Diffusive Shock Acceleration
of Cosmic Rays
Hyesung Kang, Pusan National University, KOREA
T. W. Jones, University of Minnesota, USA
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
- Astrophysical plasmas are ionized, magnetized, often shock heated,
tenuous gas.
- CRs & turbulent B fields are ubiquitous in astrophysical plasmas.
- It is important to understand the interactions btw charged particles
and turbulent B fields to understand the CR acceleration.
- Diffusive shock acceleration provides a natural explanation for
CRs.
-Recent Progresses in DSA theory:
1) injection and drift acceleration at perpendicular shocks
2) comparison with DSA theory with observation of SNRs
3) DSA simulation of 1D spherical SNRs
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Interactions between particles and fields
downstream
upstream
scattering of particles in turbulent magnetic fields
 isotropization in local fluid frame
 transport can be treated as diffusion process
|| rg
( 
, i.e. mean scattering time  crossing time )
 Vsh
streaming CRs
- drive large-amplitude Alfven waves
- amplify B field( Lucek & Bell 2000)
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Numerical Methods for the Particle Acceleration
- Full plasma simulations: follow the individual particles and B fields,
provide most complete picture, but computationally too expensive
- Monte Carlo Simulations with a scattering model:
reproduces observed particle spectrum (Ellison, Baring 90s)
applicable only for a steady-state shock
- Two-Fluid Simulations: solve for ECR + gasdynamics
computationally cheap and efficient, but strong dependence on closure
parameters (   ,  C ) and injection rate (Drury, Dorfi, KJ 90s)
- Kinetic Simulations : solve for f(p) + gasdynamics
Berezkho et al. code: 1D spherical geometry, piston driven shock ,
applied to SNRs, renormalization of space variables with diffusion length
i.e. x( p)   ( p) : momentum dependent grid spacing
Kang & Jones code: 1D plane-parallel and spherical geometry,
AMR technique, self-consistent thermal leakage injection model
coarse-grained finite momentum volume method
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Complex microphysics: particles  waves in B field
Following individual particle trajectories and evolution of fields are impractical.
 diffusion approximation (isotropy in local fluid frame is required)
 Diffusion-convection equation for f(p) = isotropic part in Kinetic simulations

f
f 1   
f

f
 (U i  uw,i )
   (U  uw ) p 
( i , j
)  Q( x, p)
t
xi 3
p xi
x j
Injection
coefficient
shock
B
u w  Alfven wav e drift speed
 xx   || cos 2  Bn    sin 2  Bn : normal to shock
n
Bn
 Bn  angle btw shock normal and
mean field lines
x
Geometry of an oblique
May 17-19, 2006
shock
KAW4@KASI.Daejeon.Korea
Parallel (Bn=0) vs. Perpendicular (Bn=90) shock
 xx   
Slide from
Jokipii (2004):
KAW3
cross- field
diffusion
 xx   ||
parallel
diffusion
Injection is efficient at parallel shocks, while it is difficult in perpendicular shocks
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Three Shock Acceleration mechanisms work together.
1) First-order Fermi mechanism: scattering across the shock
dominant at quasi-parallel shocks (Bn< 45)
2) Shock Drift Acceleration: drift along the shock surface
dominant at quasi-perpendicular shocks (Bn> 45)
3) Second-order Fermi mechanism: Stochastic process,
turbulent acceleration  add momentum diffusion term
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Diffusive Shock Acceleration in quasi-parallel shocks
Alfven waves in a converging flow
Shock front
act as converging mirrors
B
 particles are scattered by waves
 cross the shock many times
“ Fermi first order process”
p U s energy gain
~
at each crossing
p
v
vc
mean field
U2
U1
upstream
particle
downstream
shock rest frame
Converging mirrors
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Parallel diffusion coefficient
||  hrg
||
m. f. p.
1
 ||  hrg parallel
3
h 1
For completely random field (scattering within one gyroradius, h=1)
“Bohm diffusion coefficient”
1
p
 ||   B  rg 
minimum value
3
B
particles diffuse on diffusion length scale ldiff = ||(p) / Us
so they cross the shock on diffusion time tdiff = ldiff / Us= ||(p) / Us2
smallest  means shortest crossing time and fastest acceleration.
Bohm diffusion with large B and large Us leads to fast acceleration.
 highest Emax for given shock size and age for parallel shocks
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Thermal leakage injection
at quasi-parallel shocks:
due to small anisotropy in
velocity distribution in local
fluid frame,
hot thermalized
plasma
unshocked gas
suprathermal particles in nonMaxwellian tail
 leak upstream of shock
Bw
compressed waves
CRs streaming upstream
generate MHD waves
(Bell & Lucek)
 compressed and amplified
in downstream: Bw
selfgenerated
wave
Bohm diffusion is valid
B0
uniform
field
leaking
particles
Suprathermal particles leak out of thermal pool into CR population.
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Drift Acceleration in perpendicular shocks with weak turbulences
y

