The shape of the simulated peaks in (b) and (c) reflect the range ofcompressions in the sample. The solid curves includes the broadeningobserved due to the α − ǫ phase transition

0% 2% 4%

Compression (1-a /a0 )100%

lan

giS

0

2

4

6

8

10

12

20 40 60 80 100 120

Time (ns)

)%(

noisser

pm

oC

Probe time

Diffraction from uncompressed

material

Compression

Fig. 6 Experimental data andsimulated temporal profile of thecompression in the sample bothat a laser drive intensity of1 × 1013 W/cm2. The samplewas Vanadium coated with a5 µm plastic heat shield. Theexperimental data was generatedusing only the late time pulse, sothe distinct peaks is consistentwith a shocking of the sample

Simulations of the diffraction peaks using a copper back-

lighter are shown in Fig. 5c. The copper backlighter, with

its shallower penetration depth, generates sharper peaks than

an iron backlighter but the scattering signal levels will be

reduced due to the higher absorption in the iron at the copper

k-shell emission line.

Figure 6 shows recent experimental data driven at a laser

drive intensity of 1 × 1013 W/cm2. The timing of the late

x-ray pulse was set to probe the lattice as the compression

wave reached the sample. In Fig. 6 the data is obtained by a

single late x-ray pulse. The two distinct peaks is consistent

with the initial small shock that is generated by the ramped

compression profile steepening up in the plastic heat shield

layer. While a thick heat shield over the sample is desired

to keep the plasma heat from the sample, too thick a sample

will lead to an initial shock or potentially shock compression

of the sample.

5 Conclusion

Quasi-isentropic compression gives us the ability to explore

material states off the shock Hugoniot. This approach also

provides us with one key and truly unique capability, namely

access to compressions in solids that approach the terapas-

cal regime. Typically, single shock techniques melt solids at

pressures exceeding 200–400 gigapascals, and so properties

of solids beyond these pressures remain completely unex-

plored. Investigations in the terapascal regime enabled by

quasi-isentropic compression certainly will provide an un-

precedented window into the extreme states found in plane-

tary interiors.

Time-resolved X-ray diffraction methodologies will play

an unique and pivotal role in investigating the crystal lattice

and other fundamental properties of the solid during these ex-

treme conditions. For shocked solids, our group and others

throughout the community have already demonstrated that

x-ray diffraction can successfully probe atomistic phenom-

ena such as phase transformations with nanosecond temporal

resolution. However, diffraction measurements in a quasi-

isentropically compressed solid have never been demon-

strated and pose important challenges; in this work, we have

discussed a novel approach to enabling such investigations.

We have described key experimental aspects of a technique to

perform time resolved diffraction measurements, including

a new geometrical target design as well as relevant back-

lighter and timing issues. Additionally, we have explored

the need for a detailed hydrodynamic understanding of the

compression process, either through bulk measurements or

detailed simulations, that will permit quantitative analyses of

the quasi-isentropically compressed lattice. Finally, we have

presented preliminary experimental results showing in situ,

real-time lattice measurements during quasi-isentropic com-

pression and modelling to quantify the contributions from

Springer


spatial and temporal gradients. Development of this tech-

nique will continue on the Omega laser at the University

of Rochester, the Jupiter Laser Facility at LLNL, and the

Vulcan laser at the Rutherford Appleton Laboratory in the

UK.

This work was conducted under the auspices of the U.S.

DOE by the UC LLNL and LANL under Contract No.

W-7405-Eng-48. The quasi-isentropic compression experi-

ments were conducted at the University of Rochester Labo-

ratory for Laser Energetics. Additional support was provided

by LDRD program Project No. 06-SI-004 at LLNL and by

the DOE under Grants No. DEFG0398DP00212 and No.

DEFG0300SF2202, by the U.K. EPSRC under Grant No.

GR/R25699/01.

References

Asay, J.R.: The use of shock-structure methods for evaluating high-pressure material properties. Int. J. Impact Eng. 20(1–5), 27 (1997)

Bancroft, D., Peterson, E.L., Minshall, S.: Polymorphism of iron at highpressure. J. Appl. Phys. 27(3), 291–298 (1956)

Barnes, J.F., Blewett, P.J., Mcqueen, R.G., Meyer, K.A., Venable, D.:Taylor instability in solids. J. Appl. Phys. 45(2), 727–732 (1974)

Bridgman, P.W.: Collected Experimental Papers. Harvard UniversityPress, Cambridge (1964). [by] P.W. Bridgman. illus. 25 cm. Somepapers in German. “In most cases the original reprints have beenreproduced photographically, although in a few instances it provednecessary to reset a paper in type.”

Clark, S.J., Ackland, G.J., Crain, J.: Theoretical stability limit of di-amond at ultrahigh pressure. Phys. Rev. B 52(21), 15035–15038(1995)

Edwards, J., Lorenz, K.T., Remington, B.A., Pollaine, S., Colvin, J.,Braun, D., Lasinski, B.F., Reisman, D., McNaney, J.M., Gree-nough, J.A., Wallace, R., Louis, H., Kalantar, D.: Laser-drivenplasma loader for shockless compression and acceleration of sam-ples in the solid state. Phys. Rev. Lett. 92(7), 075002 (2004)

Guillot, T.: Interiors of giant planets inside end outside the solar system.Science 286(5437), 72–77 (1999)

Guillot, T.: The interiors of giant planets: models and outstanding ques-tions. Annu. Rev. Earth Planet. Sci. 33(1), 493–530 (2005)

Gupta, Y.M., Zimmerman, K.A., Rigg, P.A., Zaretsky, E.B., Savage,D.M., Bellamy, P.M.: Experimental developments to obtain real-time x-ray diffraction measurements in plate impact experiments.Rev. Sci. Instrum. 70(10), 4008–4014 (1999)

Hall, C.A.: Isentropic compression experiments on the sandia z accel-erator. Phys. Plasmas 7(5), 2069–2075 (2000)

Hawreliak, J., Colvin, J.D., Eggert, J.H., Kalantar, D.H., Lorenzana,H.E., Stolken, J.S., Davies, H.M., Germann, T.C., Holian, B.L.,Kadau, K., Lomdahl, P.S., Higginbotham, A., Rosolankova, K.,Sheppard, J., Wark, J.S.: Analysis of the x-ray diffraction signal forthe alpha-epsilon transition in shock-compressed iron: simulationand experiment. Phys. Rev. B 74(18), 184107 (2006)

Jamieson, J.C., Lawson, A.W.: X-ray diffraction studies in the 100 kilo-bar pressure range. J. Appl. Phys. 33(3), 776–780 (1962)

Kalantar, D.H., Belak, J., Bringa, E., Budil, K., Caturla, M., Colvin,J., Kumar, M., Lorenz, K.T., Rudd, R.E., Stolken, J., Allen, A.M.,Rosolankova, K., Wark, J.S., Meyers, M.A., Schneider, M.: High-pressure, high-strain-rate lattice response of shocked materials.Phys. Plasmas 10(5), 1569–1576 (2003b)

Kalantar, D.H., Belak, J.F., Collins, G.W., Colvin, J.D., Davies, H.M.,Eggert, J.H., Germann, T.C., Hawreliak, J., Holian, B.L., Kadau,K., Lomdahl, P.S., Lorenzana, H.E., Meyers, M.A., Rosolankova,K., Schneider, M.S., Sheppard, J., Stolken, J.S., Wark, J.S.: Directobservation of the alpha-epsilon transition in shock-compressediron via nanosecond x-ray diffraction. Phys. Rev. Lett. 95(7),075502 (2005)

Kalantar, D.H., Bringa, E., Caturla, M., Colvin, J., Lorenz, K.T., Kumar,M., Stolken, J., Allen, A.M., Rosolankova, K., Wark, J.S., Meyers,M.A., Schneider, M., Boehly, T.R.: Multiple film plane diagnosticfor shocked lattice measurements (invited). Rev. Sci. Instrum. 74,1929–1934 (2003a)

Kalantar, D.H., Chandler, E.A., Colvin, J.D., Lee, R., Remington, B.A.,Weber, S.V., Wiley, L.G., Hauer, A., Wark, J.S., Loveridge, A.,Failor, B.H., Meyers, M.A., Ravichandran, G.: Transient x-raydiffraction used to diagnose shock compressed Si crystals on thenova laser. Rev. Sci. Instrum. 70, 629–632 (1999)

Lorenz, K.T., Edwards, M.J., Glendinning, S.G., Jankowski, A.F., Mc-Naney, J., Pollaine, S.M., Remington, B.A.: Accessing ultrahigh-pressure, quasi-isentropic states of matter. Phys. Plasmas 12(5),056309 (2005)

Loveridge-Smith, A., Allen, A., Belak, J., Boehly, T., Hauer, A., Holian,B., Kalantar, D., Kyrala, G., Lee, R.W., Lomdahl, P., Meyers, M.A.,Paisley, D., Pollaine, S., Remington, B., Swift, D.C., Weber, S.,Wark, J.S.: Anomalous elastic response of silicon to uniaxial shockcompression on nanosecond time scales. Phys. Rev. Lett. 86(11),2349–2352 (2001)

Meyers, M.A.: Dynamic Behavior of Materials. Wiley, New York (1994)Remington, B.A., Cavallo, R.M., Edwards, M.J., Ho, D.D.M., Lasinski,

B.F., Lorenz, K.T., Lorenzana, H.E., McNaney, J.M., Pollaine,S.M., Smith, R.F.: Accessing high pressure states relevant to coreconditions in the giant planets. Astrophys. Space Sci. 298(1–2),235–240 (2005)

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Springer


DOI 10.1007/s10509-006-9247-0


Scaling Laws for Collisionless Laser–Plasma Interactionsof Relevance to Laboratory Astrophysics

D. D. Ryutov · B. A. Remington


Abstract Scaling laws for interaction of ultra-intense laser

beams with a collisionless plasmas are discussed. Special at-

tention is paid to the problem of the collective ion accelera-

tion. Symmetry arguments in application to the generation of

the poloidal magnetic field are presented. A heuristic model

for evaluating the magnetic field strength is proposed.

Keywords Laser-plasma interaction . Particle acceleration .

Magnetic field generation . Laboratory astrophysics

PACS Numbers: 52.38Kd, 52.38.Fz, 41.75.Jv

1 Introduction

Interaction of ultra-intense laser radiation with a plasma is

of significant potential interest for laboratory astrophysics.

In particular, it opens up a possibility of creating a plat-

form for studying such effects as generation of dynamically-

significant magnetic fields (Norreys et al., 2004; Moon

et al., 2005), development of various collective instabili-

ties, e.g., the filamentation instability (Silva et al., 2003;

Wei et al., 2004), particle acceleration via collective effects

(Katsouleas, 2004; Snavely et al., 2000; Mackinnon et al.,

2001), and dynamics of relativistic plasmas (Liang, 2005;

Wilks et al., 2005). Many of these effects have been studied

numerically, beginning with Wilks et al. (1992).

At these very high intensities, the Coulomb collisions of-

ten become sub-dominant, and the plasma behavior can be

reasonably well described in the collisionless approximation.

Recently, first steps have been made in developing scaling

D. D. Ryutov () · B. A. RemingtonLawrence Livermore National Laboratory, Livermore, CA 94551,USAe-mail: [email protected]

relations that would govern such situations (Pukhov et al.,

2004; Gordienko and Pukhov, 2005; Ryutov and Remington,

2006a,b).

The amplitude E0 of the incident wave defines the so called

dimensionless vector potential,

a0 ≡ eE0

mωc. (1)

Here we use the CGS-Gaussian system of units, with e and

m being the charge and the mass of the electron, c being the

speed of light, and E0 andω being the electric field amplitude

and the wave frequency. The parameter a0 is the ratio of the

electron quiver momentum normalized to mc. In the case

a0 > 1 the oscillating electrons are relativistic, with γ ∼ a0.

An intensity I ≡ cE20/8π is related to a0 by

I (W/cm2) ≈ a20

1.3 × 1018

[λ(µm)]2. (2)

For completeness, we also present the amplitude of the os-

cillating magnetic field in vacuum in terms of a0:

B0(MG) = a0

105

λ(µm). (3)

Note that, for a typical wavelength of λ ∼ 1 µm and a mildly

relativistic incident wave (a0 ∼ 1), this oscillating field is

already quite high, ∼100 MG.

One of the most efficient applications of the scaling laws

is their use for testing the applicability limits of various phys-

ical models. The main assumption of Ryutov and Remington

(2006a,b), in addition to the absence of collisions, is that

the initial plasma temperature is negligible compared to the

energies that electrons and ions (if the ion dynamics is es-

Springer


Fig. 1 The temporal dependence of the electric field in an incidentwave. The envelope function is of the form F(t/τ ), with τ being a char-acteristic temporal width. In the scaling exercise, the function F mustremain the same function of its argument, although the parameter τ mayvary from system to system

sential) acquire early in the laser pulse. Following this line,

in Section 2 we consider possible experiments that would

verify our scaling laws in the problem of the ion acceleration

in the setting of Snavely et al. (2000) and Mackinnon et al.

(2001). If the model is shown to be valid, then scaling argu-

ments can be used as a predictive tool. We present an example

of that, again using the problem of the ion acceleration, in

Section 2.

The presence of scaling laws reveals continuous symme-

tries of the problem. Of a comparable importance in a number

of cases are discrete symmetries, allowing one to make very

general conclusions regarding the geometrical properties of

the system. An example of using such symmetries in the

problem of generation of the quasi-static poloidal magnetic

field was given in Ryutov and Remington (2006a), where the

geometrical structure of such a field was established. Now, in

Section 3, we extend the analysis of Ryutov and Remington

(2006a) to add a heuristic estimate of the magnitude of the

poloidal magnetic field.

Throughout this paper we assume that the duration of the

ultra-intense pulse τ is much greater than the wave period

2π /ω (Fig. 1). The shape of the envelope function F has to

remain the same throughout the scaling exercise, although

normalization can vary.

2 Scaling for the ion acceleration experiment

In this section, we consider a standard setting for the exper-

iment on the ion acceleration with a pulsed laser. The ultra-

intense pulse hits the surface of a thin foil, where some blow-

off-plasma already created by a pre-pulse is usually present.

The incident beam generates ultra-relativistic electrons

which cannot leave the system because of a quasineutrality

constraint. These electrons start oscillating in an ambipolar

potential well. On the rear surface, the hydrogen-containing

impurities liberate hydrogen ions, which are accelerated by

this ambipolar field, roughly speaking, in the normal direc-

tion to the foil. For a thin-enough foil, the fast electrons make

many bounces inside the well, before they lose their energy.

The ion acceleration by the ambipolar electric field is not

affected by collisions. In other words, the ions can be treated

as collisionless. We note that the foil in this discussion plays

a passive role and is not involved in the process of the energy

transfer from the oscillating electrons to the ions. In order

for this to be correct, the foil has to be thin enough so as

to make energy losses of “oscillating” relativistic electrons

negligible. One can also note in passing that a very similar

setting has been studied in great detail in the problem of ion

acceleration by electrons generated in a high-current diode

and injected into vacuum (e.g., Antonsen and Ott, 1976; Ryu-

tov and Stupakov, 1976; Arzhannikov et al., 1976; see also

survey Humphries, 1980 of these early studies).

The full set of the Maxwell–Vlasov equations describing

collisionless plasmas, with relativistic electrons and non-

relativistic ions was reduced to the dimensionless form in

Ryutov and Remington (2006a,b). It was shown that, under

the conditions described above, the system is fully charac-

terized by the following six parameters:

n, L , τ, ω, E0, M/Z , (4)

which are: the density at a characteristic point of the blow-

off plasma; length-scale (e.g., spot size) of the incident beam

and the blow-off plasma; the pulse duration τ of the main

pulse; the frequency ω of the incident radiation; the maxi-

mum amplitude E0 of the electric field of the incident wave

(or, equivalently, the maximum intensity I), and the mass-to-

charge ratio for the accelerated ions. [The latter parameter

may be of interest in the context of comparing the accelera-

tion of hydrogen vs. deuterium.]

The dimensionless parameters that determine the scala-

bility between any two (or more) systems are [Ryutov and

Remington (2006a,b)]:

T ≡ ωτ ; R ≡ Lω/c; S ≡ 4πn0ec

E0ω;

U ≡√

ZeE0

Mωc. (5)

They must be held constant in order that the dimensionless

equations remain unchanged between the two systems such

that the evolution of these systems is similar. We empha-

size that we consider a system with relativistic electrons.

There exists also a similarity transformation that covers

both the non-relativistic and relativistic regimes (Ryutov and

Remington, 2006b), but it requires a constancy of one more

Springer


dimensionless parameter, a0, Equation (1), and is in this

regard more restrictive. In the case of relativistic electrons,

the constancy of the parameter S means the constancy of

the ratio of the frequency of the incident wave and the rela-

tivistic cut-off plasma frequency. This is the reason why the

parameter a0 is absorbed into the parameter S.

In addition to holding the dimensionless parameters (5)

constant, in order that the two systems behave in a scaled

fashion, the geometric similarities must also be observed,

e.g., if the characteristic length-scale L of the plasma den-

sity distribution is increased by a factor of 2, so too must

the focal spot radius be increased by the same factor. The

geometrical characteristics of the incident radiation have

to be identical between the two systems (up to the length-

scale change): the polarization must remain the same, as

well as the direction and the convergence of the incident

beam. The shape of the temporal dependence of the laser

pulse must also remain unchanged (although its duration

may change). Under such conditions, any systems for which

the dimensionless parameters (5) are kept the same, behave

identically, up to scale transformations identified in Ryutov

and Remington (2006a,b). Here we discuss, at a concep-

tual level, the possible experimental verification of the un-

derlying physics assumptions, of which the most important

are the absence of collisions and smallness of the initial

“temperature.” Within these two assumptions, the similar-

ity covers all the processes involved, in all their complexity:

distribution functions, the spatio-temporal characteristics of

the reflected waves, possible presence of the filamenta-

tion and other instabilities, magnetic field generation, and

so on.

Any observed differences may signify that either the

basic assumptions are wrong (e.g., the system is actually

collisional), or the two experiments are not perfectly similar

in terms of their geometry (including the irradiation geome-

try), or the temporal dependence of the incident radiation.

We have six input parameters (Equation (4)) subject to

four constraints S = const, R = const, T = const, and U =const (Equation (5)). To check the validity of scaling laws,

we can arbitrarily choose any two of six input parameters in

the primed system, then adjust the remaining four so as to

keep the dimensionless parameters (Equation (5)) constant.

Consider, for example, that we increase the ion mass by

a factor of 2 (switching from hydrogen to deuterium), and

increase the intensity by a factor of 4 (columns 2 and 3

in Table 1). Then, the rest of the input parameters would

have to be changed as shown in columns 4–7 of Table 1.

This ensures the constancy of the dimensionless parameters

(Equation (5)). The other parameters of the two systems

(e.g., the average energy of the accelerated ions) changes ac-

cording to the scalings formulated in Ryutov and Remington

(2006b). Specifically, in our example, the average energy

increases by a factor of 2, and the total number of accelerated

ions also increases by a factor of 2 (columns 9, 11).

One can also reverse these arguments and, if there is a

good reason to believe in the validity of the underlying as-

sumptions, use the scalings as a predictive tool. For example,

if an experimentalist is interested in generating fast protons

with a laser with a frequency two times less than in the earlier

successful experiment, he/she can do it in a scaled fashion,

thereby being able to predict all the details of a new exper-

iment. This is illustrated by Table 2 which shows that twice

as many fast ions with the same energy can be generated if

the intensity is reduced by a factor of 4, and the other “input”

parameters are changed according to the columns 4–6 of

Table 2.

Table 1 Switching to accelerating deuterium at an increased intensity

1 2 3 4 5 6 7 8 9 10 11

Quantity Ion mass Intensity Frequency Pulse Spatial Density Electron Ion Quasistatic Number of

duration scale energy energy m.f. fastions

Original system M I ω τ L n We Wi B N

“Primed” system 2M 4I ω τ L 2n 2We 2Wi 2B 2N

The relative amplitude of the harmonics remains unchanged

Table 2 A scaled experiment at a reduced frequency

1 2 3 4 5 6 7 8 9 10

Quantity Frequency Intensity Pulse Spatial Density Electron Ion Quasistatic Number of

duration scale energy energy m.f. fastions

Original system ω I τ L n We Wi B N

“Primed” system ω/2 I/4 2τ 2L n/4 We Wi B/2 2N

Springer


3 Discrete symmetries and generation of the poloidal

magnetic field

A variety of mechanisms for generation of a quasi-static mag-

netic field have been discussed in the past, including the

thermo-electric dynamo (Stamper, 1971) and a ponderomo-

tive force (Sudan, 1993). These mechanisms easily explain

the appearance of the toroidal magnetic field, as shown in

Fig. 2 Structure of the magnetic field generated by a laser beam at nor-mal incidence: (a) Toroidal magnetic field; it is generated by the currentthat flows away from the viewer near the axis and toward the viewerat the periphery. Shades of gray show the distribution of the intensityof the incident light. (b) The current streamlines (arrows) generated inthe case where the incident wave is linearly polarized, with the waveelectric field being directed along the y axis. This current pattern leadsto the generation of the poloidal magnetic field. (c) Distribution of thez-component of the magnetic field at some intermediate depth (dots –towards the observer; hatching – away from the observer). The axialmagnetic field lines are closed by a canopy of field lines near the endsof “solenoids”. The direction of the closure at the nearest end is shownby arrows

Fig. 2 for a circular beam imprint. The experiments on the

magnetic field generation carried out thus far have been in-

terpreted in terms of the toroidal field (Norreys et al., 2004;

Tatarakis et al., 2002), without much consideration given to

the possible presence of the poloidal field. However, as was

pointed out in Ryutov and Remington (2006a), the symme-

try arguments lead to the prediction that a poloidal magnetic

field can also be generated, if the incident radiation is lin-

early polarized; a peculiar structure of the currents gener-

ating this field had been predicted. In this article, we pro-

vide a more detailed description of the geometrical structure

and present a rough heuristic estimate of the poloidal

field.

We consider the following model: A linearly polarized

laser beam falls normally onto the plasma slab (Fig. 2b). The

beam imprint is assumed to be circular, with the characteristic

radius r0 exceeding the wave-length of the incident light,

r0 ≫ λ. (6)

The plasma density varies in the z direction (normal to the

slab) from zero to some constant value, which can be both

lower and higher than the critical density. The intensity of

the beam varies along z due to effects of absorption (we do

not specify the mechanism) and varying density (including

possible cut-off beyond the critical point). We assume that

the intensity of the reflected wave is small (because of the

absorption of the incident radiation, or because of a smooth-

ness of the density variation in the case where the maximum

density is sub-critical). We assume that the electric field of

the incident wave is parallel to y.

