nov. 1-3, 2006 [email protected] self-similar evolution of cosmic ray modified shocks...
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Nov. 1-3, [email protected]
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Self-Similar Evolution of Self-Similar Evolution of
Cosmic Ray Modified ShocksCosmic Ray Modified Shocks
Hyesung Kang
Pusan National University, KOREA
Nov. 1-3, [email protected]
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- CR energy flux emerged from shocks FCR= (M) Fk
Thermal E
CR E
thermalization efficiency: (M)CR acceleration efficiency: (M)
1
Vs= u1
Egas
- kinetic energy flux through shocks
Fk = (1/2)Vs3
- net thermal energy flux generated at shocks
Fth = (3/2) [P2-P1u2
= (M) Fk
ECR
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- Astrophysical plasmas are composed of thermal particles and cosmic ray particles.
- turbulent velocities and B fields are ubiquitous in astrophysical plasmas.
- Interactions among these components are important in understanding the CR acceleration.
Astrophysical Plasma
thermal ions &
electrons
Cosmic ray ions & electrons tubluentmean
tot
BB
B
turbulentmean
tot
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scattering of particles in turbulent magnetic fields isotropization in local fluid frame transport can be treated as diffusion process
streaming CRs upstream of shocks excite large-amplitude Alfven waves amplify B field ( Lucek & Bell 2000)
upstreamdownstream
Interactions btw particles and fields: examples
) timecrossing timescatteringmean i.e. ,( || sh
g
V
r
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- Full plasma simulations: follow the individual particles and B fields, provide most complete picture, but computationally very expensive (see the next talk by Hoshino)
- Monte Carlo Simulations with a scattering model: steady-state only
particles scattered with a prescribed model assuming a steady-state shock structure
reproduces observed particle spectrum (Ellison, Baring 90s)
- Two-Fluid Simulations: solve for ECR + gasdynamics Eqns
computationally cheap and efficient, but strong dependence on closure parameters ( ) and injection rate (Drury, Dorfi, KJ 90s)
- Kinetic Simulations : solve for f(p) + gasdynamics Eqns
Berezkho et al. code: 1D spherical geometry, piston driven shock , applied to SNRs, renormalization of space variables with diffusion length i.e. : momentum dependent grid spacing
Kang & Jones code: CRASH (Cosmic Ray Amr SHock code)
1D plane-parallel and spherical grid comving with a shock
AMR technique, self-consistent thermal leakage injection model
Numerical Methods to study the Particle Acceleration
C ,
)()( ppx
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In kinetic simulations Instead of following individual particle trajectories and evolution of fields
diffusion approximation (isotropy in local fluid frame is required)
Diffusion-convection equation for f(p) = isotropic part
),()()(3
1)( ,, pxQ
x
f
xp
fpuU
x
fuU
t
f
jji
iw
iiwi
lines fieldmean
and normalshock btw angle
shock tonormalt coefficiendiffusion
sincos
speeddrift eAlfven wav 22
||
Bn
BnBnxx
wu
BB
nn
Bn
Geometry of an oblique shock
shock
Injection coefficient
x
So complex microphysics of interactions are described by macrophysical models for p & Q
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upstreamin disspation eAlfven wav todue termheating Gas :
speed.Alfven is 4/
,downstreamin 0 upstream,in is speed wavewhere
),(]),([)(3
1)(
r
P-W
B
uu
pxQx
fpx
xp
fpuu
xr
fuu
t
f
cA
A
wAw
ww
Diffusion-Convection Equation with Alfven wave drift + heating
streaming CRs
- Streaming CRs generate waves upstream
- Waves drift upstream with
- Waves dissipate energy and heat the gas.
- CRs are scattered and isotropized in the
wave frame rather than the fluid frame
instead of u smaller velocity jump
and less efficient acceleration
A
generate waves
A
UU11
upstream
Aw uuu
1
Pc
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Parallel (BnBn=0) vs. Perpendicular (BnBn=90) shock
Injection is less efficient, but the acceleration is faster at perpendicular shocks
Slide from Jokipii (2004): KAW3
diffusion
field-cross xx
diffusion
parallel
|| xx
FULL MHD
+ CR terms
Gasdynamics
+ CR terms
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U2U1
Shock front
particle
downstreamupstream
shock rest frame
Diffusive Shock Acceleration in quasi-parallel shocks
Alfven waves in a converging flow act as converging mirrors particles are scattered by waves cross the shock many times
“ Fermi first order process”
v~ sU
p
p energy gainat each crossing
Converging mirrors
Bmean field
cv
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B
prgB
3
1||
Parallel diffusion coefficient
For completely random field (scattering within one gyroradius, =1) “Bohm diffusion coefficient”
minimum value
1 and 3
1
3
1
p. f. m.
||||
||
g
g
r
r
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
||
This is often considered as a valid assumption because of self-excited Alfven waves in the precursor of strong shocks.