1
 ||

1  (|| / rg )
 ||  h B ,
2
: classical scattering theory
 
h
B
1 h 2
h  || / rg  1 for weak t urbulence
    ||
B

1 
E   V B
c
x
so accelerati on is faster at perpend. shocks
BUT, to ensure that diffusion approx. is valid,
  ,min ~  B (
Vs
) ~  B  0.01
c
for 3000km/s
Particle trajectory in weakly turbulent fields
Energy gain comes mainly from drifting in the convection electric field along the

1  
shock surface (Jokipii, 1982), i.e. e = |q E L|,
E   V B
c
“Drift acceleration”
 but particles are advected downstream with field lines, so injection is difficult:
(Baring et al. 1994, Ellison et al. 1995, Giacalone & Ellison 2000)
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Diffusive Shock Acceleration at oblique shocks
||
 xx   || cos  Bn    sin  Bn
2
2
diffusion normal to shock surface


1

: classical scattering theory
 || 1  (|| / rg ) 2
but   /  || ~ 0.02  0.04 : for turbul ent field due to field line meandering
since t acc 
4
p
 2xx : accelerati on time scale
dp / dt U s
Giacalone & Jolipii 1999
the accelerati on is faster at a perpendicu lar shock
Turbulent B field with Kolmogorov spectrum
smaller xx at perpendicular shocks
 shorter acceleration time scale
 higher Emax than parallel shocks
Monte Carlo Simulation by Meli & Biermann (2006)
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Test-Particle simulation at oblique shocks : Giacalone (2005a)
(B/B)2=1
dJ/dE = f(p)p2
stronger turbulence  more efficient injection
Injection energy weakly depends on Bn
for fully turbulent fields.
~ 10 % reduction at perpendicular shocks
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Test-Particle simulation at oblique shocks : Giacalone (2005a)
(B/B)2=1
dJ/dE = f(p)p2
weak fluctuations
The perpendicular shock accelerates particles
to higher energies compared to the parallel
shock at the same simulation time .
May 17-19, 2006
Injection is less efficient, but acceleration is
faster at perpendicular shocks for weakly
turbulent fields.
KAW4@KASI.Daejeon.Korea
Hybrid plasma simulations of perpendicular shock : Giacalone (2005b)
- acceleration of thermal protons by
perpendicular shocks : thermal leakage
- Field line meandering due to large scale
turbulent B fields  increased cross-field
transport  efficient injection at shock
- thermal particles can be efficiently
accelerated to high energies by a
perpendicular shock
-injection problem for perpendicular
shocks: solved !
Particles are injected where field lines
density of particles with energies E > 10Ep
cross the shock surface
dotted
lines: field lines
May 17-19, 2006
 efficient injection
KAW4@KASI.Daejeon.Korea
Parallel vs. Perpendicular Shocks for Type Ia SNRs : ion injection
Ion injection only for quasi-parallel
shocks (polar cap regions only)
 spherical flux from paralleshock
shock calculations should be
reduced by fre ~0.2
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Determination of B amplification factor, ion injection rate, proton-to-electron
number ratio with SNR observations:
Comparison with kinetic simulation (Berezhko & Voelk)
x
Slide from Voelk (2006)
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Recent Observations of SNRs in X-ray and radio: (Voelk et al. 2005)
Cas A, SN 1006, Tyco, RCW86, Kepler, RXJ1737, …
- thin shell of X-ray emission (strong synchrotron cooling)
B field amplification through streaming of CR nuclear component
into upstream plasma (Bell 2004) is required to fit the observations
 Observational proof for dominance of hadronic CRs at SNRs
-Dipolar radiation: consistent with uniform B field configuration
- Ion injection rate : x~10-4
- Proton/electron ratio: Kp/e ~ 50-100
-~50% of SN explosion energy is transferred to CRs.

