Interaction of stellar flows with planets and moons: three ...

Interaction of stellar flowswith planets and moons:

three dimensional hybrid simulations

June 4, 2008

Stellar flows and moons and planets

1 Introduction

2 Hybrid Expanding Box (HEB) simulations

3 Global kinetic simulations of the Moon

4 Global kinetic simulations of Mercury


Introduction: Role of numerical experiments

Humans try to explain events ongoing in the surrounding nature since earlydays. Over centuries: experiment and theory.

Technological advances: today we use spacecraft measurements to improveour knowledge about processes ongoing in space environment. On their basiswe built up our theoretical models.

Advances in computer technology achieved in last decades allow us to usetheoretical knowledge to formulate studied problems in the form of numericalmodels. Extension of our theoretical/experimental capabilities.


Introduction: Simulations vs. in situ experiments

LEFT: data acquired by the electron spectrometer PEACE and by a waveanalyzer WHISPER onboard Cluster II spacecraft: waves generated in aspace plasma at fundamental frequency and twice the plasma frequency.

RIGHT: analogical data from a numerical experiment. We use a model ofelectron VDF with an electron beam. We can observe corresponding waveemissions in a numerical model.

The spacecraft must be present at the place of the interesting event.In the case of the numerical simulation we can change different input

parameters of our model and then observe, when the given phenomenareaches its maximum, or when the process is not ongoing at all.


Numerical model: Dimensionless simulation units

Physics is essentially dimensionless (important are relative rather thanabsolute values), it scales with dimensions typical for the givenproblem. In a numerical we use:

q = 1, mp = 1.

We use proton inertial length in the solar wind Λpsw = c/ωpswand the inverse of the proton cyclotron frequency Ω−1

psw as spatialand temporal units respectively.

i.e. npsw = 1, B0sw = 1, vAsw = 1.

Hermean radius (for example) in the units of the numericalmodel varies with the value of Λpsw in SI units.

Consequently: In generall - typical spatio temporal sizes ofthe studied interaction between a stellar flow and a moon orplanet depend naturally on the physical properties of the flow(its density and magnetic field in it)


Planets of our Solar systemName distance npSW BSW ΛiSW vASW radius R vsw = 300km/s dipole moment M

[AU] [cm−3] [nT] [km] [km s−1] [km] [Λi] [vA] [MEarth] [Λ3i Bsw/µ0]

Mercury 0.31 73 46 26.7 118 2 439 91 2.5 4.7 ·10−4 5.4·107

0.47 32 21 40 82 2 439 60 3.6 4.7 ·10−4 3.5·107

Venus 0.72 14 10 61 59 6 052 99 5 - -Earth 1.0 7 6 86 50 6 378 74 6 1.0 2.6·109

Mars 1.5 3.1 3.4 129 42 3 397 26 7 - -Jupiter 5.2 0.26 0.83 447 36 71 398 160 8 20.000 2.7·1013

Saturn 9.6 0.076 0.44 827 35 60 000 73 9 540 2.16·1011

Uranus 19.1 0.019 0.22 1 654 35 25 400 15 9 48 4.8·109

Neptune 30.2 0.0077 0.14 2 599 35 24 300 9 9 26 1.0·109

Pluto 39.4 35 2 500 9 unknown unknown

The Earth magnetic dipole moment is MEarth = 8 ·1022 Am2.

Mercury is the most likely (magnetized) candidate to be studied by a globalkinetic numerical model beside unmagnetized planets Venus and Mars

Effect of denser stellar flow at Mercury: from physical point of view thesize of Mercury is comparable to Earth

Mercury also changes its diameter in the stellar flow of given desity thanksto its eccentric Solar orbit

Pluto almost can not be simulated by a hybrid code and its study requires afull particle in cell code


Numerical codes

Magnetohydrodynamic (MHD) codes: traditional codes used forglobal simulations of the interaction betwee stellar flows andmoons and planets

Alow 3D modeling on global scalesNeglect kinetic effects and hence wave-particle interactions

Particle in cell (PIC): use macro-particles to describe bothelectrons and ions

Include (all) kinetics for both electrons and ionsRequire to use unrealistic parameters (for example, mass ratio)

A compromise: Hybrid model [Alan Matthews, 1994]: treatelectrons as fluid

Include ion kineticsAllow ”realistic” parameters on a global scaleDo not include electron kinetics, some types of waves, if excited,can not interact with electrons and should be artificially dumped.