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Thermal leakage injectionat quasi-parallel shocks:
due to small anisotropy in veloc
ity distribution in local fluid fra
me,
suprathermal particles in non-M
axwellian tail
leak upstream of shock B0
uniform field
self-generated
wave
leaking particles
Bw
compressed waves
hot thermalized plasma
unshocked gas
Suprathermal particles leak out of thermal pool into CR population.
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velocityflow downstream is where
)1(-exp
11)1(),(
2
1
d
Bd
B
B
ddB
d
B
dBesc
u
u
uu
uH
u
“Transparency function”: probability that particles at a given velocity can leak upstream.
e.g. esc = 1 for CRs
esc = 0 for thermal ptls
CRs
gas ptls B=0.3
B=0.25
Smaller B : stronger turbulence, difficult to cross the shock, less efficient injection
turbulent
fieldmean
)(
0
B
B
Mfcnuu
B
sd
th
d
So the injection rate is controlled by the shock Mach number and B
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02
42
1
2
1)(
3
4
),(]),([)(3
1)(
][)1()(
)()(
0)()(
0)(
p
dpppfcmP
pxQx
fpx
xp
fpuu
xr
fuu
t
f
LWx
uS
t
S
x
PuuPue
xt
e
PPuxt
ux
u
t
pc
ww
g
cgg
g
cg
g
Basic Equations for 1D plane- parallel CR shock
S = modified entropy = Pg/to follow adiabatic compression in the precursor
W= wave dissipation heating L= thermal energy loss due to injection
across the shock
outside the shock
ordinary gas dynamic Eq. + Pc terms
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CR modified shock: diffusive nature of CR pressure introduces some characteristics different from a gasdynamic shock.
- diffusion scales: td (p)= (p)/us2, ld (p)= (p)/us
wide range of scales in the problem: from pinj to pmax
numerically challenging ! not a simple jump across a shock total transition = precursor + subshock
- acceleration time scale: tacc(p) td(p)
instantaneous acceleration is not valid so time-dependent calculation is required
- particles experience different u depending on p due to the precursor velocity gradient + ld (p) f(x,p,t): NOT a simple power-law, but a concave curve
should be followed by diffusion-convection equation
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- For given shock parameters: Ms, us
the CR acceleration depends on the shock Mach number only.
So, for example, the evolution of CR modified shocks is “approximately”
similar
for two shocks with the same Ms but with different us,
if presented in units of
“Similarity” in the dynamic evolution of CR shocks
),(1
8)
)()((
32
2
2
2
1
1
21sd
s
sacc upt
M
M
u
p
u
p
uut
)(
1
82
2
ss
s
d
acc MfcnM
M
t
t
).( oft independen ith time,linearly w stretches structureshock
8
1)(
)(8 - max
2max
p
tuu
pl
u
pt s
sshock
s
- Thermal injection rate: depends on Ms and B
ssd u
p
u
pt
)( x,
)(d2
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Three runs with
(p) = 0.1 p2/(p2+1)1/2
(p) = 10-4 p
(p) = 10-6 p
at a same time t/to=10
(p) oft independen
ith t,linearly w stretches
structureshock the
8/
tul sshock
PCR,2 approaches time
asymptotic values for
t/to > 1.