As the system has two symmetry planes (xz and yz), the

quasi-static current (which is a polar vector) normal to these

planes must vanish (there is no preferential direction). Ac-

cordingly, the current pattern will look as shown in Fig. 2b.

The shape of the streamlines is identical in all four quadrants,

whereas the directions of the currents are shown by arrows. In

principle, finer structures (but possessing the same symme-

try) can also be present. The current of this form has some

finite extent along the z axis. One can think of the current

pattern as that of four solenoids of a finite length along the z

axis, with the current direction alternating from one solenoid

to another as shown in Fig. 2b. The distribution of the axial

(z) magnetic field intensity in the xy plane at some depth z

is shown in Fig. 2c by shading in dots (direction towards the

viewer), and hatching (away from the viewer). In the limit-

ing case of a small axial extent, the magnetic field structure

will be that shown in Fig. 3: this would be a field of four

current rings, with alternating direction of the currents. The

aforementioned “canopy” of the magnetic field lines shows

up, reminiscent of the field lines of a group of sunspots.

Thus far, we have been using only symmetry arguments,

without discussing the mechanism of the poloidal magnetic

Springer


Fig. 3 A more detailed structure of the magnetic field in the case of asmall axial extent of the current-carrying zone. The current loops areshown by thick thick curves (loops). The magnetic field lines are shownas the thin loops; the dashed portions of these lines correspond to thezone beneath the plane where current loops are situated

field generation. Below, we provide some initial heuristic

assessment.

In the case of a relativistic drive, the amplitude of the

electron periodic excursions in the wave field is of the order of

c/ω. For the focal spot size r0 ≫ λ, this displacement is much

less than the spot size. The presence of the radial gradient

of the intensity under such circumstances creates an average

force f acting on the electrons and varying over a scale ∼r0; for generation of the poloidal magnetic field the x and y

components of the force are important, as they can create the

electron flow pattern shown in Fig. 2b. One should note that,

for a non-relativistic drive, a0 ≪ 1, this force is potential (a

so-called Miller force, Gaponov, Miller, 1958). As we shall

see, the potential force f cannot drive a quasi-static field,

i.e., the condition of a0 > 1 (Equation (1)) is important for

our model.

By averaging the electron momentum equation over the

wave period and a spatial scale of a few wave-lengths, one

obtains an equation for the evolution of thus averaged quan-

tities:

d〈p〉dt

= 〈 f 〉 − e〈E〉 − e

c〈v〉 × 〈B〉 (7)

Here 〈v〉 represents the average velocity of the electrons,

related by the equation 〈j〉 = −en〈v〉 to the average current

density 〈 j〉. Neglecting for a moment the electron inertia (the

lhs) and the Hall effect (the last term in the rhs), and using the

Maxwell equation ∇ × 〈E〉 = −(1/c)∂〈B〉/∂t , one obtains:

∂〈B〉∂t

= −c

e∇ × 〈 f 〉 (8)

One sees that, indeed, the generation of the magnetic field

requires the presence of a solenoidal component of the pon-

deromotive force 〈 f 〉, which is absent in the case of a non-

relativistic drive. Equation (8) also shows that, after the laser

drive turns on, the magnetic field increases and reaches a

steady state by the time the drive ends. (We shall see shortly

that it may actually reach a saturation earlier, if the neglected

Hall term comes into play).

We limit ourselves to a qualitative, order-of-magnitude esti-

mate of the absolute value of the ponderomotive force. As

it is related to the radial gradient of the intensity, it can be

evaluated as

f ∼ γmc2/r0. (9)

This estimate is valid only in the relativistic domain, at γ −1 > 1. The geometrical structure of the force is similar to

the current pattern shown in Fig. 2b. In the approximation

described by Equation (8), the maximum magnetic field will

be reached at the end of the pulse and will be equal to

|〈B〉| ∼ γmc3τ

er20

(10)

This is a rough, order-of-magnitude estimate.

Let us now evaluate the possible contribution of the ne-

glected terms. The complete version of Equation (8) reads:

∂〈B〉∂t

= −c

e∇ × 〈 f 〉 + ∇ ×

[〈v〉 × 〈B〉] + c

e〈 p〉

(11)

Noting that 〈 p〉 ∼ γm 〈v〉 ∼ γm⟨

j⟩

/en ∼ (γmc/4πen)

∇ × 〈B〉, where we have used ∇ × 〈B〉 = (4π/c)〈 j〉, and

comparing the last term in the rhs with ∂ 〈B〉 /∂t , one finds

that the inertial term can be neglected if

γmc2

4πne2r20

≡ c2

ω′2per

20

≪ 1, (12)

where ω′pe is a relativistically-corrected plasma frequency (a

cut-off frequency). As the wave frequency is typically com-

parable to the cut-off frequency, this condition is automati-

cally satisfied provided condition (6) holds.

Now we assess the role of the Hall term. By using the rela-

tion 〈v〉 = −〈 j〉/en = −(c/4πen)∇ × 〈B〉, one can rewrite

Equation (11) in the following way:

∂〈B〉∂t

= −c

e∇ × 〈 f 〉 − e

mc∇ × 〈B〉 × ∇ × 〈B〉

ω2pe

. (13)

As the last term is nonlinear in B, it would lead to the sat-

uration of the magnetic field for a long enough pulse. By

balancing the forcing term, with f as in Equation (9), and

the Hall term, one finds the following rough estimate for the

saturated field:

|〈B〉|24π

∼ γmc2n. (14)

Springer


One can combine the estimate (10) and (14) in one heuristic

relation, which covers both cases of a long and short driving

pulses:

B = B0τ

τ + (2πr0/λ)(r0/c)(15)

with B0 as in Equation (3). Here we assumed that the incident

wave has a frequency near a cut-off frequency. Not surpris-

ingly, Equation (15) can be presented in terms of B0 and the

scaling parameters R and T: B = B0/[1 + R2/T ], with r0

playing the role of the length-scale.

Thus far, in the analysis of the magnetic field, we didn’t

take into account the ion motion. In reality, due to the

quasineutrality constraint, they will experience a force com-

parable to f (although, generally speaking, of a different struc-

ture). This would cause the whole plasma in the focal spot to

expand. The condition that the expansion is negligible during

the time τ , reads as:

τ <r0

c

√

M

γm Z(16)

If the opposite condition is valid, the zone of a high magnetic

field is disassembled before the end of the pulse.

It goes without saying that the poloidal field may be gen-

erated alongside the toroidal field, which we do not discuss

in this paper.

4 Discussion

In this paper we applied scaling and symmetry arguments to

study two problems in the area of the interaction of ultra-

intense light with a plasma. The first problem is that of a

collective ion acceleration by the ambipolar field where we

identified a possible experimental test of the physics model

based on two key assumptions: (1) that the system can be

described reasonably well as collisionless, for both fast elec-

trons and ions; (2) that the initial thermal spread is negligibly

small compared to the energies that they acquire early upon

arrival of the ultra-intense pulse. We discuss possible ways of

experimental verification of this model by performing prop-

erly scaled experiments conducted so as to satisfy the simi-

larity rules established in Ryutov and Remington (2006a,b).

What is very attractive with this approach is that, if the simi-

larity indeed holds, there are many experimental signatures,

starting from the spectrum and spatio-temporal behavior of

accelerated ions, through the quasi-static magnetic field evo-

lution, and ending up with the spatio-temporal dependence

of the reflected radiation. Deviations from the predictions of

the scaling laws, in a carefully performed experiment, would

mean the violation of the initial basic assumptions and would

allow one to circumscribe a parameter domain in which the

model is applicable. On the other hand, if the validity of

the model is established, one can use the scaling laws as a

predictive tool.

In the second part of the paper, based on symmetry consid-

eration, we establish a spatial structure of the poloidal mag-

netic field which may be generated alongside the toroidal

field. For a linearly-polarized wave, the axial component of

the field changes sign from one quadrant to the other. This

is a signature that can be used if an experimental attempt to

detect this field is made.

Acknowledgements The authors are grateful to B.I. Cohen, L.L.Lodestro, and T.D. Rognlien for helpful comments. Work performedunder the auspices of the U.S. DoE by UC LLNL under contract NoW-7405-Eng-48.

References

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ZHETF Lett. 24, 19 (1976)Humphries Jr., S.: Nuclear Fusion 20, 1549 (1980)Gaponov, A.V., Miller, A.: Soviet Phys. JETP 7, 168 (1958)Gordienko, S., Pukhov, A.: Phys. Plasmas 12, 043109 (2005)Katsouleas, T.: Plasma Phys. Contr. Fus. 46, B575–582 (2004)Liang, E.: Astrophys. Space Sci. 298, 211 (2005)Mackinnon, A.J., Borghesi, M., Hatchett, S., et al.: Phys. Rev. Lett. 86,

1769 (2001)Moon, S.J., Wilks, S.C., Klein, R.I., et al.: Astrophys. Space Sci. 298,

293 (2005)Norreys, P.A., Krushelnick, K.M., Zepf, M.: Plasma Phys. Contr. Fus.

46, B13–21 (2004)Pukhov, A., Gordienko, S., Kiselev, S., Kostyukov, I.: Plasma Phys.

Contr. Fus. 46, B179 (2004)Ryutov, D.D., Remington, B.A.: in: Superstrong Fields in Plasmas,

Proc. 3rd Intern. Conf. on Superstrong Fields in Plasmas, Varenna,Italy, Sept. 19–24 2005, AIP Conference Proceedings v. 287, p.341, Melville, NY (2006a)

Ryutov, D.D., Remington, B.A.: Plasma Phys. Contr. Fus. 48, L23–L31(2006b)

Ryutov, D.D., Stupakov, G.V.: Fizika Plazmy 2, 767 (1976)Silva, L.O., Fonseca, R.A., Tonge, J.W., et al.: Ap. J. Lett. 596, L121

(2003)Snavely, R.A., Key, M.H., Hatchett, S.P., et al.: Phys. Rev. Lett. 85,

2945 (2000)Stamper, J.A.: Phys. Rev. Lett. 26, 1012 (1971)Sudan, R.: Phys. Rev. Lett. 70, 3075 (1993)Tatarakis, M., Gopal, A., Watts, I., et al.: Phys. Plasmas 9, 2244 (2002)Wei, M.S., Beg, F.N., Clark, E.L., et al.: Phys. Rev. E 70, 56412 (2004)Wilks, S.C., Chen, H., Liang, E., et al.: Astrophys. Space Sci. 298, 347

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69, 1383 (1992)

Springer


DOI 10.1007/s10509-006-9274-x


Topical Issues for Particle Acceleration Mechanismsin Astrophysical Shocks

Matthew G. Baring

Received: 27 July 2006 / Accepted: 31 October 2006C© Springer Science + Business Media B.V. 2006

Abstract Particle acceleration at plasma shocks appears to

be ubiquitous in the universe, spanning systems in the he-

liosphere, supernova remnants, and relativistic jets in dis-

tant active galaxies and gamma-ray bursts. This review ad-

dresses some of the key issues for shock acceleration theory

that require resolution in order to propel our understand-

ing of particle energization in astrophysical environments.

These include magnetic field amplification in shock ramps,

the non-linear hydrodynamic interplay between thermal ions

and their extremely energetic counterparts possessing ultra-

relativistic energies, and the ability to inject and accelerate

electrons in both non-relativistic and relativistic shocks. Re-

cent observational developments that impact these issues are

summarized. While these topics are currently being probed

by astrophysicists using numerical simulations, they are also

ripe for investigation in laboratory experiments, which po-

tentially can provide valuable insights into the physics of

cosmic shocks.

Keywords Shock acceleration . Cosmic rays .

Magnetohydrodynamics . Supernova remnants .

Gamma-ray bursts . Plasma physics

1 Introduction

Supersonic flows abound in the cosmos, as do emission re-

gions exhibiting non-thermal radiation. The intimate con-

nection between the two establishes that particle accelera-

tion in astrophysical shocks is germane to many systems,

M. G. BaringDepartment of Physics and Astronomy MS-108, Rice University,P.O. Box 1892, Houston, TX 77251, U.S.A.e-mail: [email protected]

ranging from the heliosphere, to stars of various sorts ex-

pelling winds, to supernova remnants, to extragalactic jets

and gamma-ray bursts. In the case of heliospheric shocks

such as travelling interplanetary discontinuities and plane-

tary bow shock environs, we can immerse ourselves in the

plasma experiment via in situ spacecraft measurements of

non-thermal ions, electrons and turbulent magnetic fields.

While localized and therefore sparse in terms of the spatial

sampling, these observations do provide profound insights

into the complexity of the shock acceleration phenomenon.

In astrophysical sites beyond the solar system, our role is pas-

sive, as observers of signals from remote sites of acceleration.

Moreover, the information on plasma properties is subject to

a convolution with radiative processes, complicated source

morphology within our spatial resolution scale, and propa-

gational modification along the line of sight to sources.

Observationally, radio, optical and X-ray telescopes have

provided groundbreaking insights into the shock accelera-

tion phenomenon, due to advances in the angular resolution

and spectral sensitivity. In addition, the gamma-ray field is

generating a greater understanding of source energetics as

we find that many non-thermal astronomical sources emit

most of their power in the gamma-rays. Progress on the the-

oretical front builds on the observational advances, and has

turned more to computer simulations due to the dramatically

enhanced speed of computers over the last two decades. Yet

astrophysical code verification is an increasingly salient is-

sue as their complexity and computational demands burgeon.

This provides a niche for laboratory plasma experiments that

are tailored for the problem of astrophysical particle accel-

eration. Supersonic flows can be generated in controlled en-

vironments, and work along these lines focuses naturally on

using lasers to mimic blast waves and jets, supernovae and

supernova remnants (e.g. Borovsky et al., 1984; Drake et al.,

1998; Shigemori et al., 2000; Kang et al., 2001; Lebedev

Springer


et al., 2002; Woolsey et al., 2004), and probe hydrodynamic

and magnetohydronamic aspects. The question of scalabil-

ity of such laboratory findings to astrophysical systems (e.g.

Ryutov et al., 2001; see also the review of Remington et al.,

2006) is obviously of central importance. Here, an offering

on some topical issues for particle acceleration at astrophys-

ical shocks is made, to provide a basis for the community

in high energy density plasma physics to help identify ger-

mane astrophysical problems that might be well suited for

interdisciplinary investigation.

2 Magnetic field enhancements in shocks

One of the key properties of shock structure that is germane to

the acceleration of high energy cosmic rays is the strength of

the magnetic field B near the shock. In astrophysical shocks

this cannot really be measured directly, since there is gen-

erally a lack of viable spectral line diagnostics: the Zeeman

effect and cyclotron emission/absorption features are gen-

erally broadened, small or non-existent in diffuse, turbulent

shock environs. Normally, proximity of an emission region

to a stellar surface, such as in white dwarfs and neutron stars

is required to afford precise magnetic field measurements. In

heliospheric shocks, magnetometer data discern the chaotic

nature of pre- and post-shock fields, and a prominent property

appears to be (e.g. Baring et al., 1997) a shock-induced com-

pression of the field in the downstream region that is fairly

close to magnetohydrodynamic (MHD) determinations that

are derived from momentum and energy flux conservation

across the shock, i.e. the so-called Rankine-Hugoniot con-

ditions (e.g. Drury, 1983; Jones and Ellison, 1991). If the

interaction of charged particles with shock-associated field

turbulence is gyroresonant at the Doppler-shifted cyclotron

frequency (e.g., see Melrose, 1980), as is expected for Alfven

and whistler modes, then the acceleration timescale τ nat-

urally scales as the gyroperiod (i.e., τ ∼ 1/νg ∝ 1/B ; see

Forman et al., 1974; Drury, 1983) and the corresponding dif-

fusive lengthscale is comparable to the Larmor radius rg

( ∝ 1/B ). Hence the magnitude B establishes the temporal,

spatial and energy scales of acceleration at a shock, and so

is a critical parameter for the energization process.

Since the general paradigm of galactic cosmic rays (CRs)

invokes supernova remnants (SNRs) as the sites for their

production (see Drury, 1983 for a review), knowledge of the

field strength in proximity of their shocks is vital. Directional

information can be obtained on fairly large spatial scales

via radio synchrotron polarization data (e.g. see Rosenberg,

1970; Downs and Thompson, 1972; Anderson et al., 1995,

for Cassiopeia A), but values of B = |B| are not forthcom-

ing. Estimates for the field strength can be inferred by mod-

eling the continuum flux level in a given wavelength band,

but these are subject to a number of assumptions about the

medium, for example the mean density ne and the Lorentz

factors γe of the radiating electrons. The understanding of

the character of shells and interiors of SNRs has recently been

advanced by groundbreaking observations with the Chandra

X-ray Observatory, enabled by its impressive angular res-

olution coupled with its spectral capabilities. Of particular

interest is the observation of extremely narrow non-thermal

(1.2–2.0 keV) X-ray spatial profiles in selected remnants (see

Long et al., 2003; Bamba et al., 2003 for the northeast limb

of SN1006; Vink and Laming, 2003 for Cas A; for theoret-

ical modeling see Ellison and Cassam-Chenaı, 2005; Volk

et al., 2005), typically less than 5–10 arcsec across. Upstream

of these shell shocks, the X-ray emission, which is thought

to be synchrotron in origin, drops to effectively zero. These

strong brightness contrasts between the shell, and the outer,

upstream zones correspond to flux ratios exceeding R >∼50 .

The narrowness of profiles along image scans argues for the

shocks being aligned perpendicular to the sky, i.e., offering

no projectional smearing in the images. Note also that the

surface brightness angular profiles in SN1006 and Cas A are

much broader for the thermal X-rays (0.5–0.8 keV) and the

radio synchrotron than for the non-thermal X-rays.

If the synchrotron mechanism is indeed responsible for

non-thermal Chandra emission, the electrons contributing

to the Chandra signal are probably in a strongly-cooling

regime: see Baring et al. (1999) for a comprehensive discus-

sion of SNR cooling parameter space. Since the synchrotron

cooling rate for an electron scales as γ 2e B2 , then the flux ra-

tio R is approximately a measure of the ratio of B2 down-

stream (d) to upstream (u). The observed lower bounds to

R considerably exceed values R <∼ 16 expected for mag-

netohydrodynamic compression at the shocked shell; at a

plane-parallel shock with B along the shock normal there

is no field compression, while in a strong (i.e. high sonic

Mach number) perpendicular shock with B in the shock

plane, Bd/Bu ∼ 4 . Hence, the pronounced brightness con-

trast is taken as strong evidence of magnetic field am-

plification in the shock precursor/ramp upstream. Higher

fields are obviously advantageous to cosmic ray production

issues. Historically-accepted values of B ∼ 1 – 10µGauss

(i.e. 0.1–1 nanoTesla) are somewhat too small to permit ac-

celeration in SNR shocks of ages around 103 – 104 years

right up to the cosmic ray knee at ∼3 × 1015 eV (e.g. see

Lagage and Cesarsky, 1983). This problem has spawned the

suggestion (Jokipii, 1987) that relatively ineffective diffusive

transport of particles orthogonal to the mean field direction

in quasi-perpendicular regions of SNR shocks can speed up

acceleration of ions to higher energies, helping access the

knee. Yet, this enhanced rapidity is accompanied by reduced

efficiency of cosmic ray injection from thermal energies (El-

lison et al., 1995). Hence, truly larger fields provide a cleaner

path for acceleration in remnants to reach the cosmic ray knee

(e.g. Kirk and Dendy, 2001).

Springer


The X-ray observational developments have been accom-

panied by theoretical proposals of magnetic field amplifica-

tion in the upstream shock precursor. Most notable has been

the work of Lucek and Bell (2000), and subsequent papers

such as Bell (2004), Amati and Blasi (2006), and Vladimirov

et al. (2006). The idea of Lucek and Bell is that high en-

ergy cosmic rays in strong shocks could amplify B when

streaming upstream, adiabatically transferring energy to the

turbulent field by pushing against it, simultaneously deceler-

ating the upstream flow. If this process is efficient, the rate of

work done on the upstream Alfven turbulence of energy den-

sity UA naturally scales roughly with the CR pressure gra-

dient: dUA/dt = vA|∇ PCR| . Here vA = B/√

4πρ is the

Alfven speed, and PCR is the cosmic ray pressure. The as-

sociated field amplification should then scale as (δB/B)2 ∼MA PCR/ρu2

u in an upstream flow of speed uu and mass

density ρ; this then becomes very effective for high Alfvenic

Mach number (i.e. MA ≡ uu/vA ≫ 1 ), strong shocks that

generate large cosmic ray pressures. While this hypothesis

is reasonable, demonstrating it is non-trivial. Various MHD-

type simulations have been employed by Bell, such as in

Bell (2004), where large-scale currents are used to drive in-

stabilities that amplify the upstream field. The persistence

of currents on large scales is unclear, particularly due to the

action of Debye screening. Moreover, self-consistent phys-

ical connection between the cosmic rays of large Larmor

radii and the field turbulence of much shorter wavelengths

is extremely difficult to explore with MHD or plasma simu-

lations, due to the wide disparity in spatial scales involved.

This is an issue also for a growing number of particle-in-

cell (PIC) simulations (e.g. Silva et al., 2003; Hededal et al.,

2004; Nishikawa et al., 2005; see Section 4 below) used to

explore field enhancement via the Weibel instability in rela-

tivistic shocks; such developments are not that salient for the

problem of amplifying Alfven turbulence in non-relativistic

shocks, and mostly probe the inertial scales of thermal ions

and electrons defined by their plasma frequencies.

3 Non-linear feedback between the acceleration and the

hydrodynamics

Non-relativistic collisionless shocks can be highly efficient

accelerators, placing 10–50% of the bulk flow kinetic energy

into non-thermal particles. Evidence from theory, computer

simulations, and spacecraft observations supports this con-

clusion; in particular, see Ellison et al. (1990), for a study

of the Earth’s bow shock, and Drury (1983), Blandford and

Eichler (1987), and Jones and Ellison (1991) for reviews.

