Download Diffusive shock acceleration

Diffusive shock acceleration:
an introduction
Interstellar medium
Rarefied ( thermal) plasma filling the galactic space
<n> ~ 1 cm-3
(CGS units are simple)
molecular clouds: n ~ 100-1000 cm-3
warm medium:
hot medium:
magnetic field
SI:
n ~ 1 cm-3
n ~ 0.01 cm-3
<B>  3 G
<n> ~ 10-6 m-3
T ~ 10-50 K
T ~ 104 K
T ~ 106-107 K
B ~ <B> n-1/2
<B> ~ 0.3 nT
104 K  1 eV
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) ~ 10-13 erg/cm3
Particle Flux ( m2 s sr GeV )-1
Cosmic Ray Spectrum
1 particle/m2 s
„Knee”
1 particle/m2 yr
„Ankle”
1 particle/km2 yr
1 J  6 1018 eV

Energy 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)
1
1
13
6

~

3

10
s

10
years
 24
10
nc 110  3 10
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
X-ray
H-alpha
Supernova remnant Dem L71
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
B = B0 + δB
u
- compressive discontinuities: shock waves
- tangential discontinuities and velocity shear layers
-
MHD turbulence
B
Cas A
1-D shock model
for „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
δE ≠0
thermal
plasma
v~10 km/s
v~1000 km/s
CR
B
upstream
d
shock transition
layer
downstream
Particle energies downstream of the shock
evaluated from upstream-downstream Lorentz transformation
2
1

5
keV
A
(
u
/1000
km/s)
for ions
*
2
E  mv  
2
2
2.5 eV (u/1000 km/s) for electrons
where
A = mi/mH
and
for
u = u1-u2 >> vs,1
upstream sound speed
Cosmic rays (suprathermal particles)
rg,CR >> rg(E*i) ~ 10 9-10 cm ~ d
E >> E*i
(for B ~ a few μG)
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)
shock detailes
vx,i/ush
vx,e/ush
|ve|/ush
Ey
Bz/Bo
Ex
x/(c/ωpe)
suprathermal electrons
Maxwellian
I-st order Fermi
acceleration
Diffusive shock acceleration: rg >> d
u1
u2
shock compression
R = u1/u2
I order acceleration
u
p ~ p
v
where u = u1-u2
in the shock rest frame
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 1977
Axford, Leer and Skadron 1977
Bell 1978a, b
Blandford & Ostriker 1978
Acceleration time scale, e.g.:
Lagage & Cesarsky 1983 - parallel shocks
Ostrowski 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)
f
1
f
1  2 f 

 U  f  f    U p
 2  p D 
t
3
p p 
p 
plasma
advection
spatial
diffusion
adiabatic
compression
.
1
p   p  U
3
10 -2
I order: <Δp>/p ~ U/v ~
II order: <Δp>/p ~ (V/v)2 ~ 10 –8
if we consider relativistic particles with
v~c
momentum
diffusion;
„II order Fermi
acceleration”
(p)
D
2t
2
V 
p  
v
2
2
cf. Schlickeiser 1987
Diffusive acceleration at stationary planar shock
propagating along the magnetic field: B || x-axis; „parallel shock”

U  f  f 


x
outside the shock

, U  u1 or u2 ,    ||   x , f  f(x,p)
f
  f 
ui
   ||  , i  1, 2
x x  x 
+ continuity of particle density and flux at the shock
f=f(p)
the phase-space
Distribution of shock accelerated particles
f ( p)  Ap

 p

p
 p'
 1
f  ( p' )dp'
0
particles injected
at the shock
3R

R 1
Momentum distribution:
n( p )  p

, where
background particles
advected from -∞
INDEPENDENT ON ASSUMPTIONS
ABOUT LOCAL CONDITIONS
NEAR THE SHOCK
R2
  2 
R 1
Spectral index depends ONLY on the shock compression
R
 1
2
 1 2
M
5
for  
adiabatic 3
index
u1
, M
shock Mach vs ,1
,
number
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)
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)