Diffusive shock acceleration: an introduction
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Transcript of Diffusive shock acceleration: an introduction
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)