With such efficiencies, the accelerated particles acquire a siz-

able fraction of the total energy budget, influencing the shock

hydrodynamics, and therefore also the fraction of energy go-

ing into accelerated particles, in a non-linear manner. The

modified flow velocity spatial profile in the shock deviates

from the familiar step-function form in test-particle accelera-

tion scenarios, with the energetic particles pushing against the

upstream flow and decelerating it far ahead of the shock dis-

continuity. Accordingly an upstream shock precursor forms,

with declining flow velocity as the shock is approached. This

structure alters the shape of the energetic particle distribu-

tion from a power-law in momentum (e.g., Eichler, 1984;

Ellison and Eichler, 1984; Ellison et al., 1996; Berezhko

et al., 1996; Malkov, 1997; Blasi, 2002), the canonical test-

particle form where the diffusively-accelerated particle dis-

tribution samples no spatial or momentum scale. The in-

dex σ = (r + 2)/(r − 1) of this power-law dn/dp ∝ p−σ

is purely a function of the compression ratio r = uu/ud

of upstream ( uu ) to downstream ( ud ) flow speed compo-

nents normal to the shock in the shock rest frame (e.g. see

Drury, 1983; Jones and Ellison, 1991), and is independent of

the magnetic field orientation or the nature and magnitude

of the turbulence effecting diffusive transport in the shock

neighborhood.

The spatial variation of the upstream flow in strong shocks

that are efficient accelerators eliminates the scale indepen-

dence. Since the highest energy particles have greater dif-

fusive mean free paths λ (generally true for gyroresonant

interactions with MHD turbulence, and certainly so near the

Bohm diffusion limit λ ∼ rg ), they diffuse farther into the

upstream shock precursor against the convective power of the

flow, and therefore sample greater effective velocity com-

pression ratios r . Accordingly, they have a flatter distribu-

tion, yielding a distinctive concavity to the overall particle

spectrum, i.e. σ is now a declining function of momentum

p . These departures from power-law behavior amplify the

energy placed in the particles with the greatest momenta,

which in turn feeds back into the shock hydrodynamics that

modify the spatial flow velocity profile. Traveling discontinu-

ities possessing this complex feedback are termed non-linear

shocks, the non-linear label being ascribed to the interplay

between the macroscopic dynamics and the microscopic ac-

celeration process. Clearly, the possible magnetic field am-

plification in the upstream precursor that was discussed in

Section 2 contributes to the overall dynamics/energy budget

of the magnetohydrodynamic flow, and so intimately influ-

ences this non-linear aspect of astrophysical shocks.

The deviations from power-law distributions obviously

impact the radiation signatures produced by these particles,

with alterations in the fluxes expected in X-ray and TeV

gamma-ray bands in remnants, differing by as much as factors

of 3–10 from traditional test-particle predictions (e.g., see

Baring et al., 1999; Ellison et al., 2001; Berezhko et al., 2002;

Baring et al., 2005). Conclusively confirming the existence

of this non-linear spectral concavity is a major goal that is in-

herently difficult, since it demands broad, multi-wavelength

spectral coverage. There is a limited suggestion of concavity

Springer


in radio data for Tycho’s and Kepler’s SNRs (Reynolds and

Ellison, 1992), and in a multi-wavelength modeling of SN

1006 (Allen et al., 2004; see also Jones et al., 2003 for in-

ferences from radio and infra-red data from Cassiopeia A),

but this task really looks ahead to the launch of the GLAST

gamma-ray mission in late 2007, when, in conjunction with

ground-based Atmospheric Cerenkov Telescopes probing the

TeV band, it may prove possible to determine gamma-ray

spectra from SNRs spanning over 3 decades in energy.

In the meantime, an interesting astrophysical manifesta-

tion of these non-linear effects has been offered by SNR

observations by the Chandra X-ray Observatory, looking

instead at the thermal populations. Inferences of ion tem-

peratures in remnant shocks can be made using proper mo-

tion studies, or more direct spectroscopic methods (e.g.

Ghavamian et al., 2003). For the remnant 1E 0101.2-7129,

Hughes et al. (2000) used a combination of ROSAT and

Chandra data spanning a decade to deduce an expansion

speed. Electron temperatures Te are determined by line di-

agnostics, via both the widths, and the relative strengths

for different ionized species. From these two ingredients,

Hughes et al. (2000) observed that, in selected portions of

the SNR shell, 3kTe/2 ≪ 3kTp/2 ∼ m p(3uu/4)2/2 . There-

fore, the electrons were considerably cooler than would cor-

respond to equipartition with thermal protons heated in a

strong shock with an upstream flow speed of uu : the ther-

mal heating is assumed comparable to the kinematic veloc-

ity differential uu − ud ≈ 3uu/4 . The same inference was

made by Decourchelle et al. (2000) for Kepler’s remnant,

and by Hwang et al. (2002) for Tycho’s SNR. This property

of comparatively cooler electrons may be indicative of them

radiating very efficiently. Or it may suggest that the protons

are cooler (i.e. 3kTp/2 ≪ m p(3uu/4)2/2 ) than is widely as-

sumed in the test-particle theory, the conclusion drawn by

Hughes et al. (2000) and Decourchelle et al. (2000). This

effect is naturally expected in the non-linear shock accel-

eration scenario: as the highest energy particles tap signifi-

cant fractions of the total available energy, they force a re-

duction in the thermal gas temperatures. Such feedback can

profoundly influence shock layer thermalization, inducing

significant interplay with electrostatic equilibration between

low energy electrons and ions, an issue addressed in these

proceedings by Baring and Summerlin (2006). Note that non-

linear modifications may vary strongly around the shocked

shell of an SNR, since the obliquity angle Bn,u of B to

the shock normal varies considerably between different rim

locales.

4 The character of relativistic shocks

Relativistic shocks, for which the upstream flow Lorentz fac-

tor γu = 1/√

1 − (uu/c)2 considerably exceeds unity, are

less well researched than their non-relativistic counterparts,

not in small part due to their greater cosmic remoteness: they

predominantly arise in extragalactic locales like jets in active

galaxies, and gamma-ray bursts. Yet, because of such asso-

ciations, they are now quite topical. Diffusive test-particle

acceleration theory in parallel (i.e., Bn,u = 0 ) relativistic

shocks identifies two notable properties in such systems: (i)

particles receive a large energy kick E ∼ γumc2 in their

first shock crossing (e.g., Vietri, 1995), but receive much

smaller energy boosts for subsequent crossing cycles (factors

of around two: e.g., Gallant and Achterberg, 1999; Baring,

1999); (ii) a so-called ‘universal’ spectral index, σ ∼ 2.23

exists in the two limits of γu ≫ 1 and small angle scattering,

i.e., δθ ≪ 1/γu (e.g., Kirk et al., 2000; see also Bednarz and

Ostrowski, 1998; Baring, 1999; Ellison and Double, 2004).

Here, δθ is the average angle a particle’s momentum vector

deviates in a scattering event, i.e. an interaction with mag-

netic turbulence.

These characteristics are modified in parallel, mildly rela-

tivistic shocks with γu ∼ 1 . In such shocks, the distribution

dn/dp remains a power-law (scale-independence persists),

but hardens ( σ decreases) as either γu drops, or the scat-

tering angle, δθ , increases (e.g., Ellison et al., 1990; Baring,

1999; Ellison and Double, 2004; Baring, 2004), even if the

compression ration r = uu/ud is held constant (it usually

increases with declining γu due to a hardening of the Juttner-

Synge equation of state). These effects are consequences of

large kinematic energy kicks particles receive when scattered

in the upstream region after transits from downstream of the

shock. It is particularly interesting that when scattering con-

ditions deviate from fine pitch-angle-scattering regimes with

δθ ≪ 1/γu , the power-law index is dependent on δθ , with

a continuum of spectral indices being possible (Ellison and

Double, 2004; Baring, 2004). Then the nature of the turbu-

lence is extremely influential on the acceleration outcome,

so that understanding the turbulence is of paramount impor-

tance. This sensitivity of σ to the field fluctuations when

δθ >∼ 1/γu , a large angle scattering domain, contrasts the

canonical nature of σ in non-relativistic shocks mentioned

above.

In jets and gamma-ray bursts, ultra-relativistic shocks are

typically highly oblique due to the Lorentz transformation of

ambient, upstream magnetic fields to the shock rest frame.

This introduces an added dimension of variation, with in-

creasing Bn,u dramatically steepening the power-law, i.e.

increasing σ . This is naturally expected since such systems

are highly superluminal, that is, there exists no de Hoffman-

Teller (1950) shock rest frame where the flow velocities

are everywhere parallel to the mean magnetic field (which

would correspond to large scale electric fields being zero

everywhere). Therefore, relativistic shocks are much less ef-

ficient accelerators because particles convect more rapidly

away downstream from the shock (e.g. Begelman and Kirk,

Springer


1990). In oblique, relativistic shocks, σ , and indeed the ef-

ficiency of injection from the thermal particle population,

also depend on the ability of turbulence to transport parti-

cles perpendicular to the mean downstream field direction

(Ellison and Double, 2004; Niemiec and Ostrowski, 2004).

This perpendicular transport couples directly to the magni-

tude (δB/B)2 and power spectrum of field fluctuations, i.e.

the strength of the scattering. Steep spectra (σ >∼ 4) result un-

less the ratio of the diffusive mean free paths perpendicular

to and parallel to B is comparable to unity, which defines the

Bohm diffusion regime. In summation, for relativistic shocks,

the spectral index is sensitive to the obliquity Bn,u of the

shock, the nature of the scattering, and the strength of the

turbulence or anisotropy of the diffusion. These properties

are reviewed in Baring (2004).

Observational vindication of these theoretical predictions

is clearly mandated. This is not readily forthcoming, since

the only accessible information involves a convolution of

shock acceleration and radiation physics. Yet, it is clear, for

example in gamma-ray bursts (GRBs), that data taken from

the EGRET experiment on the Compton Gamma-Ray Ob-

servatory (CGRO) suggest a broad range of spectral indices

(Dingus, 1995) for the half dozen or so bursts seen at high en-

ergies. This population characteristic is commensurate with

the expected non-universality of σ just discussed. Yet it is

important to emphasize that the power-law index is not the

only acceleration characteristic germane to the GRB prob-

lem: the shapes of the particle distributions at thermal and

slightly suprathermal energies are also pertinent. This energy

domain samples particle injection or dissipational heating in

the shock layer, and is readily probed for electrons by the

spectrum of prompt GRB emission by the BATSE instru-

ment on CGRO. Tavani (1996) obtained impressive spectral

fits to several bright BATSE bursts using a phenomenologi-

cal electron distribution and the synchrotron emission mech-

anism. While there are issues with fitting low energy (i.e.<∼100 keV) spectra in about 1/3 of bursts (e.g. Preece et al.,

1998) in the synchrotron model, this radiative mechanism

still remains the most popular candidate today for prompt

burst signals.

Tavani’s work was extended recently by Baring and Braby

(2004), who provided additional perspectives, using acceler-

ation theory to underpin a program of spectral fitting of GRB

emission using a sum of thermal and non-thermal electron

populations. These fits demanded that the preponderance of

electrons that are responsible for the prompt emission consti-

tute an intrinsically non-thermal population. That is, the

contribution to the overall electron distribution that comes

from a Maxwell-Boltzmann distribution is completely dom-

inated by a non-thermal component that, to first order, can

be approximated by a power-law in energy truncated at some

minimum electron Lorentz factor. This requirement of non-

thermal dominance strongly contrasts particle distributions

obtained from acceleration simulations, as is evident in a

host of the references cited on acceleration theory above:

the non-thermal particles are drawn directly from a thermal

gas, a virtually ubiquitous phenomenon. This conflict poses

a problem for acceleration scenarios unless (i) radiative ef-

ficiencies for electrons in GRBs only become significant at

highly superthermal energies, or (ii) shock layer dissipation

in relativistic systems can suppress thermalization of elec-

trons. A potential resolution to this dilemma along the lines

of option (i) is that strong radiative self-absorption could

be acting, in which case the BATSE spectral probe is not

actually sampling the thermal electrons. It is also possible

that other radiation mechanisms such as Compton scatter-

ing, pitch-angle synchrotron, or jitter radiation may prove

more germane. Discerning the radiation mechanism(s) oper-

ating in bursts is a foremost goal of future research, and will

be facilitated by the GLAST mission, with its good sensitiv-

ity in the 5 keV–300 GeV band, in conjunction with NASA’s

current GRB flagship venture, Swift.

Option (ii) is a conjecture that has no definitive simu-

lational evidence to support it at present. The most com-

prehensive way to study dissipation and wave generation in

collisionless shocks is with PIC simulations, where particle

motion and field fluctuations are obtained as solutions of

the Newton-Lorentz and Maxwell’s equations. Rich in their

turbulence information, these have been used extensively in

non-relativistic, heliospheric shock applications, and more

recently, relativistic PIC codes have blossomed to model

shocks in various astrophysical systems. PIC simulation re-

search has largely, but not exclusively, focused on perpen-

dicular shocks, first with Gallant et al. (1992), Hoshino et

al. (1992), and then Smolsky and Usov (1996), Shimada and

Hoshino (2000), Silva et al. (2003), Nishikawa et al. (2003,

2005), Spitkovsky and Arons (2004), Hededal et al. (2004),

Liang and Nishimura (2004), Medvedev et al. (2005) and

Hededal and Nishikawa (2005). These works have explored

pair shocks, ion-doped shocks, Poynting flux-dominated out-

flows, and low-field systems with dissipation driven by the

Weibel instability, in applications such as GRBs and pulsar

wind termination shocks. PIC simulations are dynamic in

nature, and rarely achieve a time-asymptotic state. Even in

the minority of cases where there is some evidence of accel-

eration beyond true thermalization, none of these works has

demonstrated the establishment of an extended power-law

that is required in modeling emission from GRBs and active

galactic nuclei. This is perhaps due to the severely restricted

spatial and temporal scales of the simulations, imposed by

their intensive CPU and memory requirements; these limit

the modeling of realistic electron-to-proton mass ratios, full

exploration of three-dimensional shock physics such as dif-

fusive transport, and addressing the wide range of particle

momenta encountered in the shock acceleration process. In

particular, it is difficult to establish a broad inertial range

Springer


for cascading MHD turbulence when the maximum spatial

scales in the simulation are not orders of magnitude larger

than the principal ion inertial scales. Definitively observing

the injection of electrons from a thermal population to es-

tablish a truly non-thermal distribution remains a pressing

goal of plasma simulations, both for relativistic shocks and,

as has been understood for more than two decades, also their

non-relativistic cousins.

5 Conclusion

This review is by no means a complete presentation of the

topical issues for the shock acceleration problem, but it does

offer a fair sampling suitable for motivating interdisciplinary

activity. It is clear that several issues could benefit substan-

tially from input from laboratory experimentation on the high

energy density physics/astrophysics interface. One key ques-

tion is whether or not ambient magnetic fields are amplified

by both non-relativistic and relativistic shocks beyond stan-

dard MHD expectations. If so, is the amplification electro-

static in origin, or is it connected to energetic particles accel-

erated by the shock? It would be important to discern whether

there are differences between high and low Alfvenic Mach

number systems, i.e. what role the ambient magnetic field

plays in controlling the outcome. Another question concerns

whether or not suprathermal electrons and ions can actually

be seen, and whether one can identify their origin. It is salient

to ascertain if they are diffuse in nature, or if they form coher-

ent beams, both of which are seen at traveling shocks embed-

ded in the solar wind. Also, if acceleration is observed, then

identifying the role the highest energy particles have in mod-

ifying the shock hydrodynamics and the thermal structure of

the shock layer would help solve an outstanding problem that

has long been a principal goal within the cosmic ray commu-

nity. Finally, specifically concerning relativistic systems, it

would be desirable to elucidate how the distributions of any

accelerated particles seen depend on external quantities such

as the field obliquity and speed of the shock, whether thermal

electrons can be suppressed relative to accelerated ones, and

if there is an identifiable connection with the field turbulence

near the shock. These are demanding goals, yet terrestrial ex-

periments are very useful for probing global aspects of shock

problems, and in particular for extracting insights into hydro-

dynamic and MHD behavior. In order to make progress, it is

essential to prepare an experimental setup that is as tenuous

as possible, to mimic the collisionless (in the Spitzer sense)

shock environments offered throughout the cosmos. At this

juncture, exciting prospects are on the horizon for this in-

terdisciplinary forum, with contributions to be found in the

laboratory, in computer simulations, and in astronomical ob-

servations, all of which can benefit from cross-fertilization

with each other.

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DOI 10.1007/s10509-006-9242-5


Three-Dimensional Particle Acceleration in ElectromagneticCylinder and Torus

Koichi Noguchi · Edison Liang


Abstract Particle acceleration via Poynting vector with

toroidal magnetic field is studied in 3D PIC simulation of

electron-positron plasma. We choose two different initial

magnetic field configurations to compare how the particle

acceleration is affected by the expansion of electromag-

netic wave. In the cylindrical case, the electromagnetic field

strength decays as (ct)−2, and particles are accelerated in

the radial direction as well as the axial direction. Rayleigh-

Taylor instability is also observed at the center of the cylin-

der. In the torus case, the field strength decays as (ct)−3,

making the acceleration less efficient. Particles accelerated

in the axial direction by E × B force creates strong charge

separation.

Keywords Gamma ray bursts . Collapsars . PIC .

Numerical . Relativity

1 Introduction

One of the most famous models to produce a GRB and large

explosion energy from the death of a massive star is the

collapsar model (Woosley, 1993; MacFadyen and Woosley,

1999). When a massive star is collapsing to a black hole, it

accretes envelope matter at a very high rate. Woosley pointed

out that neutrinos should be emitted from the innermost re-

gion of the accretion disk through the electron-positron cap-

ture and electron-positron pair annihilation (Berezinskii and

Prilutskii, 1987; MacFadyen and Woosley, 1999). A strong

jet with a large explosion energy on the order of ∼1052 ergs,

is generated from the polar region by the pair-annihilation

K. Noguchi () · E. LiangRice University, Houston, TX 77005-1892e-mai: [email protected].

of neutrinos, and the shock wave propagates through a chan-

nel created inside of the stellar envelope. Two-dimensional

relativistic hydrodynamic simulations with nucleosynthe-

sis (MacFadyen and Woosley, 1999; Nagataki, 2000, 2001;

Nagataki et al., 2003) confirms the formation and propaga-

tion of jets inside of the stellar envelope.

Poynting flux acceleration (PFA) may be launched when

the magnetic jet head or the magnetar stripe wind emerge

from the stellar envelope surface due to the sudden de-

confinement of the magnetic field and embedded electron-

positron plasma. When the electron-positron plasma jet

is emerged out from the envelope, electromagnetic wave

expands into ambient medium with particles accelerated

by the PFA (Liang and Nishimura, 2004; Noguchi et al.,

2005). Since magnetic fields are connected to the accretion

disk associated with the collapsar, the emerged jet is colli-

mated and creates a magnetic tower (Lynden-Bell, 2003).

Three-dimensional magnetohydrodynamic global simula-

tions of a central black hole associated with low magne-

tized accretion disk (Kato et al., 2004) shows that ver-

tically inflating toroidal fields supports the magnetic jet

collimation.

In this article we present 3D PIC simulations of particle ac-

celeration driven by PFA with two different initial magnetic

field configurations. In the first model, initial toroidal mag-

netic field and particle distributions are a decreasing func-

tion of radius, and have infinite length in the axial direction.

In the second model, magnetic field and particles are dis-

tributed as a radially decreasing function as the same as the

first model, but has a finite length in the axial direction. The

first model represents the particle acceleration in the mag-

netic tower, and the second represents the magnetic jet head

emerged from the stellar envelope. Hereafter we call the first

model as the cylindrical case and the second as the torus case,

respectively.

Springer


Fig. 1 3D contour plot of magnetic field strength (left) and the magnetic field profile of the plane y = 0 (right) at tce = 520

2 Initial setup of the simulation

We use the 3D explicit PIC simulation scheme based on the

Yee algorithm (Yee, 1966). Spatial grids for the fields are

uniform in all directions, x = y = z = c/ωpe, where

ωpe is the electron plasma frequency. We employ 1283 cells

with triply periodic boundary conditions. In the simulation,

the origin is located at the center, and x direction corresponds

to the axis of the magnetic cylinder and torus.

For both the cylindrical and torus cases, the radial profiles

of initial magnetic field and particle density are the same. In

order to make the magnetic field satisfy ∇ · B = 0, the initial

magnetic field profile has toroidal component only, which is

given as

Bφ =

B0r/(2x), 0 ≤ r < 2x

2B0x/r, 2x ≤ r ≤ 8x. (1)

We note that the radial profile of Bφ corresponds to the

profile around cylindrical infinite current column with j =cB0/4πr (r < 2πx). For simplicity and self-consistency,

we assume no initial electric field and current. Electric field

and current are self-induced by magnetic field and particle

motion associated with it. In magnetic-tower scenario, mag-

netic field near the tip of a jet is almost purely toroidal, and

plasma motion is driven by magnetic field expansion.

The density profile is given as

ρ =

ρ0r/(2x), 0 ≤ r < 2x

2xρ0/r, 2x ≤ r ≤ 5x. (2)

Here, ρ0 is calculated by assuming ωpe/ce = 0.1, where

ce is the electron cyclotron frequency.

The initial temperature of plasma is a spatially uniform

Maxwellian kB Te = kB Tp = 100eV, where Te and Tp are the

electron and positron temperature, respectively. We note that

we use c/ωpe instead of electron Debye length as a unit scale

in the simulations, because EM field expands with the speed

of light rather than the thermal speed.

In the cylindrical case, Bφ and ρ are uniform in the z direc-

tion. In the torus case, however, both Bφ and ρ vanishes for

z < −2c/ωpe and z > 2c/ωpe in order to simulate the mag-

netic jet head. Initial temperature is assumed to be uniform

Maxwellian with kB Te = kB Tp,i = 100eV.

3 Results

First, we study the cylindrical case. Three-dimensional con-

tour plot of magnetic field strength and the magnetic field pro-

file of the plane y = 0 at tce = 520 are shown in Fig. 1.Only

a quarter of the whole simulation box is shown in Fig. 1a.

Electric field Ez is automatically generated by the expansion

of magnetic field, and the direction of the Poynting vector

is always in the radial direction. EM wave expands to sur-

rounding vacuum region with the speed ∼ c, carrying parti-

cles within its ponderomotive well. Electrons and positrons

are also accelerated in the negative and positive z direction

respectively, due to E × B drift, as we expected. Due to the

periodic boundary condition, no charge separation occurs in

the z direction.

Rayleigh-Taylor instability occurs in the central region

(r < 10x). The critical wave number kx for electron-

positron plasma is given by Chen (1984)

k2xv

20 < −g

ρ ′0

ρ0

, (3)

where v0 = −g/ce is the positron centrifugal drift velocity

in the x direction,ρ ′0 = dρ0/dr , and g = v2

0/r is the centrifu-

gal field strength in the radial direction by the magnetic field

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Fig. 2 Phase plots of electrons (blue) and positrons (red) in the cylindrical case at tce = 560

Fig. 3 3D contour plot of magnetic field strength (a) and the spatial distribution of electrons (blue) and positrons (red) (b) at tce = 520

curvature. The minimum wavelength λ = 2πr is ∼ 30x on

the surface of the initial plasma column, which corresponds

to the wavelength in magnetic field profile Fig. 1b. Only the

center of the column becomes unstable, since ρ ′0 rapidly goes

to zero toward the edge of magnetic field column.

