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 particlesb) DSA mechanism known to be efficient: direct evidence: heliosphere, SNRsc) B-fields larger than expected MFA connected to DSAd) 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 generationb) Scales (length, momentum) large and connected through NL interactionsc) Test-particle approximations lose essential physicsd) Plasma physics important

3) Where do we stand?a) Active work from various directions:

i. Semi-analytic solutions of diffusion-convection equationsii. Monte Carlo particle simulationsiii. Hydrodynamic fluid simulationsiv. Particle-in-cell simulations

b) All making progress on understanding plasma physics but all limited in important ways

Tycho’s SNR Cassam-Chenai et al. 2007

magnetically limited rim

magnetically limited rim

synch loss limited rim

synch loss limited rim radio X-ray

Good evidence for radiation losses and, therefore, large, amplified magnetic field. On order of 10 times higher than expected

If drop from B-field decay instead of radiation losses, expect synch radio and synch X-rays to fall off together.

Radial cuts

Chandra observations of Tycho’s SNR (Warren et al. 2005)

Sharp synch. X-ray edgesEvidence for High (amplified) B-fields in SNRs Cassam-Chenai et al. 2007

keV(CD)8.02.0(FS)H

CD

FS

R

R

SE

SE southeast

Efficient DSA: RFS/RCD ~ 1

SNR SN1006Cassam-Chenai et al (2008)

In east and south strong nonthermal emission RFS/RCD ~ 1

inefficient RFS/RCD > 1

Efficient RFS/RCD ~ 1

Evidence for efficient particle acceleration in SNRs

SNR Morphology: Forward shock close to contact discontinuity clear prediction of efficient DSA of protons

Ellison, Mobius & Paschmann 1990

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

Direct evidence at Earth Bow Shock

Thermal leakage injection in action !

Ellison, Jones & Eichler 1981

Dots are AMPTE spacecraft observations

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

u is flow speed

Q(x,p) is injection term

x is position

p is particle momentum

Charged particles gain energy by diffusing in converging flows.

Bulk K.E converted into random particle energy.

Note, for nonrelativistic shocks ONLY

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.

3 /( 1) 4 2( ) , or, ( )( ) r rf p p p N E E

From test-particle theory, in Non-relativistic shocks (Krymskii 76;

Axford, Leer & Skadron 77; Bell 78; Blandford & Ostriker 78):

3 /( 1)0( ) if r r

p skf p p v u V Power law index is:

Independent of any details of diffusion

Independent of shock Obliquity (geometry)

But, for Superthermal particles only

Ratio of specific heats, , along with Mach number, determines

shock compression, r

For high Mach number shocks:

1 (5 / 3) 14 !

1 (5 / 3) 1r

( ) is phase space density

is compression ratio

f p

r

u0 is shock speed

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)

injinj

2 4 /p

p

Ep p dp dp p

)()( 2 pfppN

injln |pp

Energy diverges if r = 4

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)

1

1

rBut

X

subshock

Flow speed

► Concave spectrum

► Compression ratio, rtot > 4

► Low shocked temp. rsub < 4

Temperature

Lose universal power law

TP: f(p) p-4

test particle shock

NL

If acceleration is efficient, shock becomes smooth from backpressure of CRs

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

p4 f

(p)

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

Why is NL DSA with MFA so hard to figure out?

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

upstreamDS

Start with NO B-field, Field is generated self-consistently (Weibel instability?), shock forms, see start of Fermi acceleration Plasma physics done self-consistently!

B-field

Density

B generated at shock

Shock Mass density

En. density in B

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)Amato, Blasi, Caprioli, Morlino, Vietri: Semi-analytic2)Bell: Semi-analytic and PIC simulations3)Berezhko, Volk, Ksenofontov: Semi-analytic 4)Malkov: Semi-analytic5)Niemiec & Pohl: PIC6)Pelletier and co-workers: MHD, relativistic shocks7)Reville, Kirk & co-workers: MHD, PIC8)Spitkovsky and co-workers; Hoshino and co-workers; other PIC simulators: Particle-In-Cell simulations, so far, mainly rel. shocks9)Vladimirov, Ellison, Bykov: Monte Carlo10)Zirakashvili & Ptuskin: Semi-analytic, MHD

11)Apologies to people I missed …

growth of magnetic turbulence energy density, W(x,k). (x position; k wavevector)

energetic particle pressure gradient. (p momentum)

VG parameterizes complicated plasma physics

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

)(res

CR ),(),(

stream kG

ppdk

dp

x

pxPVkxW

dt

d

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

Produce turbulence resonantly assuming QLT

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)

2) Resonant diffusion approximation (QLT) (e.g., Skilling 75; Bell 1978; Amato & Blasi 2006):

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-fieldc) Maximum p; part ∝ p2 (critical for Emax)

4) Scattering for thermal particles?

0

2eff ),(

2

1

8

)(dkkxW

xB

),(

3

1),( ,),(

eff

pxvpxDeB

cppx

1)(0

res0res eB

pckBrk g),(

11),(

res2

22

2 kxWe

cppx

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

upstream

DSParticle distributions and wave spectra at various positions relative to subshock

for resonant wave production

subshock

Nonlinear Shock structure, i.e., Flow speed vs. position

Position relative to subshock at x = 0[ units of convective gyroradius]

0

2eff ),(

2

1

8

)(dkkxW

xB

),(

3

1),( ,),(

effpxvpxD

eB

cppx Bohm approx. for

D(x,p)

k W(k,p)p4 f(p)

D(x,p)/p

Iterate:

D(x,p)f(x,p)

W(k,p) upstream

DS

DS

Nonlinear Shock structure

u(x)

Red: Bohm diffusion approximation

Flow speed

Beff

subshock

Amplified B-fieldB0 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.

upstream DS

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. CR current produces waves with scales short compared to CR gyro-radius

ii. CR current produces waves with scales long compared to CR gyro-radius

2) Calculation of D(x,p) once turbulence is knowna) 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)

max

thermal

9TeV10

keV

E

E

pe

11Diff Length TeV protons10

electron skin depth, (c/ )

Energy range:

Length scale (number of cells in 1-D):

Run time (number of time steps): -1

pe

14Accel Time to TeV10

Energy, length, & time scales: Requirements for PIC simulations to do “entire” DSA MFA problem in non-relativistic shocks:

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.

No B-amp

B-amp

Shocks with and without B-field amplification

The maximum CR energy a given shock can produce increases with B-amp

BUT

Increase is not as large as downstream Bamp/B0 factor !!

Monte Carlo Particle distribution functions f(p) times p4

All parameters are the same in these cases except one has B-amplification

p

4 f

(p)

For this example,

Bamp/B0 = 450G/10G = 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

protons

Riquelme & Spitkovsky 2009

3-D PIC simulation of Bell’s instability

Only Bell non-resonant instability

Resonant wave generation suppressed

Determine steady-state, shock structure with iterative, Monte Carlo technique

Position relative to subshock at x = 0

[ units of convective gyroradius ]

Upstream Free escape boundary

Unmodified shock with r = 4

Self-consistent, modified

shock with rtot ~ 11 (rsub~

3)

Energy Flux (only conserved when escaping particles taken into account)

Momentum Flux conserved (within few %)

Flow speed