1. 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

2. Tycho’s SNR Cassam-Chenai et al. 2007 magnetically 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 edges Evidence for High (amplified) B-fields in SNRs Cassam-Chenai et al. 2007

3. SE south east Efficient DSA: R FS /R CD ~ 1 SNR SN1006 Cassam-Chenai et al (2008) In east and south  strong nonthermal emission  R FS /R CD ~ 1 inefficient R FS /R CD > 1 Efficient R FS /R CD ~ 1 Evidence for efficient particle acceleration in SNRs SNR Morphology: Forward shock close to contact discontinuity  clear prediction of efficient DSA of protons

4. 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

5. Table from Caprioli et al 2009

6. Can describe DSA (in non-rel shocks) with transport equation (i.e., diffusion-convection equation) Requires assumption that v part >> u 0 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

7. 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.

8. From 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  Ratio of specific heats, , along with Mach number, determines shock compression, r For high Mach number shocks:  u 0 is shock speed So-called “Universal” power law from shock acceleration

9. BUT, 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 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 10 4 injected) But

10. X subshock Flow speed ► Concave spectrum ► Compression ratio, r tot > 4 ► Low shocked temp. r sub < 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 p 4 f(p)

11. 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 (M Sonic < ~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?

12. 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 upstream DS 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

13. 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

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): 1)Amato, Blasi, Caprioli, Morlino, Vietri: Semi-analytic 2)Bell: Semi-analytic and PIC simulations 3)Berezhko, Volk, Ksenofontov: Semi-analytic 4)Malkov: Semi-analytic 5)Niemiec & Pohl: PIC 6)Pelletier and co-workers: MHD, relativistic shocks 7)Reville, Kirk & co-workers: MHD, PIC 8)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 …

15. growth of magnetic turbulence energy density, W(x,k). (x position; k wavevector) energetic particle pressure gradient. (p momentum) V G 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 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?

16. 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 B eff 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-field c)Maximum p; part ∝ p 2 (critical for E max ) 4)Scattering for thermal particles?

17. 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

18. upstream DS Particle 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]

19. Bohm approx. for D(x,p) k W(k,p) p 4 f(p) D(x,p)/p Iterate: D(x,p) f(x,p) W(k,p) upstream DS Nonlinear Shock structure u(x)

20. Red: Bohm diffusion approximation Flow speed B eff subshock Amplified B-field B 0 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

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 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 known a)Resonant (QLT): Particles with gyro-radius ~ waves gives part ∝ p b)Non-resonant: Particles with gyro-radius >> waves gives part ∝ p 2 3) Production of turbulence and diffusion must be coupled to NL shock structure including injection and escape

22. 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., B amp >> 4 x B 0 ) 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., E max ? b)Effect of escaping particles at E max ? 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)

24. Energy range: Length scale (number of cells in 1-D): Run time (number of time steps): 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

25. 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, p max. Above p max, particles escape. b)Lengths are measured in gyroradii, so B-field and MFA importantly coupled to escape and p max 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.

26. 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 B amp /B 0 factor !! Monte Carlo Particle distribution functions f(p) times p 4 All parameters are the same in these cases except one has B-amplification p 4 f(p) For this example, B amp /B 0 = 450  G/10  G = 45 but increase in p max 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

27. Riquelme & Spitkovsky 2009 3-D PIC simulation of Bell’s instability

28. Only Bell non- resonant instability Resonant wave generation suppressed

31. 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 r tot ~ 11 (r sub ~ 3) Energy Flux (only conserved when escaping particles taken into account) Momentum Flux conserved (within few %) Flow speed