Figure 2shows phase plots at tce = 560. Fig. 2a shows

that troidal magnetic field Bφ create a current in the x di-

rection. Figures 2b and 2c indicates that the most energetic

particles are accelerated with γ ∼ 10 in both x and y di-

rections, and slow particles are not accelerated by PFA in y

and z. Figures 2d and 2e shows that strong bifurcation oc-

curs in the y direction. The first ponderomotive force well in

the front of EM pulse (y ∼ ±50x) is too weak to hold all

the energetic particles, and following ponderomotive wells

(y ∼ ± 40x and ± 30x) capture such particles slipped

out from the first well. Finally, Fig. 2f shows the expansion

is uniform in the y − z plane.

Next, we show the results of the torus case. Figure 3a

shows the three-dimensional contour plot of magnetic field

at tce = 520. In this case, EM field expands almost spheri-

cally. Different from the cylindrical case, the magnetic torus

has finite length in the x direction, resulting the charge sepa-

ration between electrons and positrons. Figure 3b shows the

spatial distribution of sample electrons (blue) and positrons

(red). Positrons and electrons are tend to move in the posi-

tive and negative x direction, respectively, creating electric

field in the x direction. Only few particles are captured and

accelerated by PFA, since the magnetic field strength drops

as (ct)−3 in the torus case, whereas (ct)−2 in the cylindrical

case.

Figure 4 shows phase plots at tce = 560. Figure 4a indi-

cates strong acceleration occurs on the edge of torus, creating

charge separation. The highest γ factor for the most energetic

particles is ∼10, which is the same order as the cylindrical

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Fig. 4 Phase plots of electrons (blue) and positrons (red) in the cylindrical case at tce = 560

case. Charge separation may prevent particles from acceler-

ating for long time, but simulation box is too small to see if

the acceleration will stop. Figures 4b and 4 c shows that the

acceleration in the x − y plane is highly non-uniform, and

the highest γ in the y direction is almost half of γ in the x

direction.

Figures 4d and 4e shows that the acceleration of parti-

cles captured in the front of EM pulse (|y| > 25x) is not

efficient, and more effective acceleration occurs in the sec-

ond well (10x < |y| < 25x). As we mentioned, mag-

netic field strength drops as r−3, and the first well is too

shallow to capture particles in it. The location of the second

well corresponds to the propagation of the initial field peak

(r = 2x). Finally, Fig. 4 f shows the expansion is uniform

in the y − z plane.

4 Summary

We studied 3D PIC simulations of particle acceleration driven

by PFA with two different initial magnetic field configura-

tions. Acceleration by PFA is robust in both cases, without

showing any instability. The efficiency, however, strongly de-

pends on how strong magnetic field is and how it expands.

In the cylindrical case, we observe acceleration of particles

in the radial direction as well as the axial direction. Acceler-

ation in the axial direction is due to E × B force, whereas in

the radial direction is by PFA. Without support from external

ambient pressure, EM wave expands indefinitely with decay-

ing the field strength proportional to (ct)−2. Bifurcation in

the phase space occurs because the ponderomotive potential

well becomes too shallow to hold particles in it. Rayleigh-

Taylor instability occurs at the center, which does not affect

the particle acceleration by PFA.

In the torus case, the expansion of magnetic field is spher-

ical rather than cylindrical, and the magnetic field strength

decays with (ct)−3. As a result, the front potential well is

too weak to hold particles, and radial acceleration by PFA

is not as efficient as the cylindrical case. Another important

difference is the charge separation between electrons and

positrons, which may terminate the acceleration in the axial

direction.

Our simulation results show that hydrodynamical and/or

MHD simulations are not sufficient to understand the accel-

eration process in collapsar jets. We are planning to run more

realistic model to simulate the acceleration and radiation of

particles by jets.

Acknowledgements This research is partially supported by NASAGrant No. NAG5-9223, NSF Grant No. AST0406882, and LLNLcontract nos. B528326 and B541027. The authors wish tothank ILSA, LANL, B. Remington and S. Wilks for usefuldiscussions.

References

Chen, F.F.: Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press,New York (1984)

Berezinskii, V.S., Prilutskii, O.F.: A&A 175, 309 (1987)Kato, Y., Mineshige, S., Shibata, K.: ApJ 605, 307 (2004)Liang, E., Nishimura, K.: Phys. Rev. Lett. 92, 175005 (2004)Lynden-Bell, D.: MNRAS 341, 1360 (2003)MacFadyen, A.I., Woosley, S.E.: ApJ 524, 262 (1999)Nagataki, S.: ApJS 127, 141 (2000)Nagataki, S.: ApJS 551, 429 (2001)Nagataki, S., Kohri, K., Ando, S., Sato, K.: Astropart. Phys. 18, 551

(2003)Noguchi, K., Liang, E., Nishimura, K.: Nuovo Ciment C 028, 381 (2005)Woosley, S.E.: ApJ 405, 273 (1993)Yee, K.S.: IEEE Trans. Antennas Propag. 14, 302 (1966)

Springer


DOI 10.1007/s10509-006-9281-y


Simulating Poynting Flux Acceleration in the Laboratorywith Colliding Laser Pulses

Edison Liang

Received: 9 May 2006 / Accepted: 21 November 2006C© Springer Science + Business Media B.V. 2007

Abstract We review recent PIC simulation results which

show that double-sided irradiation of a thin over-dense

plasma slab with ultra-intense laser pulses from both sides

can lead to sustained comoving Poynting flux acceleration

of electrons to energies much higher than the conventional

ponderomotive limit. The result is a robust power-law elec-

tron momentum spectrum similar to astrophysical sources.

We discuss future ultra-intense laser experiments that may

be used to simulate astrophysical particle acceleration.

Keywords Electron acceleration . Laser-plasma

interaction . Laboratory astrophysics

1 Introduction

Most high energy astrophysical sources (pulsars, blazars,

gamma-ray bursts, supernova remnants) emit a simple power-

law spectrum in the X-gamma-ray range. The most common

observed photon index lies in the range 2–3, which trans-

lates into an electron momentum index of 3–5 for optically

thin radiation (Rybicki and Lightman, 1979). The most pop-

ular current models for astrophysical particle acceleration

are shock acceleration (first-order Fermi), diffusive wave

acceleration, and Poynting flux acceleration by large-scale

electromagnetic fields. In earlier work (Liang et al., 2003)

we demonstrated that Poynting flux acceleration driven by

electromagnetic-dominated outflows (Liang et al., 2003) nat-

urally produces robust power-law relativistic electron spec-

tra. Poynting flux acceleration of e+e− plasmas is especially

relevant to gamma-ray bursts and pulsar winds. It is therefore

highly desirable to study particle acceleration in the labora-

E. LiangRice University, Houston, TX 77005-1892 USAe-mail: [email protected]

tory that may mimic or at least shed new light on Poynting

flux acceleration in astrophysics.

Recent advances in ultra-intense short-pulse lasers (ULs)

(Mourou et al., 1998; Ditmire, 2003) open up new fron-

tiers on particle acceleration by ultra-strong electromagnetic

(EM) fields in plasmas (Lontano et al., 2002). However, most

conventional laser acceleration schemes (e.g. laser wakefield

accelerator, plasma wakefield accelerator, plasma beat-wave

accelerator, free wave accelerator, see Tajima and Dawson,

1979; Sprangle et al., 1990; Kawata et al., 1991; Hussein et

al., 1992; Esarey et al., 1996; Woodworth et al., 1996; Pukhov

et al., 1997; Malka, 2002) involve the propagation of lasers

in an underdense plasma (ωpe = (4πne2/me)1/2 < ωo =2πc/λ, λ = laser wavelength, n = electron density). In such

schemes the acceleration gradient (energy gain/distance)

(Esarey et al., 1996; Malka, 2002) and energetic particle

beam intensity are limited by the underdense requirement.

They also do not produce a power-law electron spectrum.

Here we review PIC simulation results of a radically differ-

ent concept: comoving acceleration of overdense (ωpe > ωo)

plasmas using colliding UL pulses. In this case the acceler-

ation gradient and particle beam intensity are not limited by

the underdensity condition. This colliding laser pulses ac-

celerator (CLPA) concept may have important applications

to laboratory astrophysics since CLPA naturally produces

a power-law electron spectrum, similar to the high energy

spectra of observed astrophysical sources. Most other laser

acceleration schemes produce either exponential or quasi-

monoenergetic electron momentum distributions.

2 Colliding laser pulses accelerator

Figure 1 shows the basic idea of CLPA. Two linear polar-

ized intense laser pulses with aligned B vectors irradiate a

Springer


thin

slab of

e+e-

plasma2 opposite EM pulses

B B

Fig. 1 Schematic diagram showing the CLPA concept

thin overdense plasma slab from opposite sides. They com-

press the slab until it becomes thinner than 2 relativistic

skin depths. At that point the laser pulses “tunnel through”

and capture the surface electrons as they reemerge at the

far side of the slab. Due to plasma loading the laser pulses

slow down and stay in phase with the fastest particles, and

accelerate them continuously with self-induced comoving

J × B forces. Figure 2 shows the PIC simulation of two lin-

early polarized plane half-cycle EM pulses with parallel B,

irradiating a thin e+e− slab from opposite sides (thickness

= λ/2, initial density no = 15ncr (critical density)). Cases

with nonparallel B are more complex and are still under in-

vestigation. Each incident pulse compresses and accelerates

the plasma inward (Fig. 1a), reaching a terminal Lorentz

factor of γmax ∼ (e/ωpe)2 ∼ 40. Only ∼10% of the inci-

dent EM amplitudes is reflected because the laser reflec-

tion front is propagating inward relativistically (Kruer et al.,

Fig. 2 Evolution of two linearly polarized plane EM pulses(I (λ/µm)2 = 1021 W/cm2, cτ = λ/2) irradiating an overdense e+e−plasma (no = 15 ncr, thickness = λ/2, kT = 2.6 keV) from oppositesides. We plot magnetic field By (medium), electric field Ez (light),current density Jz (dark) and px /mc vs. x (inset) at tωo/2π = (a)1.25,

(b)1.5, (c) 1.75; (d) Snapshots of px/mec vs. x (dots) for the right-moving pulse at tωo/2π = 2.5 (black), 5 (red), 10 (blue), 22.5 (green)showing power law growth of γmax ∼ t0.45. We also show the profilesof By (medium), Ez (light) at tωo/2π = 22.5 (from Liang, 2006)

Springer


Fig. 3 Results of two Gaussian pulse trains (λ = 1µm, I =1021 W/cm2, cτ = 85 fs) irradiating a e+e− plasma (no = 9ncr, thick-ness = 2λ/π , kT = 2.6 keV). (a) early By and no/ncr (B) profiles attωo = 0; (b) time-lapse evolution of log (px /mec) vs. logx for the right-moving pulse at tωo = (left to right) 180, 400, 800, 1600, 2400, 4000,4800 showing power-law growth of γmax ∼ t0.8; (c) evolution of elec-

tron energy distribution f (γ ) vs. γ showing the build-up of power-lawbelow γmax with slope ∼ −1: tωo = (left to right) 180, 400, 800, 2400,4800. (Slope = −1 means equal number of particles per decade of en-ergy), (d) plot of γ vs. θ (= |pz |/|px |) in degrees at tωo = 4800, show-ing strong energy-angle selectivity and narrow beaming of the mostenergetic particles (from Liang, 2006)

1975). As the relativistic skin depths from both sides start to

merge (Fig. 1b), the two UL pulses interpenetrate and tunnel

through the plasma, despite ωpe > 〈γ 〉1/2ωo. Such transmis-

sion of EM waves through an overdense plasma could not be

achieved using a single UL pulse, because there the upstream

plasma is snowplowed by the laser pressure indefinitely. As

the transmitted UL pulses reemerge from the plasma, they

induce new drift currents J at the trailing edge of the pulses

(Fig. 1c), with opposite signs to the initial currents (Fig. 1b),

so that the new J × B forces pull the surface plasmas out-

ward. We emphasize that the plasma loading which slows the

transmitted UL pulses plays a crucial role in sustaining this

comoving acceleration. For a given e/ωpe the higher the

plasma density, the more sustained the comoving accelera-

tion, and a larger fraction of the plasma slab is accelerated.

This unique feature distinguishes this overdense acceleration

scheme from other underdense schemes. As slower particles

gradually fall behind the UL pulses, the plasma loading of

the UL pulses decreases with time. This leads to continuous

acceleration of both the UL pulses and the dwindling pop-

ulation of trapped fast particles. The phase space evolution

(Fig. 1d) of this colliding laser pulses accelerator (CLPA)

resembles that of the DRPA discovered earlier (Liang et al.,

2003, 2004; Nishimura et al., 2004).

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Fig. 4 Conceptual experimentalsetup for the demonstration ofthe CLPA mechanism usingthree PW lasers

3 Acceleration by colliding Gaussian laser pulse trains

Figure 3 shows the results of irradiating an overdense e+e-

slab using more realistic Gaussian pulse trains (λ = 1µm,

pulse length τ = 85fs, Ipeak = 1021 Wcm−2). We see that

γmax increases rapidly to 2200 by 1.28 ps and 3500 by

2.56 ps, far exceeding the ponderomotive limit a2o/2 (∼360).

The maximum Lorentz factor increases with time accord-

ing to γmax(t) ∼ e∫

E(t) dt/mc. E(t) is the UL electric

field comoving with the highest energy particles. E(t) de-

creases with time due to EM energy transfer to the parti-

cles, plus slow dephasing of particles from the UL pulse

peak. This leads to γmax growth slower than linear and

γmax ∼ t0.8 (Fig. 2b). In practice, γmax will be limited by the

diameter D of the laser focal spot, since particles drift trans-

versely out of the laser field after t ∼ D/c. The maximum

energy of any comoving acceleration is thus < eEo D =6 GeV(I/1021 Wcm−2)1/2(D/100µm). The asymptotic mo-

mentum distribution forms a power-law with slope ∼ −1

(Fig. 2d) below γmax, distinct from the exponential distri-

bution of ponderomotive heating (Kruer et al., 1985; Wilks

et al., 1992; Gahn et al., 1999; Wang et al., 2001; Sheng et al.,

2004). A quasi-power-law momentum distribution is formed

below γmax since there is no other preferred energy scale be-

low γmax, and the particles have random phases with respect

to the EM field profile.

4 Proposed laser experiment

An experimental demonstration of the CLPA will require a

dense and intense e+e− source. (Cowan et al., 1999, 2000)

demonstrated that such an e+e− source can be achieved

by using a PW laser striking a gold foil. Theoretical works

(Liang et al., 1998; Shen et al., 2001) suggest that e+e−densities >1022 cm−3 may be achievable with sufficient laser

fluence. Such a high density e+e− jet can be slit-collimated

to produce a ∼ micron thick e+e− slab, followed by 2-sided

irradiation with opposite UL pulses. As an example, consider

UL pulses with τ = 80 fs and intensity = 1019 Wcm−2. We

need focal spot diameter D > 600µm for the pairs to remain

inside the beam for >1 ps. This translates into ∼1 KJ energy

per UL pulse. Such high-energy UL’s are currently under

construction at many sites (Ditmire, 2003). Figure 3 shows

the artist conception of such an experimental setup.

We have also performed simulations of CLPA using

electron-ion plasmas. Results so far suggest that as long the

e-ion slab is sufficiently thin and laser pulses sufficiently in-

tense, so that the electrons can be compressed to less than two

relativistic skin depths before the lasers are reflected, the elec-

trons are accelerated by the reemerging pulses similar to the

e+e− case. However the ions lag behind the electrons due to

their inertia and are accelerated only by the charge-separation

electric field. The late-time partition between electron and

ion energies depends on the plasma density and laser inten-

sities. Note that CLPA is insensitive to the relative phases of

the two pulses. If one pulse arrives first it simply pushes the

plasma toward the other pulse until it hits. Then both pulses

compress the slab together with the same final results.

Acknowledgements EL was partially supported by NASA NAG5-9223, LLNL B537641 and NSF AST-0406882. He thanks Scott Wilksfor help with running ZOHAR and the graphics of Fig. 4, and BruceLangdon for providing the ZOHAR code.

References

Cowan, T.E. et al.: Laser Part. Beams 17, 773 (1999)Cowan, T.E. et al.: Phys. Rev. Lett. 84, 903 (2000)Ditmire, T. (ed.): SAUUL Report, UT Austin (2003)Esarey, E., Sprangle, P., Krall, J., Ting, A.: IEEE Trans. Plasma Sci. 24,

252 (1996)

Springer


Gahn, C. et al.: Phys. Rev. Lett. 83, 4772 (1999)Hussein, M.S., Pato, M.P., Kerman, A.K.: Phys. Rev. A 46, 3562 (1992)Hussein, M.S., Pato, M.P.: Phys. Rev. Lett. 68, 1992 (1991)Kawata, S., Maruyama, T., Watanabe, H., Takahashi, I.: Phys. Rev. Lett.

66, 2072 (1991)Kruer, W.L., Valeo, E.J., Estabrook, K.G.: Phys. Rev. Lett. 35, 1076

(1975)Kruer, W.L., Estabrook, K.G.: Phys. Fluids 28, 430 (1985)Liang, E., Nishimura, K., Li, H., Gary, S.P.: Phys. Rev. Lett. 90, 085001

(2003)Liang, E., Nishimura, K.: Phys. Rev. Lett. 92, 175005 (2004)Liang, E. Phys. Plasmas in press (2006)Liang, E.P., Wilks, S.C., Tabak, M.: Phys. Rev. Lett. 81, 4887 (1998)Lontano, L., et al. (eds.): Superstrong Fields in Plasmas, AIP Conf.

Proc. No. 611 AIP, NY (2002)Malka, V.: AIP Conf. Proc. No. 611, p. 303, In: Lontano, M. et al. (eds.)

AIP, NY (2002)

Mourou, G.A., Barty, C.P.J., Perry, M.D.: Phys. Today 51(1), 22(1998)

Nishimura, K., Liang, E.: Phys. Plasmas 11 (10) (2004)Pukhov, A., Meyer-ter-Vehn, J.: Phys. Rev. Lett. 79, 2686

(1997)Rybicki, G., Lightman, A.P.: Radiative Processes in Astrophysics, Wi-

ley, NY (1979)Shen, B., Meyer-ter-Vehn, J.: Phys. Rev. E 65, 016405 (2001)Sheng, Z.M., Mima, K., Zhang, J., Meyer-ter-Vehn, J.: Phys. Rev. E 69,

016407 (2004)Sprangle, P., Esary, E., Ting, A.: Phys. Rev. Lett. 64, 2011 (1990)Tajima, T., Dawson, J.M.: Phys. Rev. Lett. 43, 267 (1979)Wang, P.X., et al.: App. Phys. Lett. 78, 2253 (2001)Wilks, S.C., Kruer, W.L., Tabak, M., Langdon, A.B.: Phys. Rev. Lett.

69, 1383 (1992)Woodworth, J.G., Kreisler, M.N., Kerman, A.K.: in: Tajima, T. (ed.),

The Future of Accelerator Physics. p. 378, AIP, NY (1996)

Springer


DOI 10.1007/s10509-006-9291-9


Three-Dimensional Particle Acceleration in ElectromagneticDominated Outflows with Background Plasma and Clump

Koichi Noguchi · Edison Liang


Abstract The effect of background plasma on particle ac-

celeration via Poynting fluxes is studied in 3D PIC simu-

lation of electron-positron and electron-ion plasmas. When

a strongly magnetized ejecta at the center expands to low-

temperature electron-positron ambient plasma background

and a low-density clump, electromagnetic wave front ac-

celerates particles in the background and clump, and cap-

tures them in the Ponderomotive potential well. We do not

observe any instability, and the momentum distributions of

background and clump form a power law of slope close to

−1.5 with a sharp peak in the middle. When an ejecta expands

to the ion-electron interstellar medium (ISM), the accelera-

tion via Poynting flux is severely damped due to the charge

separation.

Keywords Gamma ray bursts . PIC . Numerical .

Relativity

1 Introduction

Gamma-ray bursts (GRBs) are the most luminous physi-

cal phenomenon in the universe, whose mechanism is still

unknown. There are two competing paradigms for the ori-

gin of the prompt GRB emissions: hydrodynamic internal

shocks (Meszaros, 2002; Piran, 2000) versus Poynting fluxes

(Lyutikov and Blackman, 2001). Both pictures require the

rapid and efficient acceleration of nonthermal electrons to

high Lorentz factors in moderate magnetic fields to radiate

gamma-rays. In the hydrodynamic internal shock scenario,

shock wave energy is mainly transferred to ions, resulting

emission with low energy peak, whereas in the Poynting flux

K. Noguchi () · E. LiangRice University, Houston, TX 77005-1892, USAe-mail: [email protected]

scenario, long-wavelength electromagnetic (EM) energy can

be directly converted into gamma-rays using the electrons or

electron-positron pairs as radiating agents.

Recent large-scale 3D PIC simulations (Hededal and

Nishikawa, 2005; Nishikawa et al., 2006) shows that rela-

tivistic jets propagating through a weakly or nonmagnetized

ambient plasma exite the Weibel instability, and that acceler-

ated electron jet in the electron-ion jet has a significant hump

above a thermal distribution, whereas electron-positron jet

does not. However, the maximum γ is around 10 in both

cases, and acceleration mainly occurs in the perpendicular

direction relative to the shock wave propagation.

The recent 2D PIC simulations (Liang et al., 2003; Liang

and Nishimura, 2004), which is particularly relevant to the

Poynting flux scenario of GRBs, shows that intense EM

pulses imbedded in an overdense plasma (EM wavelength

λ ≫ plasma skin depth c/ωpe) capture and accelerate par-

ticles via sustained in-phase Lorentz forces when the EM

pulses try to escape from the plasma. Such Poynting flux

may originate as hoop-stress-supported magnetic jets driven

by strongly magnetized accretion onto a nascent blackhole,

or as transient millisecond magnetar winds, in a collapsar

event (Zhang et al., 2003) or in the merger of two compact

objects (Ruffert and Janka, 2003).

Liang and Nishimura (2004) showed that the Poynting flux

acceleration (PFA) reproduces from first-principles many of

the unique features of GRB pulse profiles, spectra and spec-

tral evolution, and Noguchi et al. (2005) recently showed

that the mechanism is robust even with the radiation damp-

ing force.

