May 17-19, 2006 Diffusive Shock Acceleration of Cosmic Rays Hyesung Kang, Pusan National University, KOREA T. W. Jones, University of Minnesota, USA

May 17-19, Astrophysical plasmas are ionized, magnetized, often shock heated, tenuous gas. - CRs & turbulent B fields are ubiquitous in astrophysical plasmas. - It is important to understand the interactions btw charged particles and turbulent B fields to understand the CR acceleration. - Diffusive shock acceleration provides a natural explanation for CRs. -Recent Progresses in DSA theory: 1) injection and drift acceleration at perpendicular shocks 2) comparison with DSA theory with observation of SNRs 3) DSA simulation of 1D spherical SNRs

May 17-19, 2006 scattering of particles in turbulent magnetic fields  isotropization in local fluid frame  transport can be treated as diffusion process streaming CRs - drive large-amplitude Alfven waves - amplify B field( Lucek & Bell 2000) upstream downstream Interactions between particles and fields

May 17-19, Full plasma simulations: follow the individual particles and B fields, provide most complete picture, but computationally too expensive - Monte Carlo Simulations with a scattering model: reproduces observed particle spectrum (Ellison, Baring 90s) applicable only for a steady-state shock - Two-Fluid Simulations: solve for E CR + gasdynamics computationally cheap and efficient, but strong dependence on closure parameters ( ) and injection rate (Drury, Dorfi, KJ 90s) - Kinetic Simulations : solve for f(p) + gasdynamics 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: 1D plane-parallel and spherical geometry, AMR technique, self-consistent thermal leakage injection model coarse-grained finite momentum volume method Numerical Methods for the Particle Acceleration

May 17-19, 2006 Complex microphysics: particles  waves in B field Following individual particle trajectories and evolution of fields are impractical.  diffusion approximation (isotropy in local fluid frame is required)  Diffusion-convection equation for f(p) = isotropic part in Kinetic simulations B n  Bn Geometry of an oblique shock shock Injection coefficient x

May 17-19, 2006  Bn  Bn Parallel (  Bn =0) vs. Perpendicular (  Bn =90) shock Injection is efficient at parallel shocks, while it is difficult in perpendicular shocks Slide from Jokipii (2004): KAW3

May 17-19, 2006 Three Shock Acceleration mechanisms work together. 1)First-order Fermi mechanism: scattering across the shock dominant at quasi-parallel shocks (  Bn < 45) 2)Shock Drift Acceleration: drift along the shock surface dominant at quasi-perpendicular shocks (  Bn > 45) 3)Second-order Fermi mechanism: Stochastic process, turbulent acceleration  add momentum diffusion term

May 17-19, 2006 U2U2 U1U1 Shock front particle downstream upstream 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” energy gain at each crossing Converging mirrors B mean field

May 17-19, 2006 Parallel diffusion coefficient For completely random field (scattering within one gyroradius,  =1)  “Bohm diffusion coefficient” minimum value particles diffuse on diffusion length scale l diff =  || (p) / U s so they cross the shock on diffusion time t diff = l diff / U s =  || (p) / U s 2 smallest  means shortest crossing time and fastest acceleration.  Bohm diffusion with large B and large U s leads to fast acceleration.  highest E max for given shock size and age for parallel shocks

May 17-19, 2006 Thermal leakage injection at quasi-parallel shocks: due to small anisotropy in velocity distribution in local fluid frame, suprathermal particles in non- Maxwellian tail  leak upstream of shock B 0 uniform field self- generated wave leaking particles B w compressed waves hot thermalized plasma unshocked gas CRs streaming upstream generate MHD waves (Bell & Lucek)  compressed and amplified in downstream: B w  Bohm diffusion is valid Suprathermal particles leak out of thermal pool into CR population.

