Diffusive shock acceleration: an introduction

Interstellar mediumRarefied ( thermal) plasma filling the galactic space

<n> ~ 1 cm-3 (CGS units are simple)

molecular clouds: n ~ 100-1000 cm-3 T ~ 10-50 K

warm medium: n ~ 1 cm-3 T ~ 104 K hot medium: n ~ 0.01 cm-3 T ~ 106-107 K

magnetic field <B> 3 G B ~ <B> n-1/2

SI: <n> ~ 10-6 m-3 <B> ~ 0.3 nT 104 K 1 eV

Cosmic rays are energetic particles.

Primary:- protons and heavier nuclei- electrons (and positrons)

Secondary CR include also:- antiprotons, positrons, neutrinos, gamma rays

with energies much above the thermal plasma and the non-thermalenergy distribution.

In our Galaxy: PCR Pg (= nkT) PB (= B2/8) ~ 10-13 erg/cm3

Cosmic rays

Cosmic Ray Spectrum

Energy eV

„Knee”1 particle/m2 yr

Par

ticl

e F

lux

( m

2 s

sr G

eV )

-1

1 particle/m2 s

„Ankle”1 particle/km2 yr

1 J 61018 eV

CR collisions in ISM

For a high energy collision of a CR particle with the interstellar atom (nucleus) we have (n ~ 1/cm3 and the cross section ~ 10-24 cm2)

yearsscn

6131024

10103103101

1~

1

Cosmic ray sources ? Possible SNRs shock waves.

CR energy within the galactic volume

ECR = V * CR ~ 1068 cm3 * 10-13 erg/cm3 = 1055 erg

Mean CR residence time CR = 2 *107 yr

CR production required for a steady-state

ECR / CR ~ 1040 erg/s

1 SN / 100 yrs injects ~1051 erg /3*109 s 3*1041 erg/s

10% efficiency is enough

Tycho

X-ray picture from Chandra

Supernova remnant Dem L71

X-ray H-alpha

Particle acceleration in the interstellar medium

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

δE = δu/c ✕ B

- compressive discontinuities: shock waves

- tangential discontinuities and velocity shear layers

- MHD turbulence

u

B = B0 + δB

B

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 transitionlayer

d

thermalplasma

δE ≠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 μG)

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

Ex

shock detailes

x/(c/ωpe)

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: <Δp>/p ~ U/v ~ 10 -2

II order: <Δp>/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 α = 4.0 (σ = 2.0)(for CR dominated shock: γ ≈ 4/3 R ≈ 7.0 and γ ≈ 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 α 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.

Spectral index

the observed spectrum below 1015 eV -> =2.7

the escape from the Galaxy scales as ~E0.5,

thus the injection spectral index i=2.2

It is very close to the above value DSA=2.0 for M>>1

In real shocks with finite M the above value of i

very well fits the modelled effective spectral index(like by Berezkho & Voelk for SNRs)