diffusive shock acceleration: an introduction michał ostrowski astronomical observatory...
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Diffusive shock acceleration: an introduction
Michał Ostrowski
Astronomical Observatory Jagiellonian University
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
u
B = B0 + B
B
Tycho
X-ray picture from Chandra
Supernova remnant Dem L71
X-ray H-alpha
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 frontlayer
d
thermalplasma
E 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 G)
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
x
shock detailes
x/(c/pe)
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: <p>/p ~ U/v ~ 10 -2
II order: <p>/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 = 4.0 = 2.0(for CR dominated shock: 4/3 R 7.0 and 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 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.
Acceleration time scale at parallel shock
shockvu
t1
11
4
vut
2
22
4 for returning particles
For a „cycle”:
2
2
1
14
uuvt
pv
uup 21
3
4
2
2
1
1
21
3
uuuup
tptacc
v 3
1 i
i ut
vrgB 3
1min
Bohm
Minimum of tacc:
A few numbers for a (SNR-like) shock wave
B ~ 10 G , ~ rg , u = 1000 km/s (=108 cm/s)
For a particle energy E = 1 MeV electron (rg ~ 108 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 ~ 1010 cm/s tacc ~ 106 s ~ 0.1 AU ~ 1 month
E = 1 PeV (= 1015 eV) rg ~ 1018 cm , v ~ 1010 cm/s tacc ~ 1012 s ~ 1 pc ~ 105 yr
E= 1 EeV (=1018 eV) rg ~ 1021 cm , v ~ 1010 cm/s tacc ~ 1015 s ~ 1 kpc ~ 108 yr
2
~)(
~
u
vT
u
v
u
Ert g
gacc
tSNR ~ 104 yr
perpendicular
oblique
parallel
Oblique magnetic fields 0
B1
B2 > B1
shock
reflection
transmission
) 2 ,1 ( cos,
i
uu
i
iiB
For uB,1 << v the spectral index is the same as at parallel shocks !
1
3
R
R
However tacc can be substantially modified
2/1
2||,2,2
2,
2
12/1
1||,1,1
1,
2211 //)/(
3
n
n
n
nacc
uR
B
B
uuBBut
constp
xx B /||
BB
1
The absolute minimum acceleration time scale
(outside the diffusive approximation)
u
vT
u
rt g
gacc ~~min,
at quasi-perpendicular shock waves with 90
Non-linear modifications of the acceleration process
A. Self-induced scattering (Bell 1978)
Wave generation due to streaming instability upstream of the shock
www E
x
EVu
t
E)()(
for Ew – energy density of Alfvén waves with k~2/rg(p) per log p
damping coefficient
growth rate x
fvp
E
V
w
4
decaying
growing
CR density
0
x
f
B. Modification of the shock structure by CR precursor(two fluid approximation: g + CR)
dppxfvpPcr ),(3
4 3
0
01
crg PPxx
uu
t
u
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 !
From Drury & Völk 1981 – weak shock (two fluid model)
precursor
subshock
Velocity profile
Pg
Pcr
M = 2
u
Pg
Efficient acceleration in a strong shock (two fluid model)
Pg
Pcr
M = 13
R 7
u
Pg
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
I and II order acceleration at parallel shocks(with isotropic alfvénic turbulence)
plasma beta ( Pg/PB )
Alfvén velocity
(Ostrowski & Schlickeiser 1993)
Our knowledge of acceleration processes acting at non-relativisticshocks 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.
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.