In this article we report 3D PIC simulations of particle

acceleration driven by Poynting flux with low-temperature

background ambient medium and low-density clump with

newly developed 3D PIC code, and we show the power spec-

trum and radiation power strength from each particle.

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2 Initial setup of the simulation

We use the 3D explicit PIC simulation scheme based on the

Yee algorithm (Yee, 1966). Spatial grids for the fields are

uniform in all directions, x = y = z = c/ωpe, where

ωpe is the electron plasma frequency. The simulation domain

is −600x ≤ x ≤ 600x , −5y ≤ y ≤ 5y and −5z ≤z ≤ 5z with triply periodic boundary conditions.

Following Noguchi et al. (2005), the background magnetic

field B0 = [0, By, 0] is applied at the center of the simulation

box, −6x < x < 6x , −5y ≤ y ≤ 5y and −5z ≤z ≤ 5z, so that the magnetic field freely expands toward

the ambient plasma regions. The magnetic field strength By

is given by

By =

⎧

⎪

⎨

⎪

⎩

B0, |x | < 4x

B0[−|x |/(2x) + 3], 4x < |x | < 6x

0, otherwise

. (1)

We note that By has finite gradient at edges to avoid unphys-

ical particle acceleration.

In order to study the particle acceleration only from PFA,

initial electric field and current are assumed to be zero, which

seems rather too simple and artificial. However, our study

shows (Noguchi et al., 2005) that the acceleration is insen-

sitive to the initial field configuration, and the existence of

ordered Poynting vector is the key to accelerate particles via

PFA. The group velocity of EM wave front is very close to

the speed of light, and any hydrodynamical instability due to

the electric field or current non-uniformity can be ignored.

The most energetic particles are concentrated in the wave

front, and instabilities such as Weibel instability which may

occur in downstream does not affect the PFA mechanism.

The number density distribution of initial electron-

positron ejecta ρej is proportional to By in order to keep

the ratio ωpe/ce = 0.1, where ce is the electron cyclotron

frequency.

The clump and the ambient plasma consist of either

electron-positron or electron-ion. The clump is a 100x ×6y × 6z cuboid with density ρcl = 0.1ρej , whose center

is located at (−60x, 0, 0) so that the distance between the

front of the ejecta and the edge of the column is 4x . The re-

maining of the simulation box is filled by the ambient plasma

with density ρam = 0.01ρej .

The initial temperature of ejecta is assumed to be a

spatially uniform relativistic Maxwellian, kB Te = kB Tp = 1

MeV, where the subscripts e and p refer to electrons and

positrons. The temperature of the clump and the ambient

plasma is also uniform Maxwellian with kB Te = kB Tp,i =100 eV.

3 Results

First, we study the electron-positron background case. Fig-

ure 1 shows the spatial distribution of particles at tce = 650

and the phase plot at tce = 12000. The color of each parti-

cle in Fig. 1a represents the magnitude of estimated radiation

damping force using the relativistic dipole formula (Rybicki

and Lightman, 1979)

〈P〉 = 2e2

3m2c3(F2

‖ + γ 2 F2⊥), (2)

Fig. 1 The spatial distribution of particles in the ejecta (top), ambientmedium (middle) and clump (bottom) at tce = 650, and the phase plotof particles at tce = 12000 with Px − x (top), Py − x (middle) andPz − Px (bottom). The color of each particle in the left panel repre-

sents the magnitude of estimated radiation damping power 〈P〉. In theright panel, blue dots represent ejecta particles, green clump, and redambient, respectively

Springer


where F‖ and F⊥ are the parallel and perpendicular

components of the force with respect to the particle’s

velocity.

As the ejecta expands, electric field is automatically gener-

ated in the z direction, expanding the clump in the z direction.

Particles are also accelerated in the direction of the Poynting

vector (positive x direction for x > 0 and negative for x < 0),

due to the ponderomotive force. At tce = 12000, acceler-

ation by PFA still continues, and the clump is compressed

into a thin layer co-moving with the wave front. Figure 1b

shows that the highest γ in the ejecta is around 250, whereas

γ ≃ 100 in the background and the clump. As we mentioned,

there is no acceleration in the y direction, and the momentum

distribution in the y direction does not change. There is no

Fig. 2 The phase plot of electrons with ISM plasma (x > 0) and with ion-electron clump and ISM (x < 0) at tce = 10000 with Px − x (top),Py − x (middle) and Pz − Px (bottom). The meaning of colors is the same as Fig. 1

0 0.5 1 1.5 2 2.510

0

101

102

103

104

105

Log10

(E/mc2)

f(E

)

Ejecta NCEjecta CAmbient NCAmbient CClump

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.810

0

101

102

103

104

Log10

(E/mec

2)

Log

10(N

)

Ejecta NCAmbient NC Ejecta C Ambient C Clump

(b)

Fig. 3 The power spectrum of electrons in the electron-positron case (a) and the ion-electron case (b)

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charge separation in the x direction, and no instability occurs

in front of the EM pulse.

Figure 2 shows the phase plot of electrons in the ion-

electron clump and ambient plasma case at tce = 3000.

Different from the electron-positron case, the acceleration

by the PFA is strongly reduced by the charge separation.

Especially with the clump (x < 0), the initial EM field energy

is too weak to accelerate ions, and electrons are bounced

back to the center. However, electrons in the wavefront (x ≃±300) are still captured by the pontderomotive force well,

and accelerated by PFA.

Next, we compare the power spectrum of electrons in both

runs at tce = 3000. Figure 3a shows the electron-positron

case and Fig. 3b shows the electron-ion case with the clump

(C) and without the clump (NC). In the electron-positron

case, the existence of clump does not affect the acceleration

of the ejecta. Particles in both clump and ambient plasma get

acceleration, making a sharp peak at E = 1.2mc2, and form

a power law of slope close to −1.5. The ion-electron case

shows that the energy peak of the ambient plasma shifts to the

lower, and the acceleration of the ejecta is severely reduced

because of the charge separation between ions and electrons

even though the ejecta consists of electrons and positrons.

The power spectrum for the clump and background plasma

sharply drops around E = 1.3mc2, indicating that the electric

field by the charge separation slows down the accelerated

particles.

4 Summary

We studied the effect of electron-positron and ion-electron

background ambient plasma on particle acceleration via

Poynting fluxes. With electron-positron ambient plasma, the

acceleration mechanism is still robust in the electron-positron

background case, and particles in the background and clump

are also accelerated. With the ion-electron case, however, the

acceleration is severely suppressed due to the charge separa-

tion in the background and clump plasma. If the density of

interstellar medium near the ejecta is more than 10%, higher

initial magnetic field energy ωpe/ce ≫ 1 is required to cre-

ate high energy tail by PFA.

The advantage of PFA compared with the internal shock

acceleration scenario is the efficient energy transfer to high

energy particles, and less bulk heating. High energy tail of

GRBs requires gamma > 100 or more, which can be ex-

plained by the energetic particles accelerated by PFA, but not

by the internal shock. When electron-positron and electron-

ion plasma coexist, charge separation between ions and elec-

trons decelerates the acceleration. Internal shock acceleration

may take place if the initial plasma temparature is so low that

ion density is much higher than positron density, or the initial

plasma is weakly or not magnetized. We are currently work-

ing on simulations of the electron-positron with low-density

electron-ion plasma in the ejecta and background.

Longer timescale simulations are required to show the

final power distribution of particles and resulting radiation

spectrum, which remains as a future problem.

Acknowledgements This research is partially supported by NASAGrant No. NAG5-9223, NSF Grant No. AST0406882, and LLNL con-tract nos. B528326 and B541027. The authors wish to thank ILSA,LANL, B. Remington and S. Wilks for useful discussions.

References

Birdsall, C.K., Langdon, A.B.: Plasma Physics via Computer Simula-tion. McGraw-Hill (1985)

Hededal, C.B., Nishikawa, K.-I.: ApJ 623, L89 (2005)Liang, E., Nishimura, K., Li, H., Gary, S.P.: Phys. Rev. Lett. 90, 085001

(2003)Liang, E., Nishimura, K.: Phys. Rev. Lett. 92, 175005 (2004)Lyutikov, M., Blackman, E.G.: MNRAS 321, 177 (2001)Meszaros, P.: Ann. Rev. Astron. Astrophys. 40, 137 (2002)Nishikawa, K.-I., Hardee, P. E., Hededal, C. B., Fishman, G. J.: ApJ

642, 1267 (2006)Noguchi, K., Liang, E., Nishimura, K.: Nuovo Ciment C 028, 381 (2005)Piran, T.: Phys. Rep. 33, 529 (2000)Ruffert, M., Janka, H-Th.: Gamma-Ray Burst and Aftergrow Astron-

omy. In: AIP Conference Proceedings, vol. 662, p. 193 (2003)Rybicki, G.B., Lightman, A.P.: Raidiative Processes in Astrophysics.

Wiley-Interscience, New York (1979)Yee, K.S.: IEEE Trans. Antennas Propag. 14, 302 (1966)Zhang, W., Woosley, S.E., MacFadyen, A.I.: ApJ 586, 356 (2003)

Springer


DOI 10.1007/s10509-006-9234-5


3-D RPIC Simulations of Relativistic Jets: Particle Acceleration,Magnetic Field Generation, and Emission

K.-I. Nishikawa · C. B. Hededal · P. E. Hardee ·

G. J. Fishman · C. Kouveliotou · Y. Mizuno


Abstract We have applied numerical simulations and mod-

eling to the particle acceleration, magnetic field generation,

and emission from relativistic shocks. We investigate the

nonlinear stage of the Weibel instability and compare our

simulations with the observed gamma-ray burst emission. In

collisionless shocks, plasma waves and their associated in-

stabilities (e.g., the Weibel, Buneman and other two-stream

instabilities) are responsible for particle (electron, positron,

and ion) acceleration and magnetic field generation. 3-D rel-

ativistic electromagnetic particle (REMP) simulations with

three different electron-positron jet velocity distributions and

also with an electron-ion plasma have been performed and

show shock processes including spatial and temporal evolu-

tion of shocks in unmagnetized ambient plasmas. The growth

time and nonlinear saturation levels depend on the initial

jet parallel velocity distributions. Simulations show that the

K.-I. Nishikawa ()National Space Science and Technology Center, Huntsville,AL 35805; Department of Physics and Astronomy, The Universityof Alabama, Tuscaloosa, AL 35487e-mail: [email protected]

C. B. HededalDark Cosmology Center, Niels Bohr Institute, Juliane MariesVej 30, 2100 Copenhagen Ø, Denmark

P. E. HardeeDepartment of Physics and Astronomy, The University ofAlabama, Tuscaloosa, AL 35487

G. J. Fishman · C. KouveliotouNASA-Marshall Space Flight Center, National Space Science andTechnology Center, Huntsville, AL 35805

Y. MizunoNational Space Science and Technology Center/MSFC,Huntsville, AL 35805

Weibel instability created in the collisionless shocks accel-

erates jet and ambient particles both perpendicular and par-

allel to the jet propagation direction. The nonlinear fluctua-

tion amplitude of densities, currents, electric, and magnetic

fields in the electron-positron shocks are larger for smaller

jet Lorentz factor. This comes from the fact that the growth

time of the Weibel instability is proportional to the square

of the jet Lorentz factor. We have performed simulations

with broad Lorentz factor distribution of jet electrons and

positrons, which is assumed to be created by photon annihi-

lation. Simulation results with this broad distribution show

that the Weibel instability is excited continuously by the

wide-range of jet Lorentz factor from lower to higher val-

ues. In all simulations the Weibel instability is responsible

for generating and amplifying magnetic fields perpendicu-

lar to the jet propagation direction, and contributes to the

electron’s (positron’s) transverse deflection behind the jet

head. This small scale magnetic field structure contributes to

the generation of “jitter” radiation from deflected electrons

(positrons), which is different from synchrotron radiation in

uniform magnetic fields. The jitter radiation resulting from

small scale magnetic field structures may be important for

understanding the complex time structure and spectral evo-

lution observed in gamma-ray bursts or other astrophysical

sources containing relativistic jets and relativistic collision-

less shocks. The detailed studies of shock microscopic pro-

cess evolution may provide some insights into early and later

GRB afterglows.

Keywords Gamma-ray bursts . Relativistic jets . Weibel

instability . Particle acceleration . Magnetic field

generation . Methods: numerical . Particle-in-cell

simulation

Springer


1 Introduction

This report presents the study of collisionless relativistic

shocks associated with prompt gamma-ray bursts and their

afterglows. Using a 3-D relativistic particle-in-cell code we

have investigated the dynamics of relativistic shocks which

play an essential role for afterglows. There is now general

agreement among theorists that the prompt emission from a

gamma-ray burst (regardless of the central engine) requires

the formation of a highly relativistic, highly collimated jet

of out-flowing material that emits the observed prompt gam-

maray emission. Observational evidence suggests that GRBs

are produced by Doppler beamed and boosted emission from

shocks associated with a jet like flow. Within these general

scenarios it is proposed that synchrotron radiation from shock

accelerated particles in a shock-generated magnetic field pro-

duces the gamma-ray burst and produces the associated af-

terglows (e.g., Piran, 2005).

The Swift satellite is a multi-wavelength observatory de-

signed to detect GRBs and their X-ray and UV/optical af-

terglows. Thanks to its fast pointing capabilities, Swift is

revealing the early afterglow phase. The Swift X-Ray Tele-

scope (XRT) found that most X-ray afterglows fall off rapidly

for the first few hundred seconds, followed by a less rapid

decline (Tagliaferri et al., 2005). In the early afterglows of

GRB 050406 and GRB 050502b, XRT detected strong X-ray

flares: rapid brightening of the X-ray afterglow after a few

hundred seconds post-burst (Burrows et al., 2005). These

results suggest the existence of additional emission compo-

nents in the early afterglow phase besides the conventional

forward shock (blast wave) emission (Kobayashi et al., 2005).

Zhang et al. (2003) have discussed a clean recipe for con-

straining the initial Lorentz factor γ 0 of GRB fireballs by

making use of the early optical afterglow data alone. The

input parameters are ratios of observed emission quantities,

so that poorly known model parameters related to the shock

microphysics (e.g. ǫe, ǫB, etc.) largely canceled out. This

approach is readily applicable in the Swift era when many

early optical afterglows are expected to be regularly caught.

This data has been combined with other information such

as ℜB ≡ (ǫrB/ǫ

f

B )1/2 where superscripts r and f represent the

reverse and forward shock region and σ , the ratio between

the electromagnetic energy flux and the particle energy flux.

σ is closely related to the initial magnetization of the outflow

(Zhang and Kobayashi, 2005). The result is that this method

has provided, for the first time, information about the mag-

netic content of the ejecta. Such information about the initial

Lorentz factor of the fireball and whether the central engine

is strongly magnetized are helpful for the identification of

the GRB prompt emission site and mechanism, which are

currently uncertain (e.g. Zhang and Meszaros, 2002).

In collisionless shocks, plasma waves and their associated

instabilities (e.g., the Weibel, Buneman and other twostream

instabilities) are responsible for particle (electron, positron,

and ion) acceleration and magnetic field generation. Three-

dimensional relativistic particle-in-cell (PIC) simulations

have been used to study the microphysical processes in rela-

tivistic shocks. Recent PIC simulations using counterstream-

ing relativistic jets show that rapid acceleration is provided

in situ in the downstream jet, rather than by the scattering of

particles back and forth across the shock as in Fermi accel-

eration (Silva et al., 2003; Frederiksen et al., 2004; Hededal

et al., 2004; Hededal and Nishikawa, 2005; Nishikawa et al.,

2003, 2005, 2006a; Medvedev et al., 2005). Three recent in-

dependent simulation studies have now confirmed that the

relativistic counter-streaming jets excite the Weibel insta-

bility (Weibel 1959). The Weibel instability generates cur-

rent filaments and associated magnetic fields (Medvedev and

Loeb, 1999; Brainerd, 2000; Pruet et al., 2001; Gruzinov,

2001), and accelerates electrons (Silva et al., 2003; Frederik-

sen et al., 2004; Hededal et al., 2004; Hededal and Nishikawa,

2005; Jaroschek et al., 2005; Spitkovsky, 2006; Nishikawa

et al., 2003, 2005, 2006a). The current filaments and associ-

ated magnetic fields produced by the Weibel instability form

the dominant structures in a relativistic collisionless shock.

The growing current filaments generate highly nonuniform

small-scale transverse magnetic fields around the current fil-

aments. The “jitter” radiation to be expected from deflected

electrons has different properties than synchrotron radiation

(Medvedev, 2000; Medvedev, 2006; Fleishman, 2006), and

may explain the complex time evolution and/or spectral struc-

ture in gamma-ray bursts (Preece et al., 1998; Preece et al.,

2002). Rapid particle acceleration perpendicular and paral-

lel to the jet propagation direction accompanied by the non-

linear development of the filamentary structures cannot be

characterized as Fermi acceleration.

2 Simulations with 3-D remp code

Four simulations were performed using an 85×85×640 grid

with a total of 380 million particles (27 particles/cell/species

for the ambient plasma) and an electron skin depth, λce =c/ωpe = 9.6, where ωpe = (4πe2ne/me)1/2 is the electron

plasma frequency and is the grid size. In all simulations,

jets are injected at z = 25 in the positive z direction. Radi-

ating boundary conditions were used on the planes at z = 0,

zmax. Periodic boundary conditions were used on all other

boundaries (Buneman, 1993). The ambient and jet electron-

positron plasma has mass ratio me/mp ≡ me−/me+ = 1. The

ion-electron mass ration is m i = me = 20. The electron ther-

mal velocity in the ambient plasma is vth = 0.1c where c is

the speed of light.

The electron number density of the jet is 0.741nb where nb

is the ambient electron number density. The jet makes con-

tact with the ambient plasma at a 2D interface spanning the

computational domain. Here the dynamics of the propagating

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jet head and shock region is studied. Effectively, we study a

small portion of a much larger shock.

We have simulated three different initial distributions of

jet electrons (positrons). Relativistic jets are injected into

an unmagnetized ambient electron-positron plasma (initially

σ = 0). Two cases have narrow distributions of jet electrons

(positron) with (γ V|| = 5, and 15). The other case mimics

cold jet electrons and positrons created by photon annihila-

tion (4 < γ V|| < 100) (Meszaros et al., 2001; Ramirez-Ruiz

et al., 2006). For all three cases the temperature of jet parti-

cles is very cold (0:01c in the rest frame) (Nishikawa et al.,

2006b). In order to examine the dependence of theWeibel in-

stability on the jet particle species, one case with electron-ion

jet (γ v|| = 5) has been simulated (Nishikawa et al. 2006a).

Current filaments (Jz) resulting from development of the

Weibel instability behind the jet front are shown in Fig. 1

at time t = 59.8/ωpe. If the density of the ambient plasma

(ISM) is 1 cm−3, the electron plasma frequency ωpe/2π

is about 9 × 103 Hz. Therefore this simulation time corre-

sponds to 6.64 ms. The estimated radiation loss time t1/2 ∼6πm3c

µ0e4 B2γ0(Eq. 2.19 in Hededal, 2005) is at least 102 times

larger than the simulation time. Electrons are accelerated by

Fig. 1 2D images in the x–z plane at y = 43 for the electron-ion (a)electron-positron (b, c, and d) jet injected into an unmagnetized ambientelectron-positron plasma at t = 59.8/ωpe. The colors indicate the z-component of current density with Jz,x represented by arrows for γ v|| =5(a, b) and 15(c), and widely distributed pair injection 4<γv|| < 100 (d)

“radial” electric fields accompanying the current filaments.

The electrons are deflected by transverse magnetic fields (Bx ,

By) via the Lorentz force: −e(v × B), generated by current

filaments (Jz), which in turn enhance the transverse mag-

netic fields (Weibel, 1959; Medvedev and Loeb, 1999). The

complicated filamented structures resulting from the Weibel

instability have diameters on the order of the electron skin

depth λce = 9.6. This is in good agreement with the predic-

tion of λ ≈ 21/4cγ1/2th /ωpe ≈ 1.188λce = 10 (Medvedev

and Loeb, 1999; Ramirez-Ruiz, Nishikawa, and Hededal,

2006). Here, γth ∼ 1 is a thermal Lorentz factor.

The x-component of magnetic field is plotted as a func-

tion of z (along the jets) for the four different cases in Fig. 2.

(a)

(b)

(c)

(d)

60 120 180 240 300 360 420 480 540 600

60 120 180 240 300 360 420 480 540 600

60 120 180 240 300 360 420 480 540 600

60 120 180 240 300 360 420 480 540 600

Z/

Z/

Z/

Z/

x

x

Bx

B x

Fig. 2 The x-component of the magnetic field in the x–y plane along thez-direction for the electron-ion injected into an unmagnetized ambientelectron-ion plasma (a), electron-positron (b, c, d), jet injected into anunmagnetized ambient electron-positron plasma at t = 59.8/ωpe γ v|| =5 (a, b) and 15 (c), and widely distributed pair injection 4 < γv|| < 100(d). Onedimensional cuts for (a) and (c) are at x/ = 38 and y/ = 33(blue), 43 (red), and 53 ( green), and cuts are separated by about anelectron skin depth

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One-dimensional cuts for (a) and (c) are at x/ = 38 and

y/ = 33 (blue), 43 (red), and 53 ( green), and cuts are sep-

arated by about an electron skin depth. TheWeibel instability

generates transverse magnetic fields due to the current fila-

ments shown in Fig. 1. It should be noted that Fig. 2a shows

the evolution of the ion Weibel instability. However, in order

to investigate its dynamics it requires a larger system with a

longer simulation time (Spitkovsky, 2006).

Figures 2b, c, and d show similar structures: near the jet

head a narrow region has a large magnetic field energy. These

regions with higher magnetic field may correspond to the

preshock magnetic field discussed in Li and Waxman (2006).

Behind that region the magnetic fields become smaller where

the current channels created by the Weibel instability merge

nonlinearly. The particles are strongly accelerated in the per-

pendicular direction and particles are accelerated and decel-

erated in the flow direction where the magnetic field energy

is largest, and nonlinearly saturated at the some distance be-

hind the jet front. In the case with γ v|| = 5, the saturation

level is the largest (Nishikawa et al., 2006a). However, the

magnetic fields dissipate strongly in the nonlinear region.