Hybrid Expanding Box code: The motivation

Properties of collisionless plasmas are largely determined by wave-particleinteractions

A homogeneous slowly expanding plasma (without any fluctuating waveenergy evolves adiabatically. The ion parallel and perpendiculartemperatures T‖ and T⊥ satisfy the equations T‖ ∝ B and T⊥ ∝ n2/B2,respectively

In the case of a strictly radial expansion, B(0) = (1,0,0), the temperatureanisotropy evolves as T⊥/T‖ ∝ (1+ t/te)−2, where te = R0/U is thecharacteristic time of the expansion (R0 is initial distance from the Sun, Uis the (constant) solar wind speed

In slowly expanding (or compressed) plasma temperature anisotropiesnaturally develop. Temperature anisotropy (i.e. departure from Maxwellianparticle distribution function) represent a possible source of free energy formany different instabilities.

As a feed-back, the instabilities constrain the shape of particle distributionfunctions.


Solar wind plasma: Protons

[Hellinger et al., 2006]

Both panels show a colorscale plot of the relativefrequency of (βp‖,Tp⊥/Tp‖)in the WIND/SWE data(1995-2001) for the solarwind vSW < 600 km/sThe over plotted curves showthe contours of the maximumgrowth rate (in units ωcp) inthe correspondingbi-Maxwellian plasma

proton cyclotron instability(solid curves)parallel fire hose instability(dashed curves)proton mirror instability(dotted curves)oblique fire hose instability(dash-dotted curves)


Solar wind plasma: Electrons

[Stverak et al., 2008]

Panel shows a gray shadedscale plot of the relativefrequency of (βp‖,Tp⊥/Tp‖)in the Helios data for thesolar wind vSW < 600 km/sThe over plotted curves showthe contours of the maximumgrowth rate (in units ωce) inthe correspondingbi-Maxwellian plasma

whistler instability(dash-dotted curves)parallel electron fire hoseinstability (dashed curves)


Hybrid Expanding Box code: The design

The stellar plasma flow is subject to the continuous radial expansion. Itcan be also subject of compression (in a planetary magnetosheath)and/or torsion caused by a velocity shear

HEB code uses Vlasov equations in such a way that we can study agiven volume of continuously expanding plasma in a simulation box(1D or 2D) with periodic boundary conditions


Magnetosheath plasma stretching

LEFT: Evolution during the numerical simulation of plasma stretchingin the space (β‖,Tp⊥/Tp‖). The evolution (solid line) follows initiallythe adiabatic prediction. The over plotted curves show the contours ofthe maximum growth rate (in units of ωcp) (left) for the proton cyclotroninstability and (right) the mirror proton instability.

RIGHT: In situ observations made by HIA experiment

[Travnicek et al., 2007]


Magnetosheath plasma stretching

LEFT: The panel shows the total fluctuating magnetic energy δB2/B20 (solid

curve) as a function of βp‖. The dotted curve corresponds to the mirror fluctu-ating magnetic energy δB2

mir/B20 whereas the dashed curve corresponds to the

proton cyclotron fluctuating magnetic energy δB2PC/B2

0. RIGHT: The panelshows a coloured plot of the fluctuating magnetic energy δB2/B2

0 as a func-tion of βp‖ and ΘkB. The gray dashed vertical line denotes βp‖ for which thesimulation reaches the linear mirror threshold.


