1. Diffusive shock acceleration: an introduction

4. Interstellar medium
Rarefied ( thermal) plasma filling the galactic space
<n> ~ 1 cm (CGS units are simple)
molecular clouds: n ~ cm-3 T ~ K
warm medium: n ~ 1 cm T ~ 104 K
hot medium: n ~ 0.01 cm T ~ K
magnetic field <B>  3 G B ~ <B> n-1/2
SI: <n> ~ 10-6 m <B> ~ 0.3 nT
104 K  1 eV

6. Cosmic rays 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-thermal
energy distribution.
In our Galaxy: PCR  Pg (= nkT)  PB (= B2/8) ~ erg/cm3

7. Particle Flux ( m2 s sr GeV )-1
Cosmic Ray Spectrum
1 particle/m2 s
Particle Flux ( m2 s sr GeV )-1
„Knee”
1 particle/m2 yr
„Ankle”
1 particle/km2 yr
1 J  61018 eV
Energy eV

8. 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  ~ cm2)

9. Cosmic ray sources ? Possible SNRs shock waves.
CR energy within the galactic volume
ECR = V * CR ~ 1068 cm3 * 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

11. Tycho
X-ray picture from Chandra

12. Supernova remnant Dem L71
X-ray
H-alpha
Supernova remnant Dem L71

13. 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
B = B0 + δB u B

14. Cas A
1-D shock model
for „small” CR energies
from Chandra

15. Schematic view of the collisionless shock wave
( some elements in the shock front rest frame, other in local plasma rest frames ) u1 u2
δE ≠0
thermal
plasma
v~10 km/s
v~1000 km/s CR B d
shock transition
layer
upstream
downstream

16. rg,CR >> rg(E*i) ~ 10 9-10 cm ~ d (for B ~ a few μG)
Particle energies downstream of the shock
evaluated from upstream-downstream Lorentz transformation
for
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) ~ cm ~ d (for B ~ a few μG)
how to get particles with E>>E*i - particle injection problem

17. Modelling the injection process by PIC simulations. For electrons,
see e.g., Hoshino & Shimada (2002)
shock detailes
vx,i/ush
vx,e/ush
|ve|/ush Ey
Bz/Bo Ex
x/(c/ωpe)

18. suprathermal electrons
Maxwellian
I-st order Fermi
acceleration

19. Diffusive shock acceleration: rg >> d
shock compression
R = u1/u2
I order acceleration
where u = u1-u2
in the shock rest frame
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)

20. To characterize the accelerated particle spectrum one needs
information about:
„low energy” normalization (injection efficiency)
spectral shape (spectral index for the power-law distribution)
3. upper energy limit (or acceleration time scale)

21. CR scattering at magnetic field perturbations (MHD waves)
Development of the shock diffusive acceleration theory
Basic theory:
Krymsky 1977
Axford, Leer and Skadron 1977
Bell 1978a, b
Blandford & Ostriker 1978
Acceleration time scale, e.g.:
Lagage & Cesarsky parallel shocks
Ostrowski oblique shocks
Non-linear modifications (Drury, Völk, Ellison, and others)
Drury 1983 (review of the early work)

22. . Energetic particles accelerated at the shock wave:
kinetic equation for isotropic part of the dist. function f(t, x, p)
plasma
advection
spatial
diffusion
adiabatic
compression
momentum
diffusion;
„II order Fermi
acceleration” .
I order: <Δp>/p ~ U/v ~ 10 -2
II order: <Δp>/p ~ (V/v)2 ~ 10 –8
if we consider relativistic particles with v ~ c
cf. Schlickeiser 1987

23. Diffusive acceleration at stationary planar shock
propagating along the magnetic field: B || x-axis; „parallel shock”
outside the shock
+ continuity of particle density and flux at the shock
f=f(p)

24. INDEPENDENT ON ASSUMPTIONS ABOUT LOCAL CONDITIONS
the phase-space
Distribution of shock accelerated particles
particles injected
at the shock
background particles
advected from -∞
INDEPENDENT ON ASSUMPTIONS
ABOUT LOCAL CONDITIONS
NEAR THE SHOCK
Momentum distribution:

25. test particle non-relativistic
Spectral index depends ONLY on the shock compression
adiabatic
index
shock Mach
number
For a strong shock (M>>1): R = 4 and α = (σ = 2.0)
(for CR dominated shock: γ ≈ 4/ R ≈ 7.0 and γ ≈ 3.5)
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.

26. 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)