May 17-19, 2006 Energy gain comes mainly from drifting in the convection electric field along the shock surface (Jokipii, 1982), i.e.  = |q E L|,  “Drift acceleration”  but particles are advected downstream with field lines, so injection is difficult: (Baring et al. 1994, Ellison et al. 1995, Giacalone & Ellison 2000) y x Drift Acceleration in perpendicular shocks with weak turbulences Particle trajectory in weakly turbulent fields B

May 17-19, 2006 Diffusive Shock Acceleration at oblique shocks Turbulent B field with Kolmogorov spectrum smaller  xx at perpendicular shocks  shorter acceleration time scale  higher E max than parallel shocks Giacalone & Jolipii 1999 Monte Carlo Simulation by Meli & Biermann (2006)

May 17-19, 2006 Test-Particle simulation at oblique shocks : Giacalone (2005a) stronger turbulence  more efficient injection Injection energy weakly depends on  Bn for fully turbulent fields. ~ 10 % reduction at perpendicular shocks dJ/dE = f(p)p 2 (  B/B) 2 =1

May 17-19, 2006 Test-Particle simulation at oblique shocks : Giacalone (2005a) (  B/B) 2 =1 dJ/dE = f(p)p 2 weak fluctuations Injection is less efficient, but acceleration is faster at perpendicular shocks for weakly turbulent fields. The perpendicular shock accelerates particles to higher energies compared to the parallel shock at the same simulation time.

May 17-19, 2006 density of particles with energies E > 10E p dotted lines: field lines Particles are injected where field lines cross the shock surface  efficient injection Hybrid plasma simulations of perpendicular shock : Giacalone (2005b) - acceleration of thermal protons by perpendicular shocks : thermal leakage - Field line meandering due to large scale turbulent B fields  increased cross-field transport  efficient injection at shock - thermal particles can be efficiently accelerated to high energies by a perpendicular shock -injection problem for perpendicular shocks: solved !

May 17-19, 2006 Parallel vs. Perpendicular Shocks for Type Ia SNRs : ion injection Ion injection only for quasi-parallel shocks (polar cap regions only)  spherical flux from paralleshock shock calculations should be reduced by f re ~0.2

May 17-19, 2006 Slide from Voelk (2006)  Determination of B amplification factor, ion injection rate, proton-to-electron number ratio with SNR observations: Comparison with kinetic simulation (Berezhko & Voelk)

May 17-19, 2006 Recent Observations of SNRs in X-ray and radio: (Voelk et al. 2005) Cas A, SN 1006, Tyco, RCW86, Kepler, RXJ1737, … - thin shell of X-ray emission (strong synchrotron cooling)  B field amplification through streaming of CR nuclear component into upstream plasma (Bell 2004) is required to fit the observations  Observational proof for dominance of hadronic CRs at SNRs -Dipolar radiation: consistent with uniform B field configuration - Ion injection rate :  ~ Proton/electron ratio: K p/e ~ ~50% of SN explosion energy is transferred to CRs.  Consistent picture of DSA at SNRs

May 17-19, 2006 E -2.7 E -3.1 CRs observed at Earth: N(E): power-law spectrum “universal” acceleration mechanism working on a wide range of scales  DSA in the test particle limit predicts a universal power-law f(p) ~ p -q N(E) ~ E -q+2 q = 3r/(r-1) r =  2 /  1 =u 1 /u 2 this explains the universal power-law, independent of shock parameters !