The region with higher magnetic fields is nearly 10 elec-

tron skin depth (100 ≈ 10λce) (Gruzinov, 2001). Since the

case with 4 < γv|| < 100 contains jet particles with higher

Lorentz factors, the saturation level is higher in the nonlin-

ear dissipation region behind the jet front of magnetic field

energy (Fig. 2d) (Ramirez-Ruiz et al., 2006). This is due to

sequential excitement of the Weibel instability by the lower to

higher Lorentz factor. The growth time of the weibel instabil-

ity is proportional to the Lorentz factor and γsh/ωpe = 17.2

for γsh = 5 (Medvedev and Loeb, 1999). The growth time

for γsh = 30 is around 42/ωpe, which is within the simu-

lation time tωpe = 58.9. It should be noted that in the jet

front (570 < Z/ 600) where the larger magnetic field

exists as shown in Fig. 2b, c, and d, jet electrons are less

accelerated in the perpendicular direction. The spatial and

temporal structure of the magnetic field associated with ac-

celerated electrons (positrons) produces different lightcurves

and spectra; this will be investigated in the near future.

3 Discussion

We have performed self-consistent, three-dimensional rel-

ativistic particle simulations of relativistic electron-ion

and electron-positron jets propagating into unmagnetized

electron-positron ambient plasmas (initially σ = 0) for a

longer time and a larger simulation system than our pre-

vious simulations (Nishikawa et al., 2003, 2005, 2006a) in

order to investigate the nonlinear stage of theWeibel insta-

bility. The main acceleration of electrons takes place in the

downstream region. Processes in the relativistic collisionless

shock are dominated by current filament structures produced

by the Weibel instability. This instability is excited in the

downstream region behind the jet head, where electron den-

sity perturbations lead to the formation of current filaments.

On average the nonuniform electric field and magnetic field

structures associated with these current filaments decelerate

the jet electrons and positrons, while accelerating the ambi-

ent electrons and positrons, and accelerate the jet and ambi-

ent electrons and positrons in the transverse direction. The

nonlinear region current channels generated by the Weibel

instability dissipate. Dissipation levels depend on the initial

jet electron parallel velocity distributions.

A key issue is the microscopic dynamics of the reverse

shocks which play a crucial role for the generation of early

afterglows (Zhang et al., 2003; Zhang and Kobayashi, 2005;

Kobayashi et al., 2005; Fan et al., 2005). We have examined

the microscopic dynamics of relativistic shocks with three

different velocity distributions in order to understand Swift

observations and models of GRB emission. The important

parameters are the initial Lorentz factors of relativistic jets

(γ ) and the ratio between the electromagnetic energy flux

and the particle energy flux (σ ). In this report we used σ0 = 0

and have investigated the magnetization of the GRB outflows

(ℜB ≡ (ǫrB/ǫ

f

B )1/2). Figure 2 shows the x-component of the

magnetic field along the jets. We found that the values of Bx

and the spatial and temporal evolution depend on the initial

jet Lorentz factors. They will provide different lightcurves

and spectra.

A self-consistent emission calculation based on the mo-

tion of electrons (positrons) in the simulation system has

been developed. The calculation has reproduced the spectral

structure of one GRB afterglow (Hededal, 2005; Hededal and

Nordlund, 2005). In order to calculate the radiation (jitter-

like) from the particles in the electromagnetic fields gen-

erated by the Weibel instability, the retarded electric field

from a single particle is Fourier-transformed and gives the

individual particle spectrum. The individual particle spectra

are added together to produce a total spectrum over a par-

ticular simulation time span (Hededal, 2005; Hededal and

Nordlund, 2005). In order to obtain lightcurves we can cal-

culate a spectrum over short time spans, ts, relative to the

longer simulation time span. The change in power in an en-

ergy band can then be followed from one short time span to

the next giving a light curve in the energy band. It should be

noted that in this calculation very long (large) simulations

are required using small time steps in order to increase the

upper frequency limit to the spectrum. Frequency resolution

is limited by the short time span (ω=1/ts) (Hededal, 2005).

We expect that lightcurves obtained in this way will provide

reasonable evolution of the higher energy bands.

Future investigation with this newly-developed method

(Hededal, 2005; Hededal and Nordlund, 2005) will pro-

vide self-consistent lightcurves, spectra of synchrotron/jitter

emission, spectral evolution, polarization as functions of the

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viewing angle and allow their systematic comparison with

early and later afterglows observed by Swift and GLAST.

This research (K.N.) is partially supported by the NSF

awards ATM-0100997, INT-9981508, and AST-0506719,

and the National Aeronautic and Space Administration con-

tract NASA-NNG05GK73G. P. Hardee acknowledges par-

tial support by a National Space Science & Technology

(NSSTC/NASA) cooperative agreement NCC8-256 and NSF

award AST-0506666. The simulations have been performed

on IBM p690 at the National Center for Supercomputing

Applications (NCSA) which is supported by the NSF.

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DOI 10.1007/s10509-006-9204-y


Spectral Features of Photon Bubble Models of UltraluminousX-ray Sources

Justin D. Finke · Markus Bottcher


Abstract The nature of Ultraluminous X-ray Sources – X-

ray sources which exceed the Eddington luminosity for a

∼ 10M⊙ black hole – remains a mystery. One possible ex-

planation is an inhomogeneous accretion disk around a solar

mass black hole where photon transport through radiation-

pressure dominated “photon bubbles” can lead to super-

Eddington accretion. While previous studies of this model

have focused primarily on its radiation-hydrodynamics as-

pects, here we explore some observational implications of

such a model with a Monte Carlo–Fokker Planck radiation

transport code.

Keywords Accretion . Accretion disks . Methods:

numerical . Radiative transfer . X-rays: binaries

1. Introduction

In recent years, many off-nuclear X-ray sources which ex-

ceed the Eddington luminosity for a ∼10 M⊙ black hole have

been discovered in nearby galaxies (Fabbiano, 1989; Colbert

and Mushotzky, 1999; Liu and Mirabel, 2005). Dubbed

Ultraluminous X-ray Sources (ULXs), their nature is one

of the outstanding mysteries of modern astrophysics. While

some can be identified as supernova remnants or background

Active Galactic Nuclei, most seem to be the result of accre-

tion from a high-mass star onto a compact object. If the ac-

cretion is Eddington-limited, the compact object would have

a mass in the range 50–103 M⊙ – so-called intermediate-mass

black holes (IMBHs; Makishima et al., 2000; Madhusudhan

J. D. Finke () · M. BottcherAstrophysical Institute, Department of Physics and Astronomy,Ohio University, Athens, OH 45701e-mail: [email protected]

et al., 2006). Formation of such objects, however, clearly

presents theoretical difficulties.

Several models for super-Eddington luminosities have

been proposed, including anisotropic X-ray emission

from relativistic beaming in the jets of microquasars

(microblazars; Georganopoulos et al., 2002; King et al.,

2001) and the “photon bubble” model which involves accre-

tion disks with low density, optically thin cavities through

which radiation can travel more freely, leading to super-

Eddington accretion rates – and hence, luminosities – that

do not blow away the disk (Begelman, 2002). Strong beam-

ing is now considered unlikely based on recent observations

of X-ray ionization of optical nebulae associated with some

ULXs (Pakull and Mirioni, 2003; Gutierrez, 2006); however,

the photon bubble remains a viable model for accretion–

related ULXs.

2. ULXs’ X-ray spectra

Can one distinguish between IMBH and photon bubble mod-

els with ULXs’ X-ray spectra? From an accreting IMBH one

would expect a disk blackbody spectrum with a temperature

kT ∼ 0.1–0.3 keV, and possibly a high energy component

from Comptonization in a hot, thermal coronal gas, or some

other mechanism. Radiation in photon bubble cavities could

be repeatedly Compton-reflected leading to strong fluores-

cence lines and ionization edges (Ballantyne et al., 2004)

that may be observable, assuming it is not overwhelmed by

other radiation sources such as the disk’s blackbody, which

should be near that which is observed for sub-Eddington

X-ray binaries, i.e., ∼1–3 keV, or smeared out by the multi-

ple Compton scattering within the bubble.

Distinguishing models by spectral modeling is nearly im-

possible for all but the highest quality ULX data sets, and

Springer


spectral fitting to these ULX spectra gives contradictory re-

sults. Roberts et al. (2005) found that 6 of the best-observed

ULXs were fit approximately equally as well with a cool

blackbody and hard power-law as with a soft power-law

and warm blackbody. Even more surprisingly, they found

that most spectra were fit best by a cool, kT ∼ .25 keV

and a warmer kT ∼ 2 keV blackbody, a fit which exhibits

considerable challenges for any physical model. One preva-

lent interpretation is currently that of a cool disk blackbody

around an IMBH with an optically thick (τ ∼ 10) corona

producing the hard feature; such a model has considerable

problems, such as how one could observe a disk blackbody

through an optically thick plasma. The observed spectra

could also be explained by a hot, inner ADAF and a cool

outer disk, in which case the cool blackbody temperature

would not have any obvious relation to the black hole mass.

Quasi-periodic oscillations (QPOs) have been observed for

two ULXs, M82 X-1 (Strohmayer and Mushotzky, 2003) and

Holmberg IX X-1 (Dewangan et al., 2006) indicating ULXs

originate from compact objects similar to those in galactic

black hole candidates. An Fe K line has been observed in M82

X-1 (Strohmayer and Mushotzky, 2003; Agrawal and Misra,

(2006)); absorption edges have been observed in the 0.1–1

keV range in M101 ULX–1 (Kong et al., 2004); and possible

edges at ∼ 0.7 keV and ∼8 keV, and an emission feature

at ∼6 keV have been seen in Holmberg IX X-1 (Dewangan

et al., 2006). It is hoped that with better predictions of the ex-

pected X-ray spectra from photon bubbles, particularly emis-

sion lines and ionization edges characteristic for these types

of models, the nature of ULXs will be revealed.

3. Photon bubble simulations

So far, investigations of photon bubbles in ULXs have fo-

cused on radiation–hydrodynamics (see, e.g., Ruszkowski

and Begelman, 2003; Turner et al., 2005) leaving the expected

spectral features essentially unexplored. We have examined

the observational signatures of these objects with the two–

dimensional Monte Carlo/Fokker–Planck code described in

Bottcher et al. (2003) and Bottcher and Liang (2001), which

uses the Monte Carlo method of Pozdnyakov et al. (1983) for

Compton scattering and the implicit Fokker–Planck method

of Nayakshin and Melia (1998) for evolution of the electron

distribution in a thermal two-temperature plasma.

With this code, we set up a low density region sandwiched

by two high density regions in a plane parallel geometry, rep-

resenting an inhomogeneous disk at 10RS (RS = 2G M/c2 ≈3 · 106 cm for a 10M⊙ black hole). The low density region

was divided into 24 homogeneous zones, 4 radial and 6 ver-

tical, in which the Fokker–Planck equation was implicitly

and independently solved. The high density regions were

represented by blackbody (and later, blackbody + Compton

reflection) spectra inserted at the upper and lower bound-

aries, accounting for Compton reflection of these surfaces.

Photons escaping the simulation boundaries were added to

an event file for later spectral extraction. The density of the

low density region was chosen so that τT ∼ 0.1, for a length

scale of ∼0.1RS .

The MC/FP simulation was run and the spectrum incident

on the upper and lower boundaries, representing the high

density regions, was calculated. To calculate the expected

Compton reflection, this spectrum was used as an input into

the latest version of XSTAR (Kallman and Bautista, 2001).

XSTAR was run in constant pressure mode; the pressure was

calculated from the specified density by the ideal gas law. Due

to XSTAR’s limitations, the density in the high density region

was chosen to be lower than the value resulting from our stan-

dard model specifications: ne = 1 · 1017cm−3; however, the

column density was chosen to be quite large, (N = 1 · 1024

cm−2), so that, although reflected emission will not originate

from a realistic depth, the reflected spectrum should still be

consistent with a high density (ne ∼1020 cm−3), consider-

ing the spectral features will be substantially smeared out in

the course of multiple Compton scatterings in the low den-

sity regions. The area of the high density region’s surface

was 5.6 · 1014 cm2; from this and the total luminosity, the

Table 1 Ionization edges: Equivalent widths and fits

Gaussian fit parameters

Element’s K Edge EW [eV] N [1033 erg s−1 keV−1] σ [eV] E0 [keV]

Be 15 12 20 0.24

B 9.5 7.0 30 0.38

C 31 24 50 0.55

N 14 18 40 0.72

O 99 110 60 0.93

Ne 34 40 70 1.45

Mg 40 21 130 2.08

Si 95 23 150 2.82

S 140 11 180 3.67

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101

100

101

Photon Energy [keV]

1031

1032

1033

1034

1035

1036

LE

[erg

s1

keV

1]

Be BC N

O Ne MgSi

S

−

−

−

Fig. 1 Resulting X-ray spectrum of the bubble simulation with a 0.5keV blackbody representing the high density regions (solid line). It isplotted with a 0.5 keV blackbody (dotted line), and 0.5 keV blackbody+ power law fit for the soft excess + Gaussian fits to ionization edges(dashed line). See the electronic edition for a color version of this figure.

ionization parameter (ξ = L/(n R2)) was calculated. Solar

metalicities were assumed.

An example output of such a simulation with a high den-

sity temperature of kT = 0.5 keV can be seen in Fig. 1. The

spectrum was fitted with an 0.5 keV blackbody continuum, a

power law representing the soft-excess, and Gaussian fits for

the spectral features, which are believed to be K shell ion-

ization edges for various elements. Equivalent Widths and

the fit parameters are listed in Table 1. The soft-excess is

produced by Compton scattering of photons from the low

density region by the high density region. This model suc-

cessfully reproduces the major spectral features of ULXs: the

soft-excess and the high energy blackbody. It also predicts

features (i.e., the ionizationedges) that may be seen in future

observations. Results of a parameter study will be published

in a forthcoming paper (Finke and Bottcher, 2006).

Acknowledgements This work was partially supported by NASAthrough XMM-Newton GO grant no. NNG04GI50G and INTEGRALtheory grant NNG05GK59G.

References

Agrawal, V.K., Misra, R.: Astrophys. J. 638, L83 (2006)Ballantyne, D.R., Turner, N.J., Blaes, O.M.: Astrophys. J. 603, 436

(2004)Begelman, M.C.: Astrophys. J. 568, L97 (2002)Bottcher, M., Jackson, D.R., Liang, E.P.: Astrophys. J. 586, 339

(2003)Bottcher, M., Liang, E.P.: Astrophys. J. 552, 248 (2001)Colbert, E.J.M., Mushotzky, R.F.: Astrophys. J. 519, 89 (1999)Dewangan, G.C., Griffiths, R.E., Rao, A.R.: in submission, astro-

ph/0602472 (2006)Fabbiano, G.: ARA&A, 27, 87 (1989)Finke, J.D., Bottcher, M.: in preparationGeorganopoulos, M., et al.: A&A 388, L25 (2002)Gutierrez, C.M.: Astrophys. J. 640, L17 (2006)Kallman, T., Bautista, M.: Astrophys. J. Suppl. 133, 221 (2001)King, A.R., et al.: Astrophys. J. 552, L109 (2001)Kong, A.K.H., et al.: The XXII Texas Symposium on Relativistic As-

trophysics, Stanford University (2004); Kallman, T., Bautista, M.:Astrophys. J. Suppl. 133, 221 (2001)

Liu, Q.Z., Mirabel, I.F.: A&A 429, 1125 (2005)Madhusudhan, N., et al.: Astrophys. J. submitted (astro-ph/0511393)

(2006)Makishima, K., et al.: Astrophys. J. 535, 632 (2000)Nayakshin, S., Melia, F.: Astrophys. J. Suppl. 114, 269 (1998)Pakull, M.W., Mirioni, L.: RevMexAA 15, 197 (2003)Pozdnyakov, L.A., et al.: Astrophys. Space Phys. Rev. 2, 18 (1983)Roberts, T.P. et al.: Proceedings of “The X-ray Universe 2005”, San

Lorenzo de El Escorial, Spain, (2005)Ruszkowski, M., Begelman, M.C.: Astrophys. J. 586, 384 (2003)Strohmayer, T.E., Mushotzky, R.F.: Astrophys. J. 586, L61 (2003)Turner, N.J., et al.: Astrophys. J. 624, 267 (2005)

Springer


DOI 10.1007/s10509-006-9280-z


Polychromatic X-ray Beam from the Acceleration of EnergeticElectrons in Ultrafast Laser-Produced Plasmas

Felicie Albert · Kim TaPhuoc · Rahul Shah · Frederic Burgy ·

Jean Philippe Rousseau · Antoine Rousse


Abstract Polychromatic beams of hard X-rays from ultra-

fast laser plasma interaction are studied. Just as in a conven-

tional synchrotron, electrons are accelerated and wiggled, but

on a much shorter scale of a few millimeters. By focusing a

50 TW CPA laser system (30 fs duration) onto a helium gas

jet, we obtained a polychromatic collimated beam (50 mrad)

of X-ray radiation in the keV range. In addition, its perfect

synchronization with the laser system, its ultrafast duration

(≃30 fs) and its brightness (up to 108 photons/shot/solid an-

gle at 0.1% BW) will make it applicable to both X-ray science

and backlighting to address laboratory astrophysics research

issues.

Keywords Ultrafast X-ray science . Laser-Plasma source .

Electron acceleration

PACS 52.59.Px, 52.50.Dg, 52.38.Ph, 52.25.Os

1 Introduction

Laser produced X-ray sources have been widely used for

a decade to probe dense fusion plasmas, jets and shocks re-

sulting from laser plasma interactions (Hammel et al., 1993).

Such X-ray sources, mainly used for backlighting in labora-

tory astrophysics experiments (Whitlock et al., 1984; Cole

et al., 1982), are generated by focusing an intense laser

onto a solid target, resulting in line and continuum emission

(Rousse and Rischel, 2001; Workman and Kyrala, 2001).

However, those X-ray sources radiate isotropically and it

makes the collection of photons harder to realize in back-

F. Albert () · K. T. Phuoc · R. Shah · F. Burgy · J. P. RousseauA. Rousse LOA-ENSTA-Ecole Polytechnique, Chemin de lahuniere, 91761 Palaiseau, Francee-mail: [email protected]

lighting experiments. In comparison, synchrotrons produce

beams of keV X-rays with a broad and continuous spectrum,

because the rays originate from relativistic electrons oscillat-

ing in periodic magnetic structures (undulators and wigglers)

(Attwood, 1999). Here, by implementing the concepts of syn-

chrotron mechanisms and using a plasma wiggler in a fully

laser based scheme, we show that it is possible to produce a

polychromatic (1–10 keV), low divergence (50 mrad) and ul-

trafast X-ray beam (TaPhuoc et al., 2005; Rousse et al., 2004).

When an ultraintense laser (I > 1018 W/cm2) is focused

onto a gaz, the ponderomotive force, proportional to the gra-

dient of light intensity, plows the electrons away from the

strong field regions, leaving a column free of electrons in the

wake of the laser pulse (Pukhov and Meyer-ter-vehn, 2002;

Whittum, 1992). Thus, due to longitudinal space charge sepa-

ration, electrons trapped in the back of the ion channel will be

accelerated up to relativistic energies of more than 100 MeV

(Malka et al., 2002; Faure et al., 2004). Because of the trans-

verse restoring force due to the ion bubble, the accelerated

electrons displaced from the cavity axis undergo oscillations,

called betatron oscillations (Kiselev et al., 2004; Esarey et

al., 2002; Kostyukov et al., 2003). Just as in a synchrotron, a

collimated beam of keV X-ray radiation will result from the

relativistic motion of the electrons.

2 Theory

In the case of laser based synchrotron radiation, the ion chan-

nel that serves as a plasma wiggler is created by the pondero-

motive force of the laser pulse, which expels the electrons

toward the low light intensity regions, leaving an ion column

in the wake of the light pulse. This occurs for a normal-

ized vector potential a0 greater than or on the order of unity,

when relativistic effects can no longer be neglected. The laser

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produced ion channel creates a restoring force, due to space

charge separation, that can be calculated from Gauss’ law

F = −meω2pr0/2 (1)

Here, m is the electron rest mass, r0 the radius of the ion

column and ωp the plasma frequency, defined by ωp =√

nee2/mǫ0, with ne the electronic plasma density, e the el-

ementary charge, and ǫ0 the vacuum permittivity. Since the

restoring force scales with the electronic density, and that

the electronic densities used in our experiments are on the

order of ne = 1019 cm−3, this yields a very high oscillation

frequency in the plasma wiggler and reduces the constraint

on the electron beam energy compared to conventional syn-

chrotrons. For small amplitude oscillations, the electron will

produce an harmonic motion at the fundamental betatron fre-

quency ωβ = ωp/√

2γ , where γ is the relativistic Lorentz

factor of the electron defined by γ = 1/

√

1 − v2

c2 . This elec-

tron undergoing betatron oscillations in the ion channel will

emit synchrotron radiation (Kiselev et al., 2004; Esarey et al.,

2002; Kostyukov et al., 2003). In the case of an electron

slightly displaced from the axis, the fundamental wavelength

of the radiation can be approximated by λ = λβ/2γ 2. In the

case of high amplitude oscillations, high harmonics will be

radiated, and just as in a synchrotron, this can be described

by a dimensionless parameter

K = γ kβr0 = 1.33 × 10−10√

γ ne[cm−3]r0[µm] (2)

that is similar to the role of the wiggler strength parame-

ter. Here r0 is the amplitude of the betatron oscillations and

kβ = 2π/λβ . If K becomes high so that K ≫ 1, the radiation

will be emitted in many harmonics within a narrow cone of

divergence θ ≃ K/γ , and the emission frequency is a func-

tion of K (Kiselev et al., 2004; Esarey et al., 2002; Kostyukov

et al., 2003). In that case, the spectrum becomes broadband

and quasi continuous. It can be described by the synchrotron

radiation spectrum function (Jackson, 2001)

S(ω/ωc) = ω

ωc

∫ ∞

ωωc

K5/3(x)dx (3)

Here ωc represents the critical frequency beyond which there

is negligible radiation at any angle. For frequencies below

ωc and up to ω ≃ 0.29ωc, the spectrum function increases

as ω1/3 and then drops exponentially to zero. For a relativis-

tic electron wiggled in an ion core, the critical frequency is

(Kiselev et al., 2004; Esarey et al., 2002; Kostyukov et al.,

2003)

ωc = 3

2γ 3

cr0k2β ≃ 5 × 10−24γ 2ne[cm−3]r0[µm]keV (4)

(a.u

)(a

.u)

(a.u

)

a

b

c

Fig. 1 Simulated on axis X-ray spectrum for (a) K = 0.18 (γ0 = 20,r0 = 0.1 µm), (b) K = 0.94 (γ0 = 20, r0 = 0.5 µm) and (c) K = 5.6(γ0 = 20, r0 = 3 µm). If K is smaller than unity, radiation is emittedat the fundamental frequency, if K becomes high so that K ≫ 1, thespectrum becomes broadband and quasi continuous

We have simulated the theoretical on axis X-ray spectrum

(Fig. 1) using a code that calculates the electron trajectories

using a Runge–Kutta algorithm to integrate the equation of

motion. In this calculation the electron plasma density, the

electron energy and the initial transverse position r0 are the

initial conditions. The electron spectrum can be either arbi-

trary or taken from experimental data. The ion cavity is cen-

tered on the laser propagation axis (where r0 = 0). Then the

Synchrotron function from Equation (3) is used to calculate

the radiation emitted by each particle along the oscillation

path.

