Title: Magnetic%20Field%20Amplification%20in%20Diffusive%20Shock%20Acceleration
1Magnetic Field Amplification in Diffusive Shock
Acceleration Don Ellison, North Carolina State
Univ.
- Why is Diffusive Shock Acceleration (DSA) with
Magnetic Field Amplification (MFA) important? - Shocks widespread in Universe all with
nonthermal particles - DSA mechanism known to be efficient direct
evidence heliosphere, SNRs - B-fields larger than expected ? MFA connected to
DSA - Magnetic fields important beyond DSA e.g.,
control synchrotron emission - Why is DSA with MFA so hard to figure out?
- Efficient acceleration nonlinear effects on
shock structure ?? wave generation - Scales (length, momentum) large and connected
through NL interactions - Test-particle approximations lose essential
physics - Plasma physics important
- Where do we stand?
- Active work from various directions
- Semi-analytic solutions of diffusion-convection
equations - Monte Carlo particle simulations
- Hydrodynamic fluid simulations
- Particle-in-cell simulations
- All making progress on understanding plasma
physics but all limited in important ways
2Evidence for High (amplified) B-fields in SNRs
Sharp synch. X-ray edges
Cassam-Chenai et al. 2007
Tychos SNR Cassam-Chenai et al. 2007
Radial cuts
magnetically limited rim
synch loss limited rim
magnetically limited rim
synch loss limited rim
X-ray
radio
Chandra observations of Tychos SNR (Warren et
al. 2005)
If drop from B-field decay instead of radiation
losses, expect synch radio and synch X-rays to
fall off together.
Good evidence for radiation losses and,
therefore, large, amplified magnetic field. On
order of 10 times higher than expected
3Evidence for efficient particle acceleration in
SNRs
SE
south
east
inefficient RFS/RCD gt 1
Efficient RFS/RCD 1
SNR SN1006 Cassam-Chenai et al (2008)
In east and south ? strong nonthermal emission
? RFS/RCD 1
Efficient DSA RFS/RCD 1
SNR Morphology Forward shock close to contact
discontinuity ? clear prediction of
efficient DSA of protons
SE
4Direct evidence at Earth Bow Shock
Dots are AMPTE spacecraft observations
Ellison, Mobius Paschmann 1990
Observed acceleration efficiency is quite
high Dividing energy ?4 keV gives ?2.5 of
proton density in superthermal particles, and
gt25 of energy flux crossing the shock put into
superthermal protons
Thermal leakage injection in action !
Maxwellian
Ellison, Jones Eichler 1981
Bottom line Convincing evidence for efficient
Diffusive Shock Acceleration (DSA) with B-field
amplification
5Table from Caprioli et al 2009
6Can describe DSA (in non-rel shocks) with
transport equation (i.e., diffusion-convection
equation) Requires assumption that vpart gtgt
u0 to calculate the pitch angle average for
shock crossing particles Original references
Krymskii 1976 Axford, Leer Skadron 1977
Blandford Ostriker 1978 Bell 1978
Charged particles gain energy by diffusing in
converging flows. Bulk K.E converted into random
particle energy.
Note, for nonrelativistic
shocks ONLY
D(x,p) is diffusion coefficient f(x,p) is phase
distribution function u is flow speed Q(x,p) is
injection term x is position p is particle
momentum
7- Basic Ideas
- For shock acceleration to work, particle
diffusion must occur. - But, in test-particle limit, get power law
particle distribution with an index that doesnt
depend on diffusion coefficient ! (0nly on
compression ratio) - For shock acceleration to work over wide momentum
range, magnetic turbulence ( ?B/B ) must be
self-generated by accelerated particles. - If acceleration is EFFICIENT, energetic particles
modify shock structure, produce strong turbulence
(?B/B gtgt 1), and results DO depend on details of
plasma interactions.
8From test-particle theory, in Non-relativistic
shocks (Krymskii 76 Axford, Leer Skadron 77
Bell 78 Blandford Ostriker 78)
Power law index is ? Independent of any details
of diffusion ? Independent of shock Obliquity
(geometry) ? But, for Superthermal particles only
?
u0 is shock speed
Ratio of specific heats, ?, along with Mach
number, determines shock compression, r
For high Mach number shocks
?
So-called Universal power law from shock
acceleration
9BUT, Not so simple!
Consider energy in accelerated particles assuming
NO maximum momentum cutoff and r 4 (i.e., high
Mach , non-rel. shocks)
Energy diverges if r 4
But
If produce relativistic particles ? ? lt 5/3 ?
compression ratio increases If ? lt 5/3 the
spectrum is harder ? Worse energy divergence ?
Must have high energy cutoff in spectrum to
obtain steady-state, but this means particles
must escape at cutoff But, if particles escape,
compression ratio increases even more . . .
Acceleration becomes strongly nonlinear with r gt
4 !!