Moon: Motivation

When the solar wind interacts with the Moon, plasma is absorbed at thelunar surface leaving a void on the nightside referred to as the wake-tail

No atmosphere and minimal exosphere eliminates the Moon as a sourceof plasmaThe wake-tail is one of the best natural vacuums in the solar system

The solar wind IMF passes through the unmagnetized, non-conductingMoon

Crustal magnetic fields on the lunar surface may form localizedmini-magnetospheres, causing ”limb compressions” in the vicinity of thewake-tail

The lunar wake-tail is actively being studied to understand the solarwind refilling processes and its overal three dimensional structure

Today computers can acommodate 2D/3D global hybrid (kinetic)simulations of the solar wind interaction with the Moon


Moon: Observations - particles


Moon: Observations - waves

Plasma waves excited in andarround the density depletion regionobseved at ∼ 7 RL by WIND[Farrell et al, 1996]

Ion gyrofrequency waves observedfurther downstream in the wake-tail∼ 25 RL by WIND[Travnicek et al, 2005]


Moon: Observations - summary

Observations of streaming and anisotropic plasma distributions, aswell as plasma waves of various types indicate that kinetic processesare important in the refilling and formation of the wake-tail

We use 2D and 3D global hybrid (kinetic) simulations to model thesolar wind interaction with the Moon


Moon: Simulations - setup


Moon: Simulation results: 2D [Travnicek et al., 2005]

Cavity in near-moon wake; downstream a density enhacement occurs atcenter of the depletion region

Counter-streaming ion distributions in depletion region center; largetemperature anisotropies at edges

Left-hand polarized ion gyrofrequency wave fluctuations in wake-tailStellar flows and moons and planets

Moon: Simulation results: 3D [Travnicek et al., 2008]


Moon: Summary and conclusions

Wake refilling is described by a magnetized plasma to vacuum expansionwith plasma flow primarily in the magnetic field aligned direction

The density gradient at the cavity’s edge provides an electric fieldaccelerating protons towards the center of the cavity

The current associated with the plasma to vacuum expansion leads to anenhancement of the magnetic field inside the plasma depletion region

The rate of the plasma refilling process depends on the solar wind speed,plasma temperature and IMF orientation

The density cavity is filled with counterstreaming ion beams; at the cavity’sedge the plasma is anisotropic in temperature (T⊥ > T‖)

Density enhancement bump forms in center of far downstream wake due toa counterstreaming beams

Asymmetries form in the density depletion region because of the skewedIMF orientation with respect to solar wind flow direction

Left-hand polarized VLF waves near ion cyclotron frequency are generatedat 28 RL < X < 40 RL, due to an anisotropy or heat flux instability


Mercury: Observations

Mercury is smaller than Earth,however is embedded into a denserplasma

Till January, 14 2008 Mercury, hasbeen observed by single spacecraft(Mariner 10) in 70th (twoencounters)

Both encounters have shown, thatMercury has its own (weak)magnetosphere

Mercury orbits Earth on anexcentric orbit = significantchanges in its magnetosphere

New probes: MESSENGER(NASA), BepiColombo (ESA)


Mercury: Observations

Magnetometer data from two flybys by the Mariner 10established that Mercury has an intrinsic magnetic field [Ness etal., 1974].Magnetic dipole moment has been estimated to be between170 nT R3

M (≈ 2.5 ·1019 Am2) [Jackson and Beard, 1977] and349 nT R3

M (≈ 5. ·1019 Am2) [Ness et al., 1975].MESSENGER: 250 nT R3

M.The standoff distance was estimated to be between 1.3 and2.1 RM.Mariner 10 electron data indicated that a substorm-like eventmay have occurred, suggesting magnetic reconnection inmagnetotail [Siscoe et al., 1975; Eraker and Simpson, 1986].Small Hermean magnetosphere can be exposed to solar windwith speeds ranging from 200 km/s up to 600 km/s [Russell etal., 1988].MESSENGER 1st flyby: vsw = 400 km/s.


Numerical model: Global Simulations to date

All (global) numerical models employ some compromises:

MHD 3D simulations [ Kabin et al., 2001].

Hybrid models with spatial resolution of several c/ωpi [ Kalioand Janhunen, 2003] (typical Larmor radius at Mercury is 2-3c/ωpi).

Global 2D hybrid models of solar wind interacting withmagnetized obstacles have been also considered [ Omidi et al.,2002].

Scaled down model with spatial resolution of the order c/ωpi

and reasonable ratio between Larmor radius and the scaleddown size of the planet [ Travnicek et al., 2007a].