May 17-19, 2006 u 0 =(15km/s)M 0 u 0 =(150km/s)M 0 Effects of upstream CRs for low M s shocks CR acceleration efficiency  vs. Ms for plane-parallel shocks 1) The CR acceleration efficiency is determined mainly by M s 2) It increases with M s (shock Mach no.) but it asymptotes to a limiting value of  ~ 0.5 for M s > 30. 3) thermal leakage process: a fraction of  = of the incoming particles become CRs (at quasi-parallel shocks). Kang & Jones 2005

May 17-19, 2006 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 gas frame  instead of u  smaller vel jump and less efficient acceleration generate waves U1U1U1U1 upstream

May 17-19, CRASH code in 1D plane-parallel geometry = Adaptive Mesh Refinement (AMR) + shock tracking technique in the shock rest frame (thru Galilean velocity transformation) (Kang et al. 2001) - new CRASH code in 1D spherical geometry = Adaptive Mesh Refinement (AMR) + shock tracking technique in a comoving frame which expands with the shock  The shock stays in the same location (zone). RsRs R s = x s a just like Hubble expansion

May 17-19, 2006 Basic Equations for 1D spherical shocks in the Comoving Frame Wave drift + heating terms

May 17-19, 2006 SNR simulations with 1D spherical CRASH code

May 17-19, 2006 Strong nonlinear modification.

May 17-19, 2006 moderate nonlinear modification

May 17-19, 2006 = total CR number / particle no. passed though shock

May 17-19, 2006 N p : power-law like G p : non-linear concave curvature q ~ 2.2 near p inj q ~ 1.6 near p max Our results are consistent with the calculations by Berezhko et al.

May 17-19, 2006 Summary - CRs & turbulent B fields are natural byproducts of the collisionless shock formation process: they are ubiquitous in cosmic plasmas. - DSA produces a nearly universal power-law spectrum with the correct slopes. - With turbulent fields, thermal leakage injection works well even at perpendicular shocks as well as parallel shocks -, so perpendicular shocks are faster accelerators - About 50 % of shock kinetic E can be transferred to CRs for strong shocks with M s > thermal leakage process: a fraction of  = of the incoming particles become CRs at shocks. - Observations of SNRs support the dominance of CR ions (through amplified B field) and  = and K p/e ~ 100.

May 17-19, 2006 Test-Particle simulations at oblique shocks : Giacalone (2005a)

May 17-19, 2006 Numerical Tool:CRASH (Cosmic Ray Amr SHock ) Code Bohm type diffusion: for p >>1 - wide range of diffusion length scales to be resolved: from thermal injection scale to outer scales for the highest p 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 N rf =100 Kang et al. 2001

May 17-19, 2006 Time evolution of the M 0 = 5 shock structure. At t=0, pure gasdynamic shock with Pc=0 (red lines). t=0 Kang, Jones & Gieseler D Plane parallel Shock DSA simulation “CR modified shocks” - presusor + subshock - reduced P g - enhanced compression precursor No simple shock jump condition  Need numerical simulations to calculate the CR acceleration efficiency preshockpostshock

May 17-19, 2006 initial Maxwellian Concave curve CR feedback effects  gas cooling (P g decrease)  thermal leakage  power-law tail  concave curve at high E power-law tail (CRs) Particles diffuse on different l d (p) and feel different  u, so the slope depends on p. f(p) ~ p -q Evolution of CR distribution function in DSA simulation f(p): number of particles in the momentum bin [p, p+dp], g(p) = p 4 f(p) injection momenta thermal g(p) = f(p)p 4

May 17-19, 2006 electron acceleration mechanisms:  direct electric field acceleration (DC acceleration) (Holman, 1985; Benz, 1987; Litvinenko, 2000; Zaitsev et al., 2000)  stochastic acceleration via wave-particle interaction (Melrose, 1994; Miller et al., 1997)  shock waves (Holman & Pesses,1983; Schlickeiser, 1984; Mann & Claßen, 1995; Mann et al., 2001)  outflow from the reconnection site (termination shock) (Forbes, 1986; Tsuneta & Naito, 1998; Aurass, Vrsnak & Mann, 2002)

May 17-19, 2006 Thermal Leakage Injection at parallel shocks has been observed - suprathermal particle leak out of thermal pool into CR population (power-law tail)  injection rate  ~ – postshock thermal CRs preshock comparison of Monte Carlo simulations with direct measurement at Earth’s bow shock