Diffusive shock acceleration: an introduction

Michał Ostrowski

Astronomical Observatory Jagiellonian University

Particle acceleration in the interstellar medium

Inhomogeneities of the magnetized plasma flow lead to energy changes of energetic charged particles due to electric fields

E = u/c B

- compressive discontinuities: shock waves

- tangential discontinuities and velocity shear layers

- MHD turbulence

u

B = B0 + B

B

Tycho

X-ray picture from Chandra

Supernova remnant Dem L71

X-ray H-alpha

Cas A

1-D shock modelfor „small” CR energies

from Chandra

Schematic view of the collisionless shock wave( some elements in the shock front rest frame, other in local plasma rest frames )

u1 u2

B

upstream downstream

shock frontlayer

d

thermalplasma

E 0

CR

v~10 km/s v~1000 km/s

Particle energies downstream of the shock

evaluated from upstream-downstream Lorentz transformation

electronsfor km/s) /1000( eV 5.2ionsfor km/s) /1000( keV 5

2

12

22*

uuA

mvE

where A = mi/mH and u = u1-u2 >> vs,1

upstream sound speed

Cosmic rays (suprathermal particles) E >> E*i

rg,CR >> rg(E*i) ~ 10 9-10 cm ~ d (for B ~ a few G)

for

how to get particles with E>>E*i - particle injection problem

Modelling the injection process by PIC simulations. For electrons,see e.g., Hoshino & Shimada (2002)

vx,i/ush

vx,e/ush

|ve|/ush

Ey

Bz/Bo

x

shock detailes

x/(c/pe)

suprathermal electrons

Maxwellian I-st order Fermiacceleration

Diffusive shock acceleration: rg >> d

Compressive discontinuity of the plasma flow leads to acceleration of particles reflecting at both sides of the discontinuity: diffusive shock acceleration (I-st order Fermi)

u1u2

R u1/u2

v

u p~ p

in the shock rest frame

where u = u1-u2

I order acceleration

shock compression

To characterize the accelerated particle spectrum one needs

information about:

1. „low energy” normalization (injection efficiency)

2. spectral shape (spectral index for the power-law distribution)

3. upper energy limit (or acceleration time scale)

CR scattering at magnetic field perturbations (MHD waves)

Development of the shock diffusive acceleration theory

Basic theory:

Krymsky 1977Axford, Leer and Skadron 1977Bell 1978a, bBlandford & Ostriker 1978

Acceleration time scale, e.g.:

Lagage & Cesarsky 1983 - parallel shocksOstrowski 1988 - oblique shocks

Non-linear modifications (Drury, Völk, Ellison, and others)

Drury 1983 (review of the early work)

Energetic particles accelerated at the shock wave:

kinetic equation for isotropic part of the dist. function f(t, x, p)

p

fDp

pp

fpUffU

t

f 22

1

3

1

plasmaadvection

spatial diffusion

adiabatic compression

momentum diffusion;„II order Fermiacceleration”Upp

3

1.

22

2

2

)(

v

Vp

t

pD I order: <p>/p ~ U/v ~ 10 -2

II order: <p>/p ~ (V/v)2 ~ 10 –8

if we consider relativistic particles with v ~ ccf. Schlickeiser 1987

Diffusive acceleration at stationary planar shock

ffU

propagating along the magnetic field: B || x-axis; „parallel shock”

f(x,p)fuuUx x

, , or , ||21

2 ,1 , || i x

f

xx

fui

+ continuity of particle density and flux at the shock

f=f(p)

outside the shock

Distribution of shock accelerated particles

')'(')(0

1 dppfppAppfp

1

3

R

R

particles injected at the shock

background particles advected from -

1

22 where, )(

R

Rppn

INDEPENDENT ON ASSUMPTIONS ABOUT LOCAL CONDITIONS

NEAR THE SHOCK

the phase-space

Momentum distribution:

For a strong shock (M>>1): R = 4 and = 4.0 = 2.0(for CR dominated shock: 4/3 R 7.0 and 3.5)

, , 3

5for

21

1

1,

1

2sv

uM

M

R

adiabaticindex

shock Machnumber

Spectral index depends ONLY on the shock compression

Spectral shape nearly parameter free, with the index very close to the values observed or anticipated in real sources.