?Bottom line Strong shocks will be efficient
accelerators with large comp. ratios even if
injection occurs at modest levels (1 thermal ion
in 104 injected)
10Temperature
If acceleration is efficient, shock becomes
smooth from backpressure of CRs
test particle shock
Flow speed
Lose universal power law
subshock
p4 f(p)
X
NL
? Concave spectrum ? Compression ratio, rtot gt
4 ? Low shocked temp. rsub lt 4
TP f(p) ? p-4
In efficient acceleration, entire spectrum must
be described consistently, including injection
and escaping particles ? much harder
mathematically even if diffusion coefficient,
D(x,p), is assumed !
BUT, connects photon
emission across spectrum from radio to ?-rays
11Why is NL DSA with MFA so hard to figure out?
- DSA is intrinsically efficient ( ? 50 ) ?
test-particle analysis not good approximation ?
must treat back reaction of CRs on shock
structure - Magnetic field generation intrinsic part of
particle acceleration ?
cannot treat DSA and MFA separately - Strong turbulence means Quasi-Linear Theory (QLT)
not good approximation ? But QLT is our main
analytic tool - Heliospheric shocks, where in situ observations
can be made, are all small and low Mach number
(MSonic lt 10) ? dont see production of
relativistic particles or strong MFA - Length and momentum scales are currently well
beyond reach of particle-in-cell (PIC)
simulations if wish to see full nonlinear effects
? Particularly true for non-relativistic shocks - Problem difficult because TeV protons influence
injection of keV protons and electrons - To cover full dynamic range, must use approximate
methods e.g., Monte Carlo,
Semi-analytic, MHD
12Particle-in-cell (PIC) simulations (for example,
Spitkovsky 2008)
Here, relativistic, electron-positron shock
Also, this is a 2-D simulation But, good
example of state-of-art
Mass density
Shock
upstream
DS
En. density in B
Density
B generated at shock
B-field
Start with NO B-field, Field is generated
self-consistently (Weibel instability?), shock
forms, see start of Fermi acceleration
Plasma
physics done self-consistently!
13Magnetic Field Amplification (MFA) in Nonlinear
Diffusive Shock Acceleration using Monte Carlo
methods Work done with Andrey Vladimirov Andrei
Bykov Discuss only Non-relativistic shocks
14- A lot of work by many people on nonlinear
Diffusive Shock Acceleration (DSA) and Magnetic
Field Amplification (MFA) - Some current work (in no particular order)
- Amato, Blasi, Caprioli, Morlino, Vietri
Semi-analytic - Bell Semi-analytic and PIC simulations
- Berezhko, Volk, Ksenofontov Semi-analytic
- Malkov Semi-analytic
- Niemiec Pohl PIC
- Pelletier and co-workers MHD, relativistic
shocks - Reville, Kirk co-workers MHD, PIC
- Spitkovsky and co-workers Hoshino and
co-workers other PIC simulators
Particle-In-Cell simulations, so far, mainly rel.
shocks - Vladimirov, Ellison, Bykov Monte Carlo
- Zirakashvili Ptuskin Semi-analytic, MHD
- Apologies to people I missed
15First Phenomenological approach assuming
resonant wave generation (turbulence produced
with wavelengths ? particle gyro-radius)
Growth of magnetic turbulence driven by cosmic
ray pressure gradient (so-called streaming
instability) e.g., Skilling 1975, McKenzie
Völk 1982
growth of magnetic turbulence energy density,
W(x,k). (x position k wavevector)
energetic particle pressure gradient. (p
momentum)
Produce turbulence resonantly assuming QLT
VG parameterizes complicated plasma physics
Also, can produce turbulence non-resonantly
(current instability) Bells non-resonant
instability (2004) Cosmic ray current produces
turbulence with wavelengths shorter than particle
gyro-radius Cosmic ray current produces
turbulence with wavelengths longer than particle
gyro-radius e.g., Malkov Drury 2001 Reville
et al. 2007 Bykov, Osipov Toptygin 2009
Important question What are parameter regimes
for dominance?
16Once turbulence, W(x,k), is determined from CR
pressure gradient or CR current, must determine
diffusion coefficient, D(x,p) from W(x,k). Must
make approximations here
- Bohm diffusion approximation Find effective Beff
by integrating over turbulence spectrum (e.g.,
Vladimirov, Ellison Bykov 2006) - Resonant diffusion approximation (QLT) (e.g.,
Skilling 75 Bell 1978 Amato Blasi 2006) - Hybrid model for strong turbulence Different
diffusion models in different momentum ranges
applicable to strong turbulence (Vladimirov,
Bykov Ellison 2009) - Low particle momentum, p ?part constant
(set by turbulence correlation length) - Mid-range p ?part ? gyro-radius in some
effective B-field - Maximum p ?part ? p2 (critical for Emax)
- Scattering for thermal particles?