Consistent picture of DSA at SNRs
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
CRs observed at Earth:
N(E): power-law spectrum
“universal” acceleration
mechanism working on
a wide range of scales
DSA in the test particle
limit predicts a universal
power-law
E-2.7
f(p) ~ p-q
N(E) ~ E-q+2
q = 3r/(r-1)
r = r2/r1=u1/u2
E-3.1
this explains the universal power-law, independent of shock parameters !
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
CR acceleration efficiency F vs. Ms for plane-parallel shocks
Kang & Jones 2005
u0=(15km/s)M0
1) The CR acceleration efficiency is determined mainly by Ms
2) It increases with Ms (shock Mach no.)
u0=(150km/s)M0
but it asymptotes to a limiting value of F ~ 0.5 for Ms > 30.
Effects of upstream CRs
-4
-3
3) thermal leakage process: a fraction of x= 10 - 10 of
for low Ms shocks
the incoming particles become CRs (at quasi-parallel shocks).
F(t ) 
 dxECR ( x, t )
0.5r 0V t
May 17-19, 2006
3
s
,
x (t ) 
2
dx
4

p
f CR ( p)dp
 
KAW4@KASI.Daejeon.Korea
'
n
u
 o o dt
Diffusion-Convection Equation with Alfven wave drift + heating
f
f
1 
f
1 
f
 (u  u w )
 2 [r 2 (u  u w )] p
 2 [r 2 ( x, p ) ]  Q( x, p )
t
r 3r r
p r r
x
where wave speed is u w   A in upstream, u w  0 in downstream ,
 A  B / 4r is Alfven speed.
Pc
W  -A
: Gas heating term due to Alfven wav e disspation in upstream
r
- Streaming CRs generate waves upstream
- Waves drift upstream with
A
- Waves dissipate energy and heat the gas.
streaming CRs
generate
waves
- CRs are scattered and isotropized in the
wave frame rather than the gas frame
 
 
 u  u  u   instead of u
w
1
A
 smaller vel jump and less efficient
acceleration
upstream
May 17-19, 2006
A
U1
KAW4@KASI.Daejeon.Korea
- CRASH code in 1D plane-parallel geometry
= Adaptive Mesh Refinement (AMR) + shock tracking technique
in the shock rest frame (thru Galilean velocity transformation)
(Kang et al. 2001)
- new CRASH code in 1D spherical geometry
= Adaptive Mesh Refinement (AMR) + shock tracking technique
in a comoving frame which expands with the shock
 The shock stays in the same location (zone).
just like Hubble expansion
Rs = xs a
Rs
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Basic Equations for 1D spherical shocks in the Comoving Frame
2
r~ 1  (r~ )
  r~

ax
t a x
~ ~
~ 2  P
~
r
(

a
2
 ( r ) 1
g  Pc )
  r~ 2  r~  axr~

a
ax
x
a
t
~
~
~
 ( r~e~g ) 1  ( r~e~g  Pg  Pc )
a ~~
 Pc 2 ~~
~
 ( r eg  Pg )  2 r eg  axr~


a
a x ax
x
a
t
~
~
a ~  g  1 ~ ~
2 ~
S 1  (S )
  r  2 S  ~  g 1 [W  L ]

a
ax
t a x
r
1  2 g~
a ~
a g~
1  2
g~ (  u w ) g~
~
(x  )
[ x (  u w )]  ](  4 g )  3 g  2 2
[

x
a x x
a
a y
3ax x
x
a
t
~ ~  g 1 ~
~
~
4~
3 ~
3
3
~
r  ra , P  P a , P  P a , S  P / r , g  p f , y  ln( p)
g
g
c
c
g
~
L  La 3 , L  energy loss due to CR injection
~
W  Wa 3 , W  heating due to wave dissipatio n
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Wave drift +
heating terms
SNR simulations with 1D spherical CRASH code
- Supernova parameters : Eo  1051 erg (explosion energy), M SN  10M sun
- Bohm - type Diffusion
3.0 10 22
p
2
 (p)  (
cm /s)
with B  5G
B
mc
- Preexistin g CR population : f ( p)  f o p  4.5 , Pc / Pg  0.05 and 0.5 far upstream
- thermal leakage injection model
- initial conditions : Sedov - Taylor solution at ~
t 1
- warm uniform ISM condition : nH  0.3 cm 3 , To  10 4 K
initial shock Mach number : M s  130
- Normalizat ion constants
ro  6.14pc, to  1.3 103 yr, uo  4.6 103 km/s
r o  7.0 10 25 g/cm 3 , Po  1.5 10 7 erg/cm 3 ,  o  8.6 10 27 cm 2 /s
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
warm uniform ISM : nH  0.3 cm 3 , To  104 K, M s  130 at t  1
Normalizat ion Constants
ro  6.14pc,
to  1.3 103 yr,
uo  4.6 103 km/s
r o  7.0 10  25 g/cm 3 ,
Po  1.5 10 7 erg/cm 3 ,
 o  8.6 10 27 cm 2 /s
e  injection parameter
x0  6.0 10  4
x8  2.3 10 6 at the 8th level
Pc , 0  0.5Pg , 0
Strong nonlinear modification.
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
warm uniform ISM : nH  0.3 cm 3 , To  104 K, M s  130 at t  1
Normalizat ion Constants
ro  6.14pc,
to  1.3 103 yr,
uo  4.6 103 km/s
r o  7.0 10  25 g/cm 3 ,
Po  1.5 10 7 erg/cm 3 ,
 o  8.6 10 27 cm 2 /s
e  injection parameter
x0  6.0 10  4
x8  2.3 10 6 at the 8th level
Pc , 0  0.5Pg , 0
moderate nonlinear
modification
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
r1 / r 0  2
r2 / r0  8
M s  130  5
Pc , 2 /( r oU st2 )  0.4
Pg , 2 /( r oU st2 )  0.2
x  10-3  10 4
x (t ) 
2
dx
4

