1. Diffusive shock acceleration: an introduction – cont.

2. Diffusive shock acceleration
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)

3. 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

4. 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
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.

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

6. A few numbers for a (SNR-like) shock wave
B ~ 10 μG , λ ~ 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

7. perpendicular
oblique
parallel

8. Oblique magnetic fields Ψ≠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

9. Ψ1

10. The absolute minimum acceleration time scale
(outside the diffusive approximation)
„shock drift acceleration”
at quasi-perpendicular shock waves with Ψ ≈ 90°
Cross-field diffusion is required to form a power-law spectrum !

11. for Ew – energy density of Alfvén waves with k ~ 2π/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 ~ 2π/rg(p) per log p
damping coefficient
CR density
growth rate V
(decaying)
(growing)

12. Streaming instability
Accelerated particles generate waves upstream of the shock
Wave generation leads to decreasing the particle
diffusion coefficient
thus the acceleration time scale diminishes leading
to more rapid acceleration
leading to more efficient wave generation
etc., until the process saturates at B ~ B

13. (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 this simple two fluid model
up to 98% of the shock kinetic energy can be converted into CRs !

14. Simplification: hydrodynamical approach
where is an effective spatial diffusion coefficient. Then, with
PC=(C-1)EC and PG=(G-1)EG the stationary solution can be
given as a set of first integrals:

15. PG U The resulting non-linear shock solutions can be explained
in the „phase space” (PG , U) : PG U

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

17. CR spectrum in the modified shock
log n(E)
subshock acceleration
full shock transition
log E

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

19. c. Three fluid model gas + CRs + waves
wave damping heats gas
wave distribution defines 
+ nonlinearities of the two-fluid model
All the above processes are described in approximate way,
thus the considered class of multi-fluid models can be analysed
only qualitatively.

20. 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 are only approximations for real shocks

21. The second-order Fermi acceleration in turbulent MHD mediumV
Diffusion in the momentum
space
second-order
Scattering centres  MHD waves
V ~ 10 km/s (interstellar medium, solar wind)
V ~ km/s (solar flares, large scale extragalactic jets)

22. If particle distribution function f is so slowly varying in the space
that we can neglect /x terms, the transport equation takes the form:
The momentum diffusion coefficient for ultrarelativistic particles
interacting with isotropically propagating scaterers
A typically considered power-law relation (p)  p leads to
Dp p 2-

23. The shortest mean free path occurs in the „Bohm limit”:
(p) = rg  pc/eB
(this relation yields =1) . One should note that for the isotropic
diffusion the spatial diffusion coefficient
 = (1/3) c (p)
and there exists a useful relation
 Dp = V2 p2 /9

24. Acceleration time scale
The escape time scale
(the diffusion time scale to reach the system boundary at L) 2L

25. Comparison of tacc and tesc for 1 GeV particle in a few astrophysical
scenarios (for these evaluations we assume the Bohm limit with  = rg )
Interstellar medium of our Galaxy: V=10 km/s, L=1 kpc, B=3 G, rg ~ 1013 cm
tacc ~ 3(c/V) rg/c ~ 104 yr tesc ~ 3 (L/c) (L/rg) ~ yr
but in real conditions  >> rg and spatial diffusion is highly anisotropic. A role of
the CRs second-order acceleration is still disputed in the literature.
b. Shocked plasma in the supernova remnant: V=10 km/s, L=0.01 pc, B=30 G,
rg ~1012 cm
tacc ~ 103 yr tesc ~ yr
so the second-order acceleration can be important.
c. Large scale astrophysical jet: V=1000 km/s, L=Rj=1 kpc, B=300 G, rg ~ 1011 cm
tacc ~ yr tesc ~ yr
but for electrons the radiative losses time scale can be also << tesc
and the jet life-time ~107 yr

26. Diffusive shock acceleration in the presence of the second order Fermi acceleration
It is a difficult mathematical and physical problem even for stationary conditions analysed in the linear approximation:
a particle momentum distribution varies in space both upstream and downstream of the shock
the spatial diffusion is coupled to the momentum diffusion
through the diffusive terms with  = (x,p) , Dp = Dp(x,p)
problems with background and boundary conditions,
Below we consider the particle spectrum at the shock in the conditions
allowing for the power law solutions.

27. u2 u1 II order acceleration I order acceleration
where V – Alfvén velocity
I order acceleration
in the shock rest frame
where u = u1-u2

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 development
in late seventies and early eighties of last century.