3 Experimental setup

The experiment (Fig. 2) was performed in the “Salle Jaune”

of the Laboratoire d’Optique Appliquee, using a 50 TW

Sapphire doped with titanium (Ti:Al2O3) laser system based

on Chirped Pulse Amplification technology (CPA). It has a 40

nm broadband spectrum centered on λ0 = 820 nm and a 30

fs Full Width Half Maximum (FWHM) pulse duration. The

Springer


Fig. 2 Experimental setup for electron and X-ray beam characteriza-tion (top) and theoretical filters (with phosphor screen) transmissionsfor energies up to 20 keV (bottom)

laser can deliver energies up to 1.5 J on target with a linear

horizontal polarization. We used a 1 m focal length parabolic

mirror to focus the 55 mm diameter laser beam onto the edge

of a supersonic Helium gaz jet, which has been fully char-

acterized using a Mach-Zender Interferometer. It showed a

uniform and sharp edged density profile all along the jet.

Imaging of the focal plane shows a Gaussian intensity pro-

file of the laser with a beam waistw0 of 18µm which contains

50% of the total laser energy. Therefore, this system produces

a vacuum focused intensity on the order of 3 × 1018 W/cm2,

which corresponds to a normalised vector potential a0 of 1.2.

The electronic plasma density can be tuned from 1018 cm−3

to 5 × 1019 cm−3 by varying the backing pressure of the gaz

jet with a regulator. During the experiment, we characterized

the electrons accelerated above 40 MeV, by deviating them

onto a phosphor screen imaged with a visible CCD, with

permanent 1 T magnets placed in the path of the accelerated

particles. The magnetic spectrometer resolution is limited by

the dispersing power of the magnet and also by the electron

beam spatial quality. The resolution is therefore respectively

17 MeV and 6 MeV for 200 MeV and 100 MeV energies. The

X rays were observed in the forward direction on a phosphor

screen imaged with another visible CCD. Different sets of fil-

ters (Be, Al, Nb, and Sn) were along the propagation axis to

select different spectral bands (above 0.8 keV, 2 keV, 5 keV

and 10 keV respectively). This setup allowed us to make

simultaneous measurements of electron and X-ray spectra

(Fig. 2).

4 Results and discussion

The X-ray radiation was measured on axis with an average

divergence of 50 mrad at an electronic density ne = 1 × 1019

cm−3. The 500 µm Be window in front of the CCD selected

all X-ray photons above 3 keV. We also observed that the X-

ray divergence depends on the electronic density. It is a pos-

sible consequence of the variation in the betatron strength pa-

rameter K ∝ √ne as the beam divergence varies with K/γ .

Another feature of the betatron X-ray source is its size. We

have measured the radiation source size by using the shadow

of a razor blade acting as a knife edge (Fig. 3). The blade

and the X-ray CCD were placed respectively 13 cm and 2

m away from the nozzle. A geometrical relation gives then

a transverse source size of 13 µm, which is the same order

of magnitude than the laser focal spot. The spectral distri-

bution of the radiation was measured from 1 keV to 10 keV

by placing a first set of Be, Al, Sn, and Nb filters in front of

the detector. The spectral resolution was limited by the band-

widths of the filters. The spectrum decreases exponentially

from 1 to 10 keV. The total number of photons (integrated

over the bandwidths of the filters and over the divergence of

the x-ray beam) is found to be more than 108 photons (per

shot/solid angle at 0.1% BW).

Another unique feature of the x-ray beam is its intensity as

a function of the electron density of the plasma, also shown

on Fig. 4. It is found to be sharply peaked at ne = 1.1 × 1019

cm−3. Below this critical density, the x-ray signal rapidly van-

ishes mainly because the number of trapped electrons is too

low. This is confirmed in the experiment for which no elec-

trons were detected by the spectrometer. At larger densities

Fig. 3 Experimental setup and results for the source size characteriza-tion. A razor blade was placed 13 cm away from the source, in the pathof the X-ray beam. The detector recorded the images 2 m away fromthe source. Geometrical relations gave a source size of 13 µm

Springer


Fig. 4 Experimental (circles) and PIC simulation (squares) variationof x-ray intensity with electronic density. The input parameters used inthe simulation are the laser intensity (3 × 1018 W/cm2), the focal spotsize (18 µm) and the electronic plasma density (varied from 1018 cm−3

to 6 × 1019 cm−3)

the x-ray signal drops down and a plateau is reached. For

these experimental conditions, the resulting amplitude of the

plasma wave becomes too weak. The pulse must first be mod-

ulated and additional laser energy would be needed. As a re-

sult, the temperature of the electron beam decreases and its

divergence increases. In order to obtain a better description

of the X-ray properties, the laser-plasma interaction has been

simulated in this high laser intensity regime with a 3D PIC

Code (Pukhov, 2003) which has been modified to properly

model the synchrotron emission. The PIC simulations clearly

reproduce this experimental behaviour: a sharp increase of

the x-ray intensity followed by a smoother decrease of the

signal. Other laser plasma based mechanisms could poten-

tially produce polychromatic X-ray radiation. Here, unlike

in the case of non-linear Thomson scattering, the electrons

do not overlap with the light field, and experiments already

performed on the subject (TaPhuoc et al., 2003) necessitated

a normalized laser vector potential a0 = 6 to produce radi-

ation in the 100 eV range, hence a much higher laser in-

tensity would be needed to produce a keV beam. Relativis-

tic Bremsstrahlung could also produce a narrow divergence

polychromatic X-ray beam, but it would be much weaker

than what we observe at our electronic densities (Seltzer

and Berger, 1986). High Harmonic Generation (HHG) in

gaseous media can also coherently produce a bright beam

of quasi continuous light but only in the soft X-ray region

(down to 10 nm) (Tarasevitch et al., 2000; Brabec and Krausz,

2000). Recent work (Dromey et al., 2006) showed HHG

from solid targets where, unlike in gases, there is no the-

oretical prediction for a sharp cutoff in harmonic generation.

Wavelengths as short as 1.2 nm (850th order of 1,054 µm

600 fs fundamental laser light) have been obtained with this

process. However, a high intrinsic laser-pulse contrast ratio

(>1:1011) is required to observe the shortest wavelength har-

monics in this experiment while the betatron X-ray source

requires only a contrast ratio of 1:107 for our experimental

conditions.

5 Conclusion and perspectives

Laser based synchrotron radiation generates a broad-band

hard X-ray beam. It reproduces the concept of synchrotron

radiation in a plasma wiggler along a few milimeters, size of

the gas jet in which the electrons are produced, accelerated

and wiggled. It ensures a much more compact device than

conventional synchrotrons. Based on numerical simulations,

the X-ray pulse duration should be on the order of the 30

fs laser pulse duration. Nevertheless, this must be measured

experimentally. Moreover, the source is perfectly synchro-

nized with the laser system, which opens the way toward

pump-probe experiments at a femtosecond timescale. This

type of experiment was already performed with laser based

Kα sources, and has been a valuable improvement to un-

derstand the mechanisms of non thermal melting and phase

transitions (Rishel et al., 1997; Siders et al., 1999). The Be-

tatron X-ray source provides up to 108 photons/pulse/solid

angle/0.1%BW, which can be collected and focused onto a

sample by grazing incidence optics or X-ray lenses. More-

over, recent work showed that harder and brighter X-rays can

be expected from the betatron mechanism as it scales with

the electron energy that can be increased with higher laser

intensities and larger electron acceleration length (TaPhuoc

et al., 2005; Rousse et al., 2004). The main astrophysical re-

search issues that could potentially benefit from this source

are X-ray radiography and X-ray absorption to probe dense

plasmas studied in laboratory astrophysics. Hard X-rays can

indeed be used to transversely radiograph laser-driven shocks

in solid materials for the determination of equation of state

(EOS) (Koenig et al., 2005). Also, the use of short pulse

X-ray blacklighter sources is a powerful diagnostic to inves-

tigate dense plasmas in transient local thermodynamic equi-

librium (LTE) (Audebert et al., 2005). This type of plasma

can be found in various fields of fundamental and applied

research. The Betatron source has the advantage to offer a

broad white light continuum in the hard X-ray region (above

1 keV) without emission lines to probe dense plasmas studied

in laboratory astrophysics.

Acknowledgements Work supported by the European Community un-der Contract Nos. HPRI-CT-1999-00086, HPRI-CT-2000-40016, andHPRI-CT-1999-50004 (FAMTO project). F.A. also acknowledges thesupport from the HEDLA organizing committee.

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Koenig, M., Benuzzi-Mounaix, A., Ravasio, A., Vinci, T., Ozaki, N.,Lepape, S., Batani, D., Huser, G., Hall, T., Hicks, D., MacKin-non, A., Patel, P., Park, H.S., Boehly, T., Borghesi, M., Kar, S.,Romagnani, L.: Plasma Phys. Controlled Fusion 47, B441 (2005)

Kostyukov, I., Kiselev, S., Pukhov, A.: Phys. Plasmas 10, 4818 (2003)Lee, T., et al.: Chem. Phys. 299, 233 (2004)

Malka, V., Fritzler, S., Lefebvre, E., Aleonard, M.M., Burgy, F., Cham-baret, J.P., Chemin, J.F., Krushelnick, K., Malka, G., Mangles,S.P.D., Najmudin, Z., Pittman, M., Rousseau, J.P., Scheurer, J.N.,Walton, B., Dangor, A.E.: Science 298, 1596 (2002)

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Springer


DOI 10.1007/s10509-006-9279-5


Scalable Dynamics of High Energy Relativistic Electrons: Theory,Numerical Simulations and Experimental Results

T. Baeva · S. Gordienko · A. Pukhov


Abstract Similarity theory, which is necessary in order to

apply the results of laboratory astrophysics experiments to

relativistic astrophysical plasmas, is presented. The analyt-

ical predictions of the similarity theory are compared with

PIC numerical simulations and the most recent experimen-

tal data on monoenergetic electron acceleration in diluted

plasmas and high harmonic generation at overdense plasma

boundaries. We demonstrate that similarity theory is a reli-

able tool for explaining a surprisingly wide variety of lab-

oratory plasma phenomena the predictions of which can be

scaled up to astrophysical dimensions.

Keywords Similarity theory . Laser-plasma interaction .

Particle acceleration . X-ray generation

1 Introduction

The advent of chirped-pulse amplification (CPA) has opened

a new era of laser technology and laser applications (Strick-

land and Mourou, 1985). The CPA method has allowed

the construction of table-top amplifiers which can gener-

ate pulses with millijoule energies and femtosecond dura-

tions, leading to peak powers of several terrawatts (1 TW =1012 W, corresponding to the electric output of 1000 large nu-

clear power stations). With appropriate focusing, the pulses,

T. Baeva () · A. PukhovInstitut fur Theoretische Physik I, Heinrich-Heine-UniversitatDusseldorf, D-40225, Germanye-mail: [email protected]

S. GordienkoInstitut fur Theoretische Physik I, Heinrich-Heine-UniversitatDusseldorf, D-40225, Germany; L. D. Landau Institute forTheoretical Physics, Moscow, Russia

produced by these laser systems can create exotic conditions,

never before achieved in a laboratory: intensity 1020 W/cm2,

electric field 1011 V/cm, temperature 106 eV. Matter that is

exposed to these extreme conditions behaves in such a way

that gives a new insight into fundamental phenomena from

atomic, molecular and condensed matter physics (with char-

acteristic energies in the eV range), to nuclear physics, high

energy physics, astrophysics, and cosmology (with charac-

teristic energies of MeV and GeV) (Remington et al., 2000;

Takaba et al., 1999).

Cosmological and astrophysical applications of the new

laser technology (the so called “laboratory astrophysics”)

have inspired great interest mainly because of the follow-

ing reason. It is well known that the traditional astrophysical

research is based on either observation or theoretical model-

ing. Yet this approach lacks the ability to quantitatively test

models under experimental conditions where the initial and

final states are well characterized (Remington et al., 1999).

Therefore laboratory astrophysics is a new impetus for both

astrophysical and laser-plasma studies, since it allows prepar-

ing and controlling the initial states rather than make us rely

on uncontrollable observation results.

However the new laser technology not only opens new

opportunities for astrophysical research but also brings new

scientific challenges and problems. Indeed, astrophysical and

laboratory dimensions differ significantly. Consequently, a

key question is how one can re-scale laboratory-size experi-

mental results up to dimensions of astrophysical importance.

This scientific difficulty is neither new nor unique. Quite

analogous problems encounters for example the classical

(magneto-)hydrodynamics. One of the most powerful the-

oretical tools in such situations is the so called similarity

theory (Birkhoff, 1960; Sedov, 1993). The similarity allows

engineers to rescale the behavior of a physical system from

a laboratory acceptable size to a size of practical use.

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Yet not only laboratory astrophysics but also laser-plasma

interaction research is interested in developing effective

similarity theory. As a matter of fact such topical phenom-

ena as quasi-monochromatic electron acceleration in diluted

plasmas and high harmonic generation at overdense plasma

boundaries can be fully understood on the basis of the ultra-

relativistic similarity theory.

The aim of the present article is to present similarity theory

for laser-plasma interactions in the ultra-relativistic regime

and explain its application to the physical phenomena just

mentioned. Our choice of effects to discuss fell on the quasi-

monochromatic electron acceleration in diluted plasmas and

the high harmonic generation at overdense plasma bound-

aries not only because these spectacular phenomena have

found impressive numerical confirmation, but mostly be-

cause they have been recently observed experimentally and,

as a result, provide a powerful support for the applicability

of the ultra-relativistic similarity theory.

2 Ultra-relativistic similarity theory

Let us consider collisionless laser-plasma dynamics and ne-

glect the ion motion. The evolution of the electron distribu-

tion function f (t, r,p) is described by the Vlasov equation

(∂t + v · ∂r − e(E + v × B/c) · ∂p) f (t,p, r) = 0, (1)

where p = meγ v and the self-consistent fields E and B sat-

isfy the Maxwell equations.

We suppose that the laser pulse vector potential at the

time t = 0 short before entering the plasma is A(t=0) =a((y2 + z2)/R2, x/cτ ) cos(k0x), where k0 = ω0/c is the

wavenumber, R is the focal spot radius and τ is the pulse

duration. If one fixes the laser envelope a(r⊥, x), then the

laser-plasma dynamics depends on four dimensionless pa-

rameters: the laser amplitude a0 = max |ea/mec2|, the fo-

cal spot radius k0 R, the pulse duration ω0τ , and the plasma

density ratio ne/nc, where nc = meω20/4πe2 is the critical

density.

Now we are going to show that in the ultra-relativistic

regime when a20 ≫ 1, the number of independent dimension-

less parameters reduces to three: k0 R, ω0τ and S, where the

similarity parameter S is

S = ne

a0nc

. (2)

Let us introduce the new dimensionless variables

t = S1/2ω0t, r = S1/2k0r, p = p/meca0, (3)

A = eA

mec2a0

, E = S−1/2eE

mecω0a0

, B = S−1/2eB

mecω0a0

,

and the new distribution function f defined as

f = ne

(meca0)3f (t, p, r, a0, S, R, τ ), (4)

where R = S1/2k0 R and τ = S1/2ω0τ .

The normalized distribution function f is a universal one

describing the interaction of the given laser pulse with a fixed

initial plasma profile. It satisfies the equations

[∂t + v · ∂r − (E + (v × B)) · ∂p] f = 0, (5)

∇r · E = 4π (1 + ρ), ∇r · B = 0, (6)

∇r × B = 4π j + ∂t E, ∇r × E = −∂t B,

where v = p/

√

p2 + a−20 is the electron velocity, ρ =

−∫

f dp, j = −∫

v f dp and the initial condition for the

vector potential is

A(t = 0) = a((y2 + z2)/R, x/τ ) cos(S−1/2 x), (7)

with the slow envelope a such that max |a| = 1.

Equations (5, 6) together with the initial condition (7) still

depend on the four dimensionless parameters R, τ , S and a0.

However, the parameter a0 appears only in the expression for

the electron velocity. In the limit a20 ≫ 1 one can write

v = p/

√

p2 + a−20 ≈ v = p/ |p| (8)

Consequently, for the ultra-relativistic amplitude a20 ≫ 1, the

laser-plasma dynamics does not depend separately on a0 and

ne/nc. Rather, they converge into the single similarity pa-

rameter S.

The ultra-relativistic similarity means that for different in-

teraction cases with the same value of S, plasma electrons

move along similar trajectories. The number of these elec-

trons Ne, their momenta p, and the plasma fields scale as

p ∝ a0; Ne ∝ a0; (9)

φ,A,E,B ∝ a0 (10)

for ω0τ = const, k0 R = const and S = const.

The ultra-relativistic similarity is valid for arbitrary S-

values. The S parameter appears only in the initial condition

(7) so that S−1/2 plays the role of a frequency parameter since

it separates relativistically overdense plasmas with S ≫ 1

from underdense ones with S ≪ 1.

Notice that the similarity theory developed in this section

neglects the ion motion, therefore it is applicable only for

short laser pulses (compare with Ruytov and Remington,

2006)

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3 Scalings for laser wake-field acceleration(LWFA)

In this section we apply the ultra-relativistic similarity the-

ory developed in Section 2 to the special case of under-

dense plasma and consider laser wake-field acceleration,

particularly in the bubble acceleration regime (Pukhov and

Meyer-ter-Vehn, 2002). Here we present only the main

consequences of the similarity theory and their numer-

ical confirmation. Mathematically rigorous derivation of

these results and their heuristic consideration can be found

in Gordienko and Pukhov (2005), Pukhov and Gordienko

(2006).

Before starting with the scalings for the bubble accel-

eration regime let us first recall its main characteristics.

When a short relativistically intense laser pulse enters under-

dense plasma, it is able to expel all electrons behind it-

self. Thus a cavity (bubble), free of electrons but con-

taining positively charged ions, is generated and propa-

gates in the plasma just behind the pulse. The electrons

from the rear side of the bubble penetrate the cavity, get

trapped and accelerated up to high energies. A distinctive

feature of the bubble acceleration regime is that the spec-

trum of the high energy electrons is quasi-monochromatic

what was first predicted numerically (Pukhov and Meyer-ter-

Vehn, 2002) and then observed experimentally (Faure et al.,

2004).

On the basis of the ultra-relativistic theory two families

of scalings describing the bubble acceleration regime have

been obtained. The first family describes trapped and accel-

erated electrons in the bubble regime with the same values

of all of the ultra-relativistic similarity theory dimension-

less parameters S, ω0τ , k0 R. From the similarity theory pre-

sented in Section 2 (see for details Gordienko and Pukhov,

2005; Pukhov and Gordienko, 2006) follows that if one keeps

S = const, then both the number of trapped monoenergetic

electrons Nmono and the largest energy they can reach Emono

grow as the dimensionless vector potential a0 of the laser

pulse, i.e.

Nmono ∝ a0, Emono ∝ a0. (11)

In order to describe the second family we use the fact that for

the bubble acceleration regime S ≪ 1. In this case, S can be

considered as a small parameter and quite general scalings

for laser-plasma interactions can be obtained. It follows from

the theory that in the optimal configuration the laser pulse

has the focal spot radius kp R ≈ √a0 and the duration τ ≤

R/c. Here, kp = ωp/c is the plasma wavenumber and ω2p =

4πnee2/me is the plasma frequency.

The central result of the ultra-relativistic similarity the-

ory is that the bubble regime of electron acceleration is

stable, scalable and the scaling for the maximum energy

Emono of the monoenergetic peak in the electron spectrum

is

Emono ≈ 0.65 mec2

√

P

Prel

cτ

λ. (12)

Here, P is the laser pulse power, Prel = m2ec5/e2 ≈ 8.5 GW

is the natural relativistic power unit, and λ = 2πc/ω0 is the

laser wavelength. The scaling (12) assumes that the laser

pulse duration satisfies the condition cτ < R. The scaling

for the number of accelerated electrons Nmono in the mo-

noenergetic peak is

Nmono ≈ 1.8

k0re

√

P

Prel

, (13)

where re = e2/mec2 is the classical electron radius, and k0 =2π/λ. The acceleration length Lacc scales as

Lacc ≈ 0.7cτ

λZ R, (14)

where Z R = π R2/λ ≈ a0λ2p/4πλ is the Rayleigh length.

The parametric dependencies in the scalings (12)–(14)

follow from the analytical theory. The numerical pre-factors

are taken from 3D PIC simulations. These pre-factors may

change depending on the particular shape of the pulse enve-

lope. However, as soon as the envelope of the incident laser

pulse is defined, the pre-factors are fixed. The parametric

dependencies, on the other hand, are universal and do not

depend on the particular pulse shape.

To check the analytical scalings, we performed

3D particle-in-cell simulations with the code VLPL

(Pukhov, 1999). In the simulations, we use a circu-

larly polarized laser pulse with the envelope a(t, r⊥) =a0 cos(π t/2τ ) exp(−r⊥2/R2), which is incident on a plasma

with uniform density ne. We used grid steps hx = 0.07λ,

h y = hz = 0.5 λ, and 4 particles/cell.

First, we check the basic ultra-relativistic similarity with

S = const. We choose the laser pulse duration τ = 8 · 2π/ω0.

The laser radius is R = 8 λ, where λ = 2πc/ω0 is the laser

wavelength. The laser pulse aspect ratio is cτ/R = 1 in this

case.

We fix the basic similarity parameter to the value Si =10−3 and perform a series of four simulations with (i) ai

0 =10, ni

e = 0.01nc; (ii) aii0 = 20, nii

e = 0.02nc; (iii) aiii0 = 40,

niiie = 0.04nc; (iv) aiv

0 = 80, nive = 0.08nc. Taking the laser

wavelength λ = 800 nm, one can calculate the laser pulse

energies in these four cases: W i = 6 J; W ii = 24 J; W iii =96 J; W iv = 384 J. These simulation parameters correspond

to the bubble regime of electron acceleration (Pukhov and

Meyer-ter-Vehn, 2002), because the laser pulse duration τ is

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Fig. 1 Electron energy spectra obtained in the simulations (i)-(iv) (seetext). The control points 1–5 were taken after the propagation distancesL1 = 200λ, L2 = 400λ, L3 = 600λ, L4 = 800λ, L5 = 1000λ. Thespectra evolve similarly. The monoenergetic peak positions scale ∝ a0

and the number of electrons in a 1% energy range also scales ∝ a0 inagreement with the analytic scalings (9)

shorter than the relativistic plasma period√

a0ω−1p . We let the

laser pulses propagate the distance Lib = 1000 λ through the

plasma in the all four cases. At this distance, the laser pulses

are depleted, the acceleration ceases and the wave breaks.