p
f ( p ) dp
 
'
n
u
 o o dt
= total CR number
/ particle no.
passed though shock
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
N p   f (r , p) p 2 r 2 dr ,
Gp  N p p2
q
d (ln N p )
d ln p
N p : power-law like
G p : non-linear
concave curvature
q ~ 2.2 near pinj
q ~ 1.6 near pmax
Our results are consistent
with the calculations by
Berezhko et al.
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Summary
- CRs & turbulent B fields are natural byproducts of the collisionless
shock formation process: they are ubiquitous in cosmic plasmas .
- DSA produces a nearly universal power-law spectrum with the
correct slopes.
- With turbulent fields, thermal leakage injection works well even at
perpendicular shocks as well as parallel shocks
-   /  ||  1 , so perpendicular shocks are faster accelerators
- About 50 % of shock kinetic E can be transferred to CRs for strong
shocks with Ms > 30.
- thermal leakage process: a fraction of x = 10-4 - 10-3 of the incoming
particles become CRs at shocks.
- Observations of SNRs support the dominance of CR ions (through
amplified B field) and x = 10-4 - 10-3 and Kp/e ~ 100.
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Test-Particle simulations at oblique shocks : Giacalone (2005a)
 o  3U1 : initial speed



B  B  B :
random fluctuatio ns
with P(k) are imposed.
particle trajector ies
are followed.
Model Parameters ;
 B 

 ,  Bn
 B 
2
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Numerical Tool:CRASH (Cosmic Ray Amr SHock ) Code
2
p
Bohm type diffusion:  ( p) 
 p for p >>1
p2 1
- wide range of diffusion length scales to be resolved:
from thermal injection scale to outer scales for the highest p
1) Shock Tracking Method (Le Veque & Shyue 1995)
- tracks the subshock as an exact discontinuity
2) Adaptive Mesh Refinement (Berger & Le Veque 1997)
- refines region around the subshock with multi-level grids
Nrf=100
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Kang et al. 2001
t=0
“CR modified shocks”
- presusor + subshock
- reduced Pg
- enhanced compression
-1D Plane parallel Shock
DSA simulation
postshock preshock
Time evolution of the M0 = 5
shock structure.
At t=0, pure gasdynamic shock
with Pc=0 (red lines).
No simple shock jump condition
Need numerical simulations to
calculate the CR acceleration
efficiency
precursor
Kang, Jones & Gieseler 2002
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Evolution of CR distribution function in DSA simulation
f(p): number of particles in the momentum bin [p, p+dp], g(p) = p4 f(p)
CR feedback effects
gas cooling (Pg decrease)
initial Maxwellian
thermal leakage
thermal
 power-law tail
 concave curve at high E
power-law tail (CRs)
f(p) ~ p-q
g(p) = f(p)p4
3u1
q( p) 
u1  u2
Particles diffuse on different ld(p)
and feel different u,
so the slope depends on p.
Concave curve
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
electron acceleration mechanisms:

direct electric field acceleration (DC acceleration)
(Holman, 1985; Benz, 1987; Litvinenko, 2000;
Zaitsev et al., 2000)

stochastic acceleration via
wave-particle interaction
(Melrose, 1994; Miller et al., 1997)

shock waves
(Holman & Pesses,1983; Schlickeiser, 1984;
Mann & Claßen, 1995; Mann et al., 2001)

outflow from the reconnection site
(termination shock)
(Forbes, 1986; Tsuneta & Naito, 1998;
Aurass, Vrsnak & Mann, 2002)
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
Thermal Leakage Injection at parallel shocks has been observed
- suprathermal particle leak out of thermal pool into CR population (power-law tail)
 injection rate x ~ 10-4 – 10-3
postshock
preshock
CRs
May 17-19, 2006
KAW4@KASI.Daejeon.Korea
comparison of
Monte Carlo
simulations
with direct
measurement
at Earth’s bow
shock