Numerical model: Scaling down the model

We used a scaled down model of Mercury with a magneticmoment M = 25,000BswΛ3

psw4π/µ0 = 3.76 ·1019/ε Am2 forboth studies, where the scaling factor ε equals to ≈ 166 and≈ 111 for the high and low solar wind pressure casesrespectively.

Consequently we use RM = 16.36Λpsw and RM = 12.48Λpsw forthe two studied cases.

The downscaling preserves the stand-off distance of themagnetopause predicted by Rmp = (2B2

eq/(µ0Pram,sw))1/6RM,where Beq is the magnetic field at the equator of the planet withradius RM and Pram,sw is solar wind ram pressure nswmpv2

sw,where mp is the proton mass (mp = 1 in simulation units).

Although scaled down, the radius RM is always sufficiently largerthan the local Larmor radius rL.


Numerical model: Initial and boundary conditions

IMF B0sw = (Bx,By,0), B0sw = 1 makes an angle of ϕ =−30

with respect to the −X axis (i.e., with respect to the solar windflow direction). The dipolar field is defined by

BM =(M/r3)Bsw [−2 sinλer + cosλeλ ] .

No dipole tilt is applied.We keep ∂B/∂ t = 0 and E = 0 in the interior of the planet.A resistivity layer η ≈ 0.8exp(−h2/h2

0) (where h is radialdistance from the surface, h0 = 3Λpsw), is applied near theplanet’s surface.We have also injected H+ ions with density of the ordernp ∼ 10−4 nsw isotropically from Mercury’s surface in both caseswith velocity vp ∼ 0.05vA perpendicular to the surface.Both numerical experiments reached times ∼ 40Ω−1

psw which issufficiently higher than the system transit time.


Simulation parameters

High Pram,sw Low Pram,sw

Spatial resolution 4x/4y =4z 0.4/1.0 ΛpswTime step 4t 0.02 Ω−1

pswβe 1.0Number of pcles / cell 70βp 0.5Solar wind velocity (MA) 5.0 3.0Orientation of IMF Θ 30

Radius of the planet (90/60 Λpsw) 16.4 Λpsw 12.5 ΛpswTotal duration 40.0-80.0 Ω−1

p

np = 73 cm−3, Bsw = 46 nT, i.e. RM = 91c/ωpi, vAsw = 118 km/sin relatively fast solar wind (i.e. vsw = 600 km/s ≈ 5 vAsw).

np = 32 cm−3, Bsw = 21 nT, i.e. RM = 60c/ωpi, vAsw = 82 km/sin relatively slow solar wind (namely vsw = 250 km/s ≈ 3 vAsw).


Results: Virtual measurements

We can perform 3D measurements of any basic plasma entity: B,ni, E, j, T‖, T⊥, vth, rL, . . .

Derived entities: curvature of magnetic field, electric andmagnetic drifts (to understand particle motion).

Full 3D picture of the overal structure of the magnetosphere:from foreshock to magneto-tail (to understand energydistribution).

Energetic budget on boundaries.

(multispacecraft) flybys on non-gravitationally driventrajectories: plasma flow lines (cf. HEB).

Role of initial conditions: parameters in of the solar wind and/orthe obstacle itself

Fourier analysis (waves modes) and their origin, propagation,dumping.


Mercury: Density H+


Mercury: Temperature


Protons ejected from Hermean surface

We injected H+ ions into the system from the planetary surface toinclude a zero-order model of ions sputtered of the surface. Theseions were injected with density of the order np ∼ 10−4 nswisotropically from Mercury’s surface in both cases with velocityvp ∼ 0.05vA perpendicular to the surface.


Protons ejected from Hermean surface:vpsw = 5vAsw vs. vpsw = 3vAsw

Density of protons ejected from surface (in units of 10−4 npsw)


Local Larmor radius rL


Temperature anisotropy: Motivation

Aim of the scaled down model: preserve wave-particleinteractions. Temperature anisotropy Ap = T⊥/T‖ = Ap(β‖) shallbe bounded by regions unstable with respect to the anisotropydriven instabilities.

