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Diffusive Shock Acceleration of Cosmic Rays

Diffusive Shock Acceleration of Cosmic Rays. Hyesung Kang, Pusan National University, KOREA T. W. Jones, University of Minnesota, USA. - Astrophysical plasmas are ionized, magnetized, often shock heated, tenuous gas. - CRs & turbulent B fields are ubiquitous in astrophysical plasmas.

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Diffusive Shock Acceleration of Cosmic Rays

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  1. Diffusive Shock Acceleration of Cosmic Rays Hyesung Kang, Pusan National University, KOREA T. W. Jones, University of Minnesota, USA KAW4@KASI.Daejeon.Korea

  2. - 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 KAW4@KASI.Daejeon.Korea

  3. Interactions between particles and fields scattering of particles in turbulent magnetic fields isotropization in local fluid frame transport can be treated as diffusion process downstream upstream streaming CRs - drive large-amplitude Alfven waves - amplify B field( Lucek & Bell 2000) KAW4@KASI.Daejeon.Korea

  4. Numerical Methods for the Particle Acceleration • 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 ECR + 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 KAW4@KASI.Daejeon.Korea

  5. Injection coefficient • 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 partin Kinetic simulations shock B n QBn x Geometry of an oblique shock KAW4@KASI.Daejeon.Korea

  6. Parallel (QBn=0) vs. Perpendicular (QBn=90) shock Slide from Jokipii (2004): KAW3 Injection is efficient at parallel shocks, while it is difficult in perpendicular shocks KAW4@KASI.Daejeon.Korea

  7. Three Shock Acceleration mechanisms work together. • First-order Fermi mechanism: scattering across the shock dominant at quasi-parallel shocks (QBn< 45) • Shock Drift Acceleration: drift along the shock surface dominant at quasi-perpendicular shocks (QBn> 45) • Second-order Fermi mechanism: Stochastic process, turbulent acceleration  add momentum diffusion term KAW4@KASI.Daejeon.Korea

  8. DiffusiveShockAcceleration 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” Shock front mean field B particle energy gain at each crossing U2 U1 upstream downstream shock rest frame Converging mirrors KAW4@KASI.Daejeon.Korea

  9. Parallel diffusion coefficient • For completely random field (scattering within one gyroradius, h=1) • “Bohm diffusion coefficient” minimum value • particles diffuse on diffusion length scaleldiff = k||(p)/ Us • so they cross the shock on diffusion timetdiff = ldiff / Us= k||(p) / Us2 • smallest k means shortest crossing time and fastest acceleration. • Bohm diffusion with large B and large Us leads to fast acceleration. • highest Emax for given shock size and age for parallel shocks KAW4@KASI.Daejeon.Korea

  10. 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 hot thermalized plasma unshocked gas Bw compressedwaves B0 uniform field • CRs streaming upstream • generate MHD waves • (Bell & Lucek) •  compressed and amplified • in downstream: Bw • Bohm diffusion is valid self-generated wave leaking particles Suprathermal particles leak out of thermal pool into CR population. KAW4@KASI.Daejeon.Korea

  11. y x Drift Acceleration in perpendicular shocks with weak turbulences B Particle trajectory in weakly turbulent fields • Energy gain comes mainly from drifting in the convection electric field along the shock surface (Jokipii, 1982), i.e.De = |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) KAW4@KASI.Daejeon.Korea

  12. DiffusiveShockAcceleration at oblique shocks Giacalone & Jolipii 1999 • Turbulent B field with Kolmogorov spectrum • smaller kxx at perpendicular shocks •  shorter acceleration time scale • higher Emaxthan parallel shocks Monte Carlo Simulation by Meli & Biermann (2006) KAW4@KASI.Daejeon.Korea

  13. Test-Particle simulation at oblique shocks : Giacalone (2005a) (DB/B)2=1 dJ/dE = f(p)p2 stronger turbulence  more efficient injection Injection energy weakly depends on QBn for fully turbulent fields. ~ 10 % reduction at perpendicular shocks KAW4@KASI.Daejeon.Korea

  14. Test-Particle simulation at oblique shocks : Giacalone (2005a) (DB/B)2=1 dJ/dE = f(p)p2 weak fluctuations The perpendicular shock accelerates particles to higher energies compared to the parallel shock at the same simulation time . Injection is less efficient, but acceleration is faster at perpendicular shocks for weakly turbulent fields. KAW4@KASI.Daejeon.Korea

