Magnetic Field Amplification in Diffusive Shock Acceleration
Don Ellison, North Carolina State Univ.
1) Why is Diffusive Shock Acceleration (DSA) with Magnetic Field Amplification
(MFA) important?
a) Shocks widespread in Universe: all with nonthermal particles
b) DSA mechanism known to be efficient: direct evidence: heliosphere, SNRs
c) B-fields larger than expected  MFA connected to DSA
d) Magnetic fields important beyond DSA: e.g., control synchrotron emission
2) Why is DSA with MFA so hard to figure out?
a) Efficient acceleration: nonlinear effects on shock structure  wave generation
b) Scales (length, momentum) large and connected through NL interactions
c) Test-particle approximations lose essential physics
d) Plasma physics important
3) Where do we stand?
a) Active work from various directions:
i. Semi-analytic solutions of diffusion-convection equations
ii. Monte Carlo particle simulations
iii. Hydrodynamic fluid simulations
iv. Particle-in-cell simulations
b) All making progress on understanding plasma physics but all limited in
important ways
Evidence for High (amplified)
B-fieldsSNR
in SNRs
Tycho’s
Cassam-Chenai et al. 2007
magnetically
limited rim
Sharp synch. X-ray edges
Cassam-Chenai et al. 2007
Radial cuts
synch loss
limited rim
magnetically
limited rim
synch loss
limited rim
radio
X-ray
Chandra observations of Tycho’s 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
east
Efficient
RFS/RCD ~ 1
SE
south
inefficient
RFS/RCD > 1
Evidence for
efficient particle
acceleration in SNRs
SNR SN1006
Cassam-Chenai et al (2008)
RFS
RCD
In east and south
 strong nonthermal
emission
 RFS/RCD ~ 1
H (FS)
0.20.8 keV(CD)
Efficient DSA: RFS/RCD ~ 1
SNR Morphology: Forward shock
close to contact discontinuity
 clear prediction of efficient DSA
of protons
SE
Direct evidence at Earth Bow Shock
Ellison, Mobius & Paschmann 1990
Dots are AMPTE
spacecraft observations
Observed acceleration
efficiency is quite high:
Dividing energy 4 keV gives
2.5% of proton density in
superthermal particles, and
>25% of energy flux
crossing the shock put into
superthermal protons
Maxwellian
Thermal leakage
injection in action !
Ellison, Jones & Eichler 1981
Bottom line: Convincing
evidence for efficient Diffusive
Shock Acceleration (DSA) with
B-field amplification
Table from Caprioli et al 2009
Can describe DSA (in non-rel shocks) with transport equation (i.e.,
diffusion-convection equation)
Requires assumption that vpart >> u0 to calculate the pitch angle average
for shock crossing particles
Original references: Krymskii 1976; Axford, Leer & Skadron 1977; Blandford &
Ostriker 1978; Bell 1978
D(x,p) is diffusion coefficient
f(x,p) is phase distribution function
Charged particles gain energy by diffusing in
converging flows.
u is flow speed
Bulk K.E converted into random particle
energy.
Q(x,p) is injection term
Note, for nonrelativistic shocks ONLY
x is position
p is particle momentum
Basic Ideas:
1) For shock acceleration to work, particle diffusion must occur.
2) But, in test-particle limit, get power law particle distribution with an
index that doesn’t depend on diffusion coefficient ! (0nly on
compression ratio)
3) For shock acceleration to work over wide momentum range,
magnetic turbulence ( B/B ) must be self-generated by accelerated
particles.
4) If acceleration is EFFICIENT, energetic particles modify shock
structure, produce strong turbulence (B/B >> 1), and results DO
depend on details of plasma interactions.
From test-particle theory, in Non-relativistic shocks (Krymskii 76;
Axford, Leer & Skadron 77; Bell 78; Blandford & Ostriker 78):
f ( p)  p
3r /( r 1)
u0  Vsk
if v p
Power law index is:
 Independent of any details of diffusion
f ( p ) is phase space density
r is compression ratio
 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:

