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Self-Similar Evolution of Cosmic Ray Modified Shocks

Self-Similar Evolution of Cosmic Ray Modified Shocks. Hyesung Kang Pusan National University, KOREA. Thermal E. CR E. - kinetic energy flux through shocks F k = (1/2) r 1 V s 3. - net thermal energy flux generated at shocks F th = (3/2) [P 2 -P 1 (r 2 /r 1 ) g ] u 2

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Self-Similar Evolution of Cosmic Ray Modified Shocks

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  1. Self-Similar Evolution of Cosmic Ray Modified Shocks Hyesung Kang Pusan National University, KOREA EANAM2006@KASI.Daejeon.Korea

  2. Thermal E CR E