  15. 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 beefficiently accelerated to highenergies by a perpendicularshock • injection problem for perpendicular shocks: solved ! Particles are injected where field lines cross the shock surface  efficient injection density of particleswith energies E >10Ep dotted lines: field lines KAW4@KASI.Daejeon.Korea

  16. 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 fre ~0.2 KAW4@KASI.Daejeon.Korea

  17. Determination of B amplification factor, ion injection rate, proton-to-electron number ratio with SNR observations: Comparison with kinetic simulation (Berezhko & Voelk) x Slide from Voelk (2006) KAW4@KASI.Daejeon.Korea

  18. 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 : x~10-4 - Proton/electron ratio: Kp/e ~ 50-100 • ~50% of SN explosion energy is transferred to CRs. •  Consistent picture of DSA at SNRs KAW4@KASI.Daejeon.Korea

  19. 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 E-2.7 f(p) ~ p-q N(E) ~ E-q+2 q = 3r/(r-1) r = r2/r1=u1/u2 E-3.1 this explains the universal power-law, independent of shock parameters ! KAW4@KASI.Daejeon.Korea

  20. CR acceleration efficiency F vs. Ms for plane-parallel shocks Kang & Jones 2005 u0=(15km/s)M0 1) The CR acceleration efficiency is determined mainly by Ms 2) It increases with Ms (shock Mach no.) but it asymptotes to a limiting value of F ~ 0.5 for Ms > 30. 3) thermal leakage process: a fraction of x= 10-4 - 10-3 of the incoming particles become CRs (at quasi-parallel shocks). u0=(150km/s)M0 Effects of upstream CRs for low Ms shocks KAW4@KASI.Daejeon.Korea

  21. U1 generate waves Diffusion-Convection Equation with Alfven wave drift + heating • - 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 streaming CRs upstream KAW4@KASI.Daejeon.Korea

  22. - 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). just like Hubble expansion Rs = xs a Rs KAW4@KASI.Daejeon.Korea

  23. Wave drift + heating terms Basic Equations for 1D spherical shocks in the Comoving Frame KAW4@KASI.Daejeon.Korea

  24. SNR simulations with 1D spherical CRASH code KAW4@KASI.Daejeon.Korea

  25. Strong nonlinear modification. KAW4@KASI.Daejeon.Korea

  26. moderate nonlinear modification KAW4@KASI.Daejeon.Korea

  27. = total CR number / particle no. passed though shock KAW4@KASI.Daejeon.Korea

  28. N p : power-law like G p : non-linear concave curvature q ~ 2.2 near pinj q ~ 1.6 near pmax Our results are consistent with the calculations by Berezhko et al. KAW4@KASI.Daejeon.Korea

  29. 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 Ms > 30. - thermal leakage process: a fraction of x = 10-4 - 10-3 of the incoming particles become CRs at shocks. - Observations of SNRs support the dominance of CR ions (through amplified B field) and x = 10-4 - 10-3 and Kp/e ~ 100. KAW4@KASI.Daejeon.Korea

  30. Test-Particle simulations at oblique shocks : Giacalone (2005a) KAW4@KASI.Daejeon.Korea

  31. 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 Kang et al. 2001 Nrf=100 KAW4@KASI.Daejeon.Korea

  32. “CR modified shocks” • - presusor + subshock • - reduced Pg • enhanced compression t=0 -1D Plane parallel Shock DSA simulation postshock preshock Time evolution ofthe M0 = 5shock structure. At t=0, pure gasdynamic shock with Pc=0 (red lines). • No simple shock jump condition • Need numerical simulations to calculate the CR acceleration efficiency precursor Kang, Jones & Gieseler 2002 KAW4@KASI.Daejeon.Korea

  33. Evolution of CR distribution function in DSA simulationf(p): number of particles in the momentum bin [p, p+dp], g(p) = p4 f(p) • CR feedback effects • gas cooling (Pg decrease) • thermal leakage • power-law tail • concave curve at high E initial Maxwellian thermal power-law tail (CRs) f(p) ~ p-q Particles diffuse on different ld(p) and feel different Du, so the slope depends on p. g(p) = f(p)p4 Concave curve injection momenta KAW4@KASI.Daejeon.Korea

  34. 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) KAW4@KASI.Daejeon.Korea

  35. preshock postshock Thermal Leakage Injection at parallel shocks has been observed- suprathermal particle leak out of thermal pool into CR population (power-law tail) injection rate x ~ 10-4 – 10-3 thermal comparison of Monte Carlo simulations with direct measurement at Earth’s bow shock CRs KAW4@KASI.Daejeon.Korea

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