Examining temperature anisotropy in our model:

Can provide useful indication whether in our simulationswave-particle interactions are resolved properly

Provides predictions about the compression/expansion of theHermean magnetospheric plasma at different locations andpossibly excited anisotropy driven instabilities: enhancement ofHEB observations


Temperature anisotropy profile Ap = T⊥/T‖


HEB simulation and Cluster II observations

LEFT: Magnetosheath plasma in a HEB simulationRIGHT: Magnetosheath plasma observed by Cluster II(from [Travnicek et al., 2007b])


Small planet: Solar wind plasma

We use for the solar wind βp = 0.5 in both simulations


Small planet: Subsolar magnetosheath plasma


Small planet: Quasi-parallel magnetosheath plasma


Small planet: Quasi-perpendicular magnetosheath plasma


Small planet:Radiation belt plasma


Small planet: Magnetospheric and magnetosheath plasma

Two-fluid MHD predicts T⊥ ∝ B, T‖ ∝ n2/B2. Temperature anisotropiesemmerge naturally when plasma is compressed, stretched, expanding.

Convention: a > 0: expansion, a < 0: compression (special case: a‖ -stretching)

Calculate symmetric tensor field Tij = 1/2(∂ui/∂xj +∂uj/∂xi) (i.e.evaluate Tij in each point of the simulation domain).

Diagonalize Tij and find eigenvectors v1, v2, v3 and eigenvalues a1, a2, a3.

Calculate the stretching rate a‖ ∼ ∂u‖/∂x‖, and two expansion rates ac,a‖×c. The direction c is given by the vector of magnetic fieldline curvature.

Similarly, calculate the antisymmetric tensor fieldT?

ij = 1/2(∂ui/∂xj−∂uj/∂xi) and obtain a measure of the velocity shear:1/3

√a2 +b2 + c2, where a, b, c are non-diagonal elements of the

antisymmetric tensor T?ij .


Small planet: Stretching/Expansion of the plasma for β = 1


Small planet: Stretching/Expansion of the plasma for β = 4


Small planet: Torsion of the plasma for β = 1 and β = 4


Setup of another numerical experiment

nsw = Bsw = vA,sw = c/ωpi,sw = 1 (unit)Spatial resolution 4x/4y =4z 0.4/1.0 c/ωpi,swTime step 4t 0.02 Ω−1

p,swSpatial size of the system Lx/Ly = Lz 260 / 265 c/ωpi,swTotal duration 64.0 Ω−1

p,swβp / βe 1.0 / 1.0Number of macro-particles per cell 70 (total ∼ 3.2·109)Solar wind velocity (MA) vsw 4.0 vA,swOrientation of IMF - 16 (Bz = 0)Mercury’s magnetic moment MM = 250 nT (= Beq) (no tilt)

Radius of Mercury RM

[25000

Bsw

Beq

]1/3

c/ωpi,sw

Simulation coordinate system: X+ ‖ to the solar wind flow,Y+ ‖ to Mercury’s orbital speed, Z+ ‖ towards north pole(antiparallel with respect to MM)


MESSENGER (virtual) flybyH+ density with overploted (virtual) MESSENGER trajectory(equatorial plane):


MESSENGER in situ observations

[Slavin et al., 2008]


Simulation results: density and magnetic field


Simulation results: S/C trajectory in the (β‖,T⊥/T‖) space


Mercury: Conclusions

We have carried out a case study which confirms several phenomenaobserved by the MESSENGER spacecraft observed in situ and helps tointerpret the nature of the wave-particle effects.

Inbound flyby: whistler waves in the foot of the nearly strictlyperpendicular shock, mirror waves downstream the quasi-perpendicularbow shock, sheath turbulence, signatures of the presence of Mercury’s beltin magnetic field measurements.

The numerical results support presence of Kelvin-Helmholtz vorticesformed near the inbound magnetopause.

Quasi-parallel magnetosheath is filled with a cascade of large aplitudemirror-like structures formed in a cascade of magnetic field line stretchingand compression.

Outbound magnetosheath suggests presence of two transition layers onepossibly in the form of a slow mode or intermediate shock.


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