r
f ( p)  p 3r /( r 1)  p 4 ,
  1 (5 / 3)  1

4!
  1 (5 / 3)  1
(or,
N ( E )  E 2 )
So-called “Universal” power law from shock acceleration
BUT, Not so simple!
Consider energy in accelerated particles assuming NO maximum
momentum cutoff and r ~ 4 (i.e., high Mach #, non-rel. shocks)


pinj
4

E p p dp 
2
 dp / p

N ( p)  p f ( p)
2
pinj

pinj
 ln p |
Energy diverges if r = 4
But
 1
r
 1
If produce relativistic particles   < 5/3  compression ratio increases
If  < 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 > 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)
Temperature
If acceleration is efficient, shock becomes
smooth from backpressure of CRs
test particle shock
Flow speed
p4 f(p)
Lose universal
power law
subshock
X
NL
TP: f(p) 
► Concave spectrum
p-4
► Compression ratio, rtot > 4
► Low shocked temp. rsub < 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
Why is NL DSA with MFA so hard to figure out?
1) DSA is intrinsically efficient (  50% )  test-particle analysis not good
approximation  must treat back reaction of CRs on shock structure
2) Magnetic field generation intrinsic part of particle acceleration
 cannot treat DSA and MFA separately
3) Strong turbulence means Quasi-Linear Theory (QLT) not good approximation
 But QLT is our main analytic tool
4) Heliospheric shocks, where in situ observations can be made, are all “small”
and low Mach number (MSonic < ~10)  don’t see production of relativistic
particles or strong MFA
5) 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
a) Problem difficult because TeV protons influence injection of keV protons
and electrons
6) To cover full dynamic range, must use approximate methods:
e.g., Monte Carlo, Semi-analytic, MHD
Particle-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
DS
Shock
upstream
Mass
density
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!
Magnetic Field Amplification (MFA) in
Nonlinear Diffusive Shock Acceleration
using Monte Carlo methods
Work done with Andrey Vladimirov & Andrei Bykov
Discuss only Non-relativistic shocks
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):
1)
2)
3)
4)
5)
6)
7)
8)
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
9) Vladimirov, Ellison, Bykov: Monte Carlo
10) Zirakashvili & Ptuskin: Semi-analytic, MHD
11) Apologies to people I missed …
First: 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)
d

 PCR ( x, p ) dp 
 VG 
 dt W ( x, k )
x
dk  p pres ( k )

stream
Produce
turbulence
resonantly
assuming QLT
VG parameterizes complicated plasma physics
Also, can produce turbulence non-resonantly (current instability):
Bell’s 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?
Once 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:
1) Bohm diffusion approximation: Find effective Beff by integrating over
turbulence spectrum (e.g., Vladimirov, Ellison & Bykov 2006)

Beff2 ( x ) 1
  W ( x, k )dk
8
20
 ( x, p ) 
cp
,
eBeff
1
D ( x , p )  v ( x , p )
3
2) Resonant diffusion approximation (QLT) (e.g., Skilling 75; Bell 1978; Amato & Blasi
2006):
1 p 2c 2
1
 ( x, p )  2 2
 e W ( x, kres )
kresrg ( B0 )  kres
pc
1
eB0
3) Hybrid model for strong turbulence: Different diffusion models in different
momentum ranges applicable to strong turbulence (Vladimirov, Bykov & Ellison 2009)
a) Low particle momentum, p; part ~ constant (set by turbulence
correlation length)
b) Mid-range p; part ∝ gyro-radius in some effective B-field
c) Maximum p; part ∝ p2 (critical for Emax)
4) Scattering for thermal particles?
One 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
Nonlinear Shock structure, i.e., Flow speed vs. position
DS
upstream
Position relative to subshock at x = 0
[ units of convective gyroradius]
Particle distributions and
wave spectra at various
positions relative to subshock
for resonant wave
production
subshock

