1. Diffusive shock acceleration: an introduction
Michał Ostrowski
Astronomical Observatory Jagiellonian University

2. 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
B = B0 + B u B

3. Tycho
X-ray picture from Chandra

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

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

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

7. rg,CR >> rg(E*i) ~ 10 9-10 cm ~ d (for B ~ a few G)
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 G)
how to get particles with E>>E*i - particle injection problem

8. 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 x
x/(c/pe)

9. suprathermal electrons
Maxwellian
I-st order Fermi
acceleration

10. 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)

11. 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)

12. 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)

13. . 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: <p>/p ~ U/v ~ 10 -2
II order: <p>/p ~ (V/v)2 ~ 10 –8
if we consider relativistic particles with v ~ c
cf. Schlickeiser 1987

14. 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)

15. 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:

16. 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  =   = 2.0
(for CR dominated shock:   4/ R  7.0 and   3.5)
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.

17. Acceleration time scale at parallel shock
for returning particles
For a „cycle”:
shock
Minimum of tacc:
Bohm

18. A few numbers for a (SNR-like) shock wave
B ~ 10 G ,  ~ rg , u = 1000 km/s (=108 cm/s)
tSNR ~ 104 yr
For a particle energy E = 1 MeV
electron (rg ~ 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 ~ cm/s tacc ~ 106 s
~ 0.1 AU ~ 1 month
E = 1 PeV (= 1015 eV) rg ~ 1018 cm , v ~ cm/s tacc ~ s
~ 1 pc ~ 105 yr
E= 1 EeV (=1018 eV) rg ~ 1021 cm , v ~ cm/s tacc ~ s
~ 1 kpc ~ 108 yr

19. perpendicular
oblique
parallel

20. Oblique magnetic fields 0
reflection
B2 > B1
transmission B1
shock
For uB,1 << v the spectral index is the same as
at parallel shocks !
However tacc can be substantially modified

21. 1 

22. The absolute minimum acceleration time scale
(outside the diffusive approximation)
at quasi-perpendicular shock waves with   90

23. for Ew – energy density of Alfvén waves with k~2/rg(p) per log p
Non-linear modifications of the acceleration process
A. Self-induced scattering (Bell 1978)
Wave generation due to streaming instability upstream of the shock
for Ew – energy density of Alfvén waves with k~2/rg(p) per log p
damping coefficient
CR density
growth rate
decaying
growing

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

25. M = 2 Pg u From Drury & Völk 1981 – weak shock (two fluid model) Pg
precursor
subshock
Pcr
Velocity profile

26. M = 13 Pg u Efficient acceleration in a strong shock (two fluid model)
Pcr

27. Conclusions from non-linear computations:
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

28. I and II order acceleration at parallel shocks
(with isotropic alfvénic turbulence)
plasma beta (  Pg/PB )
Alfvén velocity
(Ostrowski & Schlickeiser 1993)

29. Our knowledge of acceleration processes acting at non-relativistic
shocks 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.

30. 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.