Figure 1 (i)–(iv) shows the evolution of the electron en-

ergy spectra for these four cases. One sees that the energy

spectra evolve quite similarly. Several common features can

be identified. First, a monoenergetic peak appears after the

acceleration distance L ≈ 200 λ. Later, after a propagation

distance of L ≈ 600 λ, the single monoenergetic peak splits

into two peaks. One peak continues the acceleration towards

higher energies, while the other one decelerates and finally

disappears. Comparing the axes scales in Fig. 1, we conclude

that the scalings (9) hold with good accuracy.

At the end we want to emphasize that theoretical and nu-

merical results just presented are in agreement with the ex-

perimental data (Faure et al., 2004).

4 Similarity and high harmonic generation at plasma

boundaries

In this section we explain the physical picture of high har-

monic generation (HHG) at the boundary of over-dense

plasma with emphasis on its close relation to ultra-relativistic

similarity theory and present its numerical confirmation. De-

tailed derivation of the analytical results can be found in

Gordienko et al. (2004), Baeva et al. (2006) and Baeva (2005),

here we mainly concentrate on the physical origin of high

harmonic spectrum universality the roots of which lie in the

similarity properties of collisionless plasma dynamics.

Let us consider a short laser pulse of ultra-relativistic in-

tensity, interacting with the sharp surface of an overdense

plasma slab. Under the assumption that the incident laser

pulse is short we can neglect the slow ion dynamics and con-

sider only the electron motion.

Since the laser pulse is both ulra-short and ultra-intense

we can neglect the Coulomb collisions and related to them

resistivity. Indeed, for T ≤ mec2 the collision frequency is

ν ∝ ωpe

(

e2n1/3e

T

)3/2

ln, (15)

where ωpe is the plasma frequency, T is the electron tem-

perature and ln ≈ 15 is the Coulomb logarithm. Since the

laser pulse duration τ ∝ 2π/ω0, for T ∝ mec2 Equation (15)

gives rise to

τν ∝ 2π

ω0

ωpe ln

(

e2n1/3e

T

)3/2

≈ 10−5 ≪ 1 (16)

for a laser pulse of wavelength 800 nm and typical solid state

density (e2n1/3e ≈4 eV). Consequently on the time scale of the

laser we can neglect the Coulomb collisions and use colli-

sionless description by means of the relativistic Vlasov equa-

tion. Notice that for the case of interest the kinetic energy of

electrons is much larger than mec2 and the role of collisions

is even less.

Note that only the surface area of very hot plasma is rele-

vant to high harmonics generation. In spite of the rather high

density of this plasma layer, Coulomb collisions in it are neg-

ligible. The cold plasma below the hot plasma layer could also

be important if the “return current”, generated to guarantee

plasma quasineutrality, plays a significant role. However, the

similarity theory used for the theory of high harmonic gener-

ation (Section 2) demonstrates that the typical electric field in

the area where important physical processes take place scales

as the laser field (S = const). Since this field is very large,

even if collisions were taken into account, all electrons would

be deep in the run-away regime, in which resistivity is de-

scribed by kinetic effects in collisionless plasma. Therefore

the high harmonic generation at the boundary of overdense

plasma is treated by means of the Vlasov equation in what

follows.

The plasma electrons are driven by the laser light pres-

sure and the restoring electrostatic force connected to the

Coulomb attraction to the ions. As a consequence, the plasma

surface oscillates and the electrons gain normal momentum

components pn .

Since the plasma is overdense, the incident electromag-

netic wave is not able to penetrate it. In other words there is

electric current along the plasma surface. For this reason, the

momenta of the electrons in the skin layer have, apart from

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the component normal to the plasma surface, tangential com-

ponents pτ .

According to the relativistic similarity theory (see

Equation (9)), both the normal and tangential components

of electron momentum scale as the dimensionless electro-

magnetic potential: pn,pτ ∝ a0 if the similarity parameter

S is fixed. Consequently, there is a finite angle between the

full electron momenta and the plasma surface for most of

the times. An important point is that the typical value of this

angle does not decrease if a0 increases for S = const.

Since we consider a laser pulse with ultra-relativistic in-

tensity, the motion of the electrons is ultra-relativistic. In

other words, their velocity is approximately c. Though the

motion of the plasma surface is qualitatively different. Its

velocity vs is not ultra-relativistic for most of the times but

smoothly approaches c only when the tangential electron

momentum vanishes.

Using simple algebra one can demonstrate that the γ -

factor of the surface γs also shows specific behavior. It has

sharp peaks at those times for which the velocity of the sur-

face approaches c. Thus, while the velocity function vs is

characterized by its smoothness, the hallmarks of γs are its

quasi-singularities.

When vs reaches its maximum and γs has a sharp peak,

high harmonics of the incident wave are generated and can be

seen in the reflected radiation. Physically this means that the

high harmonics are due to the collective motion of bunches of

fast electrons moving towards the laser pulse (Baeva, 2005).

These harmonics have two very important properties.

First, their spectrum is universal: the exact motion of the

plasma surface can be very complicated, since it is affected

by the shape of the laser pulse and can differ for different plas-

mas. Yet the qualitative behavior of vs and γs at the vicinity

of the surface velocity maxima is universal and since high

harmonics are generated around these moments of time the

spectrum of the high harmonics does not depend on details

of the particular surface motion.

This observation leads to the conclusion (Gordienko et al.,

2004; Baeva, 2005) that the high harmonic spectrum contains

two qualitatively different parts: a power-law decay and an

exponential decay. In its power-law part the spectrum decays

as

In ∝ 1/n8/3, (17)

up to a critical harmonics number that scales as γ 3max, where In

is the intensity of the nth harmonic. Here γmax is the maximal

γ -factor of the point, where the component of the electric

field tangential to the surface vanishes.

The second important feature of the high harmonics is

that they are phase-locked. This observation is of particular

value, since it allows for the generation of attosecond and

even sub-attosecond pulses (Gordienko et al., 2004).

Fig. 2 Spectra of the reflected radiation for the laser amplitudes a0 =5, 10, 20. The broken line marks the universal scaling I ∝ ω−8/3

Fig. 3 Electron distribution function. The helix represents the electronsurface motion in the laser field. The reddish downward spikes stayfor the surface relativistic motion towards the laser. These spikes areresponsible for the zeptosecond pulse generation

In order to check our analytical results, we perform a num-

ber of 1d PIC simulations with the VLPL code (Pukhov,

1999). A Gaussian laser pulse with a = a0 exp(−t2/τ 2L ) was

incident onto a plasma layer with a step density profile. Fig-

ure 2 shows spectra of the reflected radiation for laser ampli-

tudes a0 = 5, 10, 20, duration ωτL = 4π and plasma density

Ne = 30Nc, which roughly corresponds to the solid hydro-

gen or liquid helium.

The log-log scale of Fig. 2 reveals the power-law scaling

of the spectral intensity In ∝ 1/n8/3. The critical harmonic

number nc, where the power-law scaling changes into the

exponential decay increases for higher laser amplitudes.

Let us take a closer look at the particular case a0 = 20 (the

red line a) in Fig. 2). In this case, the power-law spectrum

extends above the harmonic number 2000, and zeptosec-

ond pulses (1 zs = 10−21 s) can be generated. As one sees

from the electron distribution function f (t, x, px ), Fig. 3,

the maximum surface γ -factor γmax ≈ 25 is achieved at the

time t ≈ 6. The temporal profile of the reflected radiation is

shown in Fig. 4. When no spectral filter is applied, Fig. 4a,

a train of attosecond pulses is observed. However, when we

apply a spectral filter selecting harmonics above n = 300, a

train of much shorter pulses is recovered, Fig. 4b. Figure 4c

zooms to one of these pulses. Its full width at half maximum

is about 300 zs. At the same time its intensity normalized

to the laser frequency is huge (eEzs/mcω)2 ≈ 14 that would

correspond to the intensity Izs ≈ 2 × 1019 W/cm2.

Spectrum of high harmonics generated at an overdense

plasma boundary in the ultra-relativistic regime including

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Fig. 4 Zeptosecond pulse train: (a) temporal structure of the reflectedradiation; (b) zeptosecond pulse train seen after spectral filtering; (c)one of the zeptosecond pulses zoomed, its FWHM duration is about300 zs

about 850 harmonics with In ∝ n−p, p = 2.5(+0.2,−0.3)

has recently been observed experimentally (Dromey et al.,

2006), thus confirming the predictions of the ultra-relativistic

similarity theory. The scaling of the spectrum roll-over

predicted analytically as proportional to γ 3max was also

experimentally confirmed in a series of experiments produc-

ing keV-photons (Zepf, 2006).

In conclusion, the ultra-relativistic similarity theory pre-

sented in this article has proved to be efficient in explaining

laboratory plasma phenomena and provides a way to scale

them up to astrophysical dimensions.

References

Baeva, T., et al.: Phys. Rev. E 74, 1 (2006)Baeva, T.: Diploma thesis: “Attosecond phenomena in laser-condensed

matter interaction”, Dusseldorf University (2005)Birkhoff, G.: Hydrodynamics. University Press, Princeton, NJ (1960)Dromey, B., Zepf, M., Gopal, A., et al.: Nature Phys. 2, 456 (2006)Faure, J., et al.: Nature 431, 541 (2004)Gordienko, S., et al.: Phys. Rev. Lett. 93, 115003 (2004)Gordienko, S., Pukhov, A.: Phys. Plasmas 12, 043109 (2005)Pukhov, A.: J. Plasma Phys. 61, 425 (1999)Pukhov, A., Gordienko, S.: Phil. Trans. R. Soc. A 364, 623 (2006)Pukhov, A., Meyer-ter-Vehn, J.: Appl. Phys. B 74, 355 (2002)Remington, B., Drake, R.P., Takabe, H., Arnett, A.: Science 284, 1488

(1999)Remington, B., et al.: Phys. Plasma 7, 1641 (2000)Ryutov, D.D., Remington, B.: Plasma Phys. Control. Fussion 48, L23

(2006)Sedov, L.I.: Similarity and dimentional methods in mechanics, 10th

edn. CRC Press, Boca Reton, FL (1993)Strickland, D., Mourou, G.: Opt. Comm. 56, 219 (1985)Takaba, H., et al.: Plasma Phys. Control. Fussion 41, A75 (1999)Zepf, M.: International Conference on the Interaction of Atoms,

Molecules and Plasmas with Intense Ultrashort Laser Pulses 1–5 Szeged, Hungary (2006)

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DOI 10.1007/s10509-007-9386-y


Proton Radiography of Megagauss Electromagnetic FieldsGenerated by the Irradiation of a Solid Target by an UltraintenseLaser Pulse

Sebastien Le Pape · Daniel Hey · Pravesh Patel · Andrew Mackinnon · Richard Klein ·

Bruce Remington · Scott Wilks · Dmitri Ryutov · Steve Moon · Marc Foord

Received: 12 May 2006 / Accepted: 22 January 2007C© Springer Science + Business Media B.V. 2007

Abstract Laser generated protons have been used to probe

the temporal and spatial evolution of megagauss magnetic

fields. Grid deflectometry techniques have been applied to

proton radiography to obtain precise measurements of pro-

ton beam angles caused by magnetic fields in laser produced

plasmas. Data are presented in two different regimes of in-

teractions at ultra high intensity (1020 W/cm2) where hots

electrons are supposed to be responsible of the B field, and

at lower intensity (1017 W/cm2) and later time where the

gradients of temperature and density are responsible of the

B field.

Keywords Laser generated-proton beam . Magnetic field .

Deflectometry

Introduction

There are a number of outstanding issues induced by large

magnetic fields in the evolution of complex physical phenom-

ena, including the formation of black holes and neutron stars

and the acceleration of cosmic rays (Shapiro and Teukolsky,

1983). Magnetic fields of the megagauss order are generated

by the interaction of a high intensity laser with a solid target

(Stamper, 1991; Tatarakis et al., n.d.). These fields are pre-

dicted to exist in a localized region near the critical density

surface. Such spontaneous fields can be generated by sev-

eral mechanisms including: (i) non parallel temperature and

density gradients in the ablated plasma (Stamper, 1991), (ii)

the ponderomotive force associated with the laser radiation

S. Le Pape () . D. Hey . P. Patel . A. Mackinnon . R. Klein .

B. Remington . S. Wilks . D. Ryutov . S. Moon . M. FoordLawrence Livermore National Laboratory 7000 East AvenueLivermore 94550 California USAe-mail: [email protected]

(Sudan, 1993) or (iii) the current of fast electrons gener-

ated during the interaction (Wilks et al., 1992; Pukhov and

Meyer-ter-Vehn, 1996). The localization of these fields near

the critical density makes them very challenging to probe.

Optical probe used for Faraday rotation are refracted on steep

density gradients, so that the optical beam can only probe

the outer part of the plasma and the lower amplitude B fields

(around 10 MG) (Borghesi et al., 1998). Recently magnetic

fields of 700(±100) megagauss were inferred from polar-

ization shifts of low order VUV harmonics induced by the

Cotton–Mouton effect (Wagner et al., 2002, n.d.). However

this technique does not provide any spatial information on

the B fields structure. The use of laser driven proton deflec-

tometry thus seems to be the ideal technique to probe fast

evolving B fields localized near the critical density.

Laser driven proton deflectometry

This technique exploits the spatial and temporal characteris-

tics of a laser driven proton source. The proton beam is gen-

erated by focusing an ultra-intense laser (Clark et al., 2000a;

Maksimchuk et al., 2000; Snavely et al., 2000; Zepf et al.,

2003) on a thin metallic foil (typically 4.5 microns gold foil).

The protons that are present at the surface of the foil as part of

the surface contaminants are accelerated via the space charge

force induced by the hot electrons that are directly acceler-

ated by the laser pulse. Experiments conducted previously

on the same installation (Allen et al., 2004) have shown that

most of the protons are emitted from the back surface of the

foil. The protons are typically accelerated to energies of a few

hundred of keV to 25 MeV for an intensity of 1019 W/cm2.

The proton beam generated is temporally short (in the order

of a ps), highly laminar and hence equivalent to a virtual

point. In proton imaging, a point projection of the probed

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Proton target mesh

Interactiontarget

Radiochromic film

Proton beam

Ll

CPA

CPA

Fig. 1 Experimental setup ofthe proton deflectometryexperiments

region is obtained with a spatial resolution set by the virtual

source size. The magnification is given by M = (L + 1)/1

(see Fig. 1). This technique is mainly sensitive to field gra-

dients, which are detected via proton density modulations

in the probe beam cross section. In proton deflectometry a

mesh (MacKinnon et al., 2004) is additionally inserted be-

tween the proton target and the interaction target in order to

preimprint a periodical pattern on the probe beam. From the

mesh distortions, the field strength can then be calculated.

B field measurement from the proton target

The experiment was realized in the new Titan facility at the

Lawrence Livermore National Laboratory. The laser delivers

around 130 J on target in 1 ps at λ = 1053 micron. It is

focused by a F/3 off axis parabola on a focal spot of about 10

microns diameter, leading to an intensity of 1020 W/cm2. The

laser is focused on a thin gold target of 4.5 microns. A 1000 lpi

mesh is placed between the target and the radiochromic film

pack. Figure 2a and b, present 18 and 22.5 MeV protons for

magnifications of M = 87 and M = 147 respectively. The

magnification is modified by changing the distance between

the proton source and the mesh (from 1000 microns to 500

microns). On these images, the meshes clearly present strong

distortions that might be induced by the presence of a B

field. Previous work (Clark et al., 2000b) has explained ring

structures observed in the proton beam by the presence of a

B field in the bulk of the target. Indeed during the interaction

of the short ultra intense laser pulse with the thin foil, a large

current of hot electrons is driven in the foil by the laser pulse.

This current is then responsible for the sheath at the back of

the foil that accelerates the protons. This flow of hot electrons

also induces large B fields in the target bulk and at the back

of the target. LSP (Welch et al., n.d.) simulations have been

realized to model the acceleration of the protons from a 5

microns gold foil irradiated by intensity around 1019 W/cm2.

Figure 3 presents these simulations. A large magnetic field

(10 MG) can be seen at the back of the foil and the B fields

persist after the electron pulse and are maximum near the

edge of the laser spot. The electron pulse lasts 100 fs whereas

the B field lasts up to 375 fs. In this work the assumption

was made that the protons were emitted from the front of

the foil. However, further works have shown that most of

the generated protons are emitted from the back on the foil

(Allen et al., n.d.). Their trajectory could then be sensitive to

the presence of B fields in that region. Figure 4a shows the

respective amplitude of the fields (electric and magnetic) as

a function of the distance to the target. It shows that the B

fields are really high (10 MG) when close to the target (less

than a micron) and that the electric fields are much weaker

in amplitude but extend over a longer distance (around 10

microns). Figure 4b shows their respective influence on the

proton trajectory. The presence of a strong B field at the back

Fig. 2 (a) 18.5 MeV protonswith a magnification of 87. Themesh looks straight on the outerpart on the beam and stronglydistorted at its center. (b) 22.5MeV protons with amagnification of 147. When themagnification is increased mostof the mesh elements lookdistorted

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Fig. 3 B field as a function of time (T0 is the time where the 100 fselectron burst is launched in the gold target, i.e 125 fs is 25 fs after theend of the electron burst). (a) 125 fs, (b) 250 fs and (c) 375 fs. The laser

comes from the left, the target is a gold foil of 5 microns. The imageshows only half of the focal spot, it is symmetric around y = 0. Thefocal spot is 10 microns diameter; the laser pulse duration is 100 fs

1 107

5 106

0

-5 106

-1 107

-1.5 107

-2 107

-2.5 107

-3 107

0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 0.0013

Distance Z (cm)

0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 0.0013

a

Distance Z (cm)

b

0.06

0.05

0.04

0.03

0.02

0.01

0

Er and BΘ Particle trajectory velocities

Part

icle

velo

cit

y (

γβ)

Bθ bump

Er acceleration

Vr (x 10)

Surface Ezacceleration <1 µm Vz

Fig. 4 (a) Particle velocities asa function of the distance to thetarget. Vz is the velocity in thedirection normal to the target,Vr is the radial velocity. (b)Magnitude of the electric andmagnetic field as a function ofthe distance to the target

of the target foil induces a bump of the radial velocity of the

protons on the first microns and then a constant acceleration

is induced by the electric field. The distortions observed on

the mesh on Fig. 2a are not observed over the entire proton

beam but are localized on a portion of the beam only. The B

fields are localized near the focal spot (Fig. 3) as the electric

field that generated the proton acceleration is localized at the

back of the proton foil on a much larger area that corresponds

to the source size. A simple model of proton trajectories as

they pass trough the B fields has been used to estimate the B

field strength. For 22.5 MeV protons, a 500 MG field is used

to fit the mesh distortions.

Two beams experiment

At the same time a two beams experiment has been set up to

probe the B fields generated on a second target. This experi-

ment has been realized on the Calisto facility at the Lawrence

Livermore National Laboratory. The laser delivers 10 J in

100 fs at 800 nm. The beam is then split in two, using a

doughnut mirror with a one-inch hole in the center. The re-

flected beam is then focused by an F/4 off axis parabola on

a thin foil of gold that will generate the proton beam. The

second beam that goes through the mirror is focused by an

F/6 off axis parabola on a thin foil of aluminum to create

the probed plasma (Fig. 5). The plasma is probed face-one.

A 1000 lines per inch mesh is placed at 1.5 mm from the

proton target. LSP hybrid PIC code simulations (Town et al.,

2005) show that, face-on, the protons are sensitive only to B

field, while side-on, the protons are sensitive to E field. The

timing of the laser has been adjusted so that the 3.5 MeV

protons arrive to the foil 90 ps (± 25 ps) after the laser pulse.

Figure 6 shows an image of the protons probing the magnetic

field. The aluminum foil edges are clearly visible on the ra-

diochromic film; the mesh is blurred at the position of the

foil because of the scattering of the protons in the aluminum.

The B field in the interaction plasma induces the ring pattern

that is visible on the film. As the time delay between the two

pulses is large (90 ps), the probed B fields are not created

by the flow of hot electrons. Indeed those B fields last only

about 10 ps after the laser pulse (Sandhu et al., n.d.). The B

fields probed at this late time are induced by the temperature

and density gradients that are present in the plasma. These

B fields are then much weaker than those induced by the hot

electrons current. A first estimation using the same simple

model of proton raytrace gives a B field amplitude of 0.01 G.

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Fig. 5 Setup of the two beams experiment. The incoming beam issplited in two using a doughnut mirror, the dashed beam creates theprobed plasma and the other one creates the proton beam. A timingslide allows to adjust the delay time between the two beams

Fig. 6 Typical imaging data at probing time t = 90 ps. This imageshows the proton beam probing the plasma created in the Aluminum foil.The 1000 lpi mesh, placed between the proton target and the interactiontarget, is visible on the image. The presence of the aluminum foil causesthe blurring of the mesh due to the scattering of the proton is the foil.The annular structure observed in the proton beam is induced by thepresence of B field in the probed plasma

Conclusion

In this paper we present an attempt to use laser driven proton

beams to probe high magnitude B fields created by a short

laser pulse. With a laser driven proton beam, high magni-

tude electric and magnetic fields can be probed near critical

density with a temporal resolution of a picosecond. The first

experiment presents data on self-generated B fields during

the proton acceleration processes. The preliminary results

show the presence of large B fields at the back of the target.

LSP simulations confirm that a large B field (10 MG) on a

really small distance (less than a micron) has a strong influ-

ence on the radial velocity of the protons. From the mesh

distortions, the B field amplitude is estimated at 500 MG.

Further experiments changing the intensity on target or the

target material will be conducted to confirm the first mea-

surement. The second experiment presents data from a two

beams setup. In this experiment the B fields are generated on

a second target and are thus decoupled from those generated

during the proton generation processes. Data are recorded at

late time, i.e. when the B fields are induced by temperature

and density gradients. The first results estimate B fields in

the order of 0.01 MG. Further experiments will be carried

out to measure the B fields at shorter time, i.e. when the B

fields are generated by the hot electrons current.

Acknowledgement This work has been performed under the auspicesof the U.S DoE LLNL under contract No W-7405-Eng-48.

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HIGH ENERGY DENSITY LABORATORY ASTROPHYSICSastro · 2015-11-24 · Astrophys Space Sci (2007) 307:1...

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