1
but max p
At t=0, Ms=20 gasdynamic shock
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Numerical Tool:CRASH (Cosmic Ray AmrNumerical Tool:CRASH (Cosmic Ray Amr SHock ) Code SHock ) Code
Bohm type diffusion: for p >>1 - wide range of diffusion length scales to be resolved:
from thermal injection scale (pinj/mc=10-2) to outer scales for the highest p (~106)
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
pp
pp
1)(
2
2
Kang et al. 2001
1024/
108 typically
level, grideach at
refinement twooffactor a
010
max,
xx
lg
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Time evolution of the M0 = 5 s
hock structure.At t=0, pure gasdynamic shock
with Pc=0 (red lines).
t=0
Kang, Jones & Gieseler 2002
-1D Plane parallel Shock DSA simulation
“CR modified shocks”- presusor + subshock- reduced Pg
- enhanced compression
precursor
No simple analytic shock jump condition
Need numerical simulations to calculate the CR acceleration efficiency
preshock postshock
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Evolution from a gasdynamic shock to a CR modified shock.1) initial states : a gasdynamic shock at x=0 at t=0
- T0= 104 K and us= (15 km/s) Ms , 10<Ms<80
- T0= 106 K and us= (150 km/s) Ms, 2<Ms<30
2) Thermal leakage injection :
- more turbulent B smaller B smaller injection
- pure injection model : f0 = 0
- power-law pre-existing CRs: fup(p) = f0 (p/pinj) -5
3) B field strength :
1)(
2
2
p
pp o4) Bohm type diffusion:
mcppp o of unitsin is where,)(or
BBB / where,25.02.0 0B
ISMfor 1~ ICM,for 1.0~
10 ,/
speeddrift wavedetermines ,4/ w
thB
A
EE
Bu
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Pcr,2 reaches to an asymptotic value,
The shock structure stretches linearly with t.
the shock flow becomes self-similar.
efficiency CR
constant
5.0
),()(
2,
30
t
tP
tV
txdxEt
c
s
CR
CR energy/shock kin. E.
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dtun
dppfpdxt
oo
CR
'
2 )(4)(
02t
01s
/
/
ration compressioshock
Pc,2 increases with Ms
but asymptotes to 50% of
shock ram pressure.
Fraction of injected CR particles is higher for higher Ms.
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4ln
)(ln
, ),( 4
pd
Gdq
dxppxfG
p
p
G p : non-linear concave curvature
q ~ 4.2 near pinj
q ~ 3.6 near pmax
f( xs, p)p4 at the shock
at t/to = 10
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CR acceleration efficiency vs. Ms for plane-parallel shocks-The CR acceleration efficiency is determined mainly by Ms . It increases with Ms, but it asymptotes to a limiting value of ~ 0.5 for Ms > 30.-larger ( larger A/cs): less efficient acceleration due to the wave drift in precursor
- larger weaker turbulence: more efficient injection, and less efficient acceleration- pre-existing CRs: higher injection: more CRs- these dependences are weak for strong shocks
tV
txdxEt
s
CR
305.0
),()(
Pre-exist Pc
B=0.25
B=0.2
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From DSA simulations using our CRASH code for parallel shocks:
-Thermal leakage injection rate is controlled mainly by Ms and the level of downstream turbulent B fields. (a fraction of = 10-4 - 10-3 of the incoming particles become CRs.)- The CR acceleration rate depends on Ms, because acc/td = fcn(Ms) in other words u/u = fcn(Ms)- The postshock CR pressure reaches a stable value after a balance between fresh in
jection/acceleration and advection/diffusion of the CR particles away from the shock is established.
-The shock structure broadens as lshock ~us t/8, linearly with time, independent of the diffusion coefficient.
So the evolution of CR shocks becomes approximately ``self-similar” in time.
It makes sense to define the CR energy ratio for the acceleration efficiency
- Ms increases with Ms, depends on B, EB/Eth (wave drift speed),
but it asymptotes to a limiting value of ~ 0.5 for Ms > 30.
SUMMAY
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Plasma simulations at oblique shocks : Giacalone (2005a)
Injection rate weakly depends on Bn for fully turbulent fields.
~ 10 % reduction at perpendicular shocks
(B/B)2=1
The perpendicular shock accelerates particles to higher energies compared to the parallel shock at the same simulation time .
parallel
perp.
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Observational example:particle spectra in the Solar wind(Mewaldt et al 2001)
-Thermal+ CR populations-suprathermal particles leak out of thermal pool into CR population
CRgas
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initial Maxwellian
Concave curve
CR feedback effects
gas cooling (Pg decrease)
thermal leakage
power-law tail
concave curve at high Epower-law tail (CRs)
Particles diffuse on different ld(p) and feel different u,
so the slope depends on p.
f(p) ~ p-q
21
13)(
uu
upq
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)
injection momenta
thermal
g(p) = f(p)p4