Diffusive shock acceleration theory in its simplest

test particle non-relativistic version became a basis of most studies considering energetic particle

populations in astrophysical sources.

Acceleration time scale at parallel shock

shockvu

t1

11

4

vut

2

22

4 for returning particles

For a „cycle”:

2

2

1

14

uuvt

pv

uup 21

3

4

2

2

1

1

21

3

uuuup

tptacc

v 3

1 i

i ut

vrgB 3

1min

Bohm

Minimum of tacc:

A few numbers for a (SNR-like) shock wave

B ~ 10 G , ~ rg , u = 1000 km/s (=108 cm/s)

For a particle energy E = 1 MeV electron (rg ~ 108 cm , v ~ 1010 cm/s) tacc ~ 102 s proton (rg ~ 1011 cm , v ~ 109 cm/s) tacc ~ 104 s ~ 0.1 day

E = 1 GeV rg ~ 1012 cm , v ~ 1010 cm/s tacc ~ 106 s ~ 0.1 AU ~ 1 month

E = 1 PeV (= 1015 eV) rg ~ 1018 cm , v ~ 1010 cm/s tacc ~ 1012 s ~ 1 pc ~ 105 yr

E= 1 EeV (=1018 eV) rg ~ 1021 cm , v ~ 1010 cm/s tacc ~ 1015 s ~ 1 kpc ~ 108 yr

2

~)(

~

u

vT

u

v

u

Ert g

gacc

tSNR ~ 104 yr

perpendicular

oblique

parallel

Oblique magnetic fields 0

B1

B2 > B1

shock

reflection

transmission

) 2 ,1 ( cos,

i

uu

i

iiB

For uB,1 << v the spectral index is the same as at parallel shocks !

1

3

R

R

However tacc can be substantially modified

2/1

2||,2,2

2,

2

12/1

1||,1,1

1,

2211 //)/(

3

n

n

n

nacc

uR

B

B

uuBBut

constp

xx B /||

BB

1

The absolute minimum acceleration time scale

(outside the diffusive approximation)

u

vT

u

rt g

gacc ~~min,

at quasi-perpendicular shock waves with 90

Non-linear modifications of the acceleration process

A. Self-induced scattering (Bell 1978)

Wave generation due to streaming instability upstream of the shock

www E

x

EVu

t

E)()(

for Ew – energy density of Alfvén waves with k~2/rg(p) per log p

damping coefficient

growth rate x

fvp

E

V

w

4

decaying

growing

CR density

0

x

f

B. Modification of the shock structure by CR precursor(two fluid approximation: g + CR)

dppxfvpPcr ),(3

4 3

0

01

crg PPxx

uu

t

u

is included into the Euler equation:

and the resulting velocity profile u(x) into CR kinetic equation

Possible efficient acceleration: in the two fluid model up to 98% of the shock kinetic energy can be converted into CRs !

From Drury & Völk 1981 – weak shock (two fluid model)

precursor

subshock

Velocity profile

Pg

Pcr

M = 2

u

Pg

Efficient acceleration in a strong shock (two fluid model)

Pg

Pcr

M = 13

R 7

u

Pg

c. Three fluid model – gas + CRs + waves

wave damping heats gas, wave distribution defines

Conclusions from non-linear computations:

- CRs can produce perturbations required for efficient acceleration

- possible efficient acceleration at high Mach shocks

- spectrum flattening at high CR energies

- a value of the upper energy cut-off important for shock modification (divergent energy spectra at high energies)

- test particle spectra only an approximation for real shocks

I and II order acceleration at parallel shocks(with isotropic alfvénic turbulence)

plasma beta ( Pg/PB )

Alfvén velocity

(Ostrowski & Schlickeiser 1993)

Our knowledge of acceleration processes acting at non-relativisticshocks is still very limited. There are basic problems with

- energetic particle injection processes (electrons !)

- existence of stationary solutions for efficient shock acceleration

- description of processes forming or reprocessing MHD turbulence near the shock

- the time dependent solutions

- the upper energy cut-offs, when compared with measurements

- CR electron spectral indices observed in objects like SNRs

etc.

Problems to be solved are usually difficult, often being

highly non-linear and/or 3D and/or non-stationary.

Progress in studies of the diffusive shock acceleration

is very slow since an initial rapid theory developement

in late seventies and early eighties of last century.