17One Example from many (Vladimirov et al
2006) Calculate shock structure, particle
distributions amplified magnetic field Assume
resonant, streaming instabilities for magnetic
turbulence generation Assume Bohm approximation
for diffusion coefficient
18Nonlinear Shock structure, i.e., Flow speed vs.
position
Particle distributions and wave spectra at
various positions relative to subshock for
resonant wave production
DS
upstream
subshock
Position relative to subshock at x 0 units of
convective gyroradius
19Bohm approx. for D(x,p)
p4 f(p)
k W(k,p)
DS
D(x,p)/p
DS
upstream
u(x)
Iterate
Nonlinear Shock structure
20Red Bohm diffusion approximation
upstream
DS
subshock
Flow speed
Beff
Amplified B-field B0 x 70
More complete examples will include Combined
resonant non-resonant wave
generation more realistic diffusion
calculations dissipation of wave energy to
background plasma cascading of turbulence etc.
21- Summary of nonlinear effects
- (1) Thermal injection (2) shock structure
modified by back reaction of accelerated
particles (3) Turbulence generation (4)
diffusion in self-generated turbulence (5)
escape of maximum energy particles -
- Production of turbulence, W(x,k) (assuming
quasi-linear theory) - Resonant (CR streaming instability) (e.g.,
Skilling 75 McKenzie Volk 82 Amato Blasi
2006) - Non-resonant current instabilities (e.g., Bell
2004 Bykov et al. 2009 Reville et al 2007
Malkov Diamond this conf.) - CR current produces waves with scales short
compared to CR gyro-radius - CR current produces waves with scales long
compared to CR gyro-radius - Calculation of D(x,p) once turbulence is known
- Resonant (QLT) Particles with gyro-radius
?waves gives ?part ? p - Non-resonant Particles with gyro-radius gtgt
?waves gives ?part ? p2 - Production of turbulence and diffusion must be
coupled to NL shock structure including injection
and escape
22- Conclusions
- Shocks and shock acceleration important in many
areas of astrophysics Shocks accelerate
particles and generate turbulence - DSA process can be efficient, i.e., 50 of shock
energy may go into rel. CRs ! - Good evidence B-field, at shock, amplified well
above compressed ambient field (i.e., Bamp gtgt 4 x
B0) - Resonant and non-resonant wave generation
instabilities both at work - Complete NL problem currently beyond PIC
simulation capabilities, but PIC is only way to
study full plasma physics (critical for injection
process) - Several approximate techniques making progress
Semi-analytic, MC, MHD - Important problems where work remains
- What are maximum energy limits of shock
acceleration, i.e., Emax? - Effect of escaping particles at Emax?
- Electron to proton (e/p) ratio? (GeV/TeV
emission from SNRs) - Realistic shock geometry, i.e., shock obliquity?
(SN1006)
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24Energy, length, time scales Requirements for
PIC simulations to do entire DSA ?? MFA problem
in non-relativistic shocks
Energy range
Length scale (number of cells in 1-D)
Run time (number of time steps)
Problem difficult because TeV protons influence
injection and acceleration of keV protons and
electrons NL feedback between TeV keV Plus,
important to do PIC simulations in 3-D (Jones,
Jokipii Baring 1998) PIC simulations will only
be able to treat limited, but very important,
parts of problem, i.e., initial B-field
generation, particle injection To cover full
dynamic range, must use approximate methods
e.g., Monte Carlo,
Semi-analytic, MHD
25- Escaping particles in Nonlinear DSA
- Highest energy particles must scatter in
self-generated turbulence. - At some distance from shock, this turbulence will
be weak enough that particles freely stream away. - As these particles stream away, they generate
turbulence that will scatter next generation of
particles - In steady-state DSA, there is no doubt that the
highest energy particles must decouple and escape
No other way to conserve energy. - In any real shock, there will be a finite length
scale that will set maximum momentun, pmax. Above
pmax, particles escape. - Lengths are measured in gyroradii, so B-field and
MFA importantly coupled to escape and pmax - The escape reduces pressure of shocked gas and
causes the overall shock compression ratio to
increase (r gt 7 possible). - Even if DSA is time dependent and has not reached
a steady-state, the highest energy particles in
the system must escape. - In a self-consistent shock, the highest energy
particles wont have turbulence to interact with
until they produce it. - Time-dependent calculations (i.e., PIC sims.)
needed for full solution.
26Shocks with and without B-field amplification
Monte Carlo Particle distribution functions f(p)
times p4
The maximum CR energy a given shock can produce
increases with B-amp BUT Increase is not as large
as downstream Bamp/B0 factor !!
protons
No B-amp
B-amp
p4 f(p)
For this example, Bamp/B0 450?G/10?G 45 but
increase in pmax only x5
Maximum electron energy will be determined by
largest B downstream. Maximum proton energy
determined by some average over precursor
B-field, which is considerably smaller
All parameters are the same in these cases except
one has B-amplification
27Riquelme Spitkovsky 2009
3-D PIC simulation of Bells instability
28Only Bell non-resonant instability Resonant wave
generation suppressed
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31Determine steady-state, shock structure with
iterative, Monte Carlo technique
Upstream Free escape boundary
Unmodified shock with r 4
Self-consistent, modified shock with rtot 11
(rsub 3)
Flow speed
Momentum Flux conserved (within few )
Energy Flux (only conserved when escaping
particles taken into account)
Position relative to subshock at x 0 units of
convective gyroradius