Bohm approx. for
D(x,p)
Beff2 ( x ) 1
  W ( x, k )dk
8
20
 ( x, p ) 
cp
,
eBeff
1
D ( x , p )  v ( x , p )
3
p4 f(p)
k W(k,p)
DS
D(x,p)/p
u(x)
W(k,p)
upstream
DS
Iterate:
f(x,p)
D(x,p)
Nonlinear Shock structure
Red: 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.
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
1) Production of turbulence, W(x,k) (assuming quasi-linear theory)
a) Resonant (CR streaming instability) (e.g., Skilling 75; McKenzie & Volk 82; Amato & Blasi
2006)
b) Non-resonant current instabilities
(e.g., Bell 2004; Bykov et al. 2009; Reville et al 2007;
Malkov & Diamond this conf.)
i.
ii.
CR current produces waves with scales short compared to CR gyroradius
CR current produces waves with scales long compared to CR gyroradius
2) Calculation of D(x,p) once turbulence is known
a) Resonant (QLT): Particles with gyro-radius ~ waves
gives part ∝ p
b) Non-resonant: Particles with gyro-radius >> waves gives part ∝ p2
3) Production of turbulence and diffusion must be coupled to NL shock structure
including injection and escape
Conclusions
1) Shocks and shock acceleration important in many areas of astrophysics: Shocks
accelerate particles and generate turbulence
2) DSA process can be efficient, i.e., ~50% of shock energy may go into rel. CRs !
3) Good evidence B-field, at shock, amplified well above compressed ambient field
(i.e., Bamp >> 4 x B0)
4) Resonant and non-resonant wave generation instabilities both at work
5) Complete NL problem currently beyond PIC simulation capabilities, but PIC is only
way to study full plasma physics (critical for injection process)
6) Several approximate techniques making progress: Semi-analytic, MC, MHD
7) Important problems where work remains:
a) What are maximum energy limits of shock acceleration, i.e., Emax?
b) Effect of escaping particles at Emax?
c) Electron to proton (e/p) ratio? (GeV/TeV emission from SNRs)
d) Realistic shock geometry, i.e., shock obliquity? (SN1006)
e) Heavy element acceleration? (CR knee region)
f) How do details of plasma physics influence results? (e.g., injection efficiency;
saturation of Bell’s instability; spectral shape at maximum energy)
Energy, 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):
Emax
TeV

 109
Ethermal keV
Diff Length TeV protons
1011
electron skin depth, (c/pe )
Accel Time to TeV
pe-1
1014
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
Escaping particles in Nonlinear DSA:
1) Highest energy particles must scatter in self-generated turbulence.
a) At some distance from shock, this turbulence will be weak enough that
particles freely stream away.
b) As these particles stream away, they generate turbulence that will scatter next
generation of particles
2) In steady-state DSA, there is no doubt that the highest energy particles
must decouple and escape – No other way to conserve energy.
a) In any real shock, there will be a finite length scale that will set
maximum momentun, pmax. Above pmax, particles escape.
b) Lengths are measured in gyroradii, so B-field and MFA importantly
coupled to escape and pmax
c) The escape reduces pressure of shocked gas and causes the overall
shock compression ratio to increase (r > 7 possible).
3) Even if DSA is time dependent and has not reached a steady-state, the
highest energy particles in the system must escape.
a) In a self-consistent shock, the highest energy particles won’t have
turbulence to interact with until they produce it.
b) Time-dependent calculations (i.e., PIC sims.) needed for full solution.
Shocks with and without
B-field amplification
Monte Carlo Particle distribution functions
f(p) times p4
protons
No B-amp
The maximum CR energy a given
shock can produce increases with
B-amp
p4 f(p)
BUT
B-amp
Increase is not as large as
downstream Bamp/B0 factor !!
For this example,
Bamp/B0 = 450G/10G = 45
but increase in pmax only ~ x5
All parameters are the same in these
cases except one has B-amplification
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
Riquelme & Spitkovsky 2009
3-D PIC
simulation of
Bell’s instability
Only Bell nonresonant instability
Resonant wave
generation
suppressed
Upstream Free
escape boundary
Determine steady-state, shock structure with iterative,
Monte Carlo technique
Unmodified shock with r = 4
Flow speed
Self-consistent, modified
shock with rtot ~ 11
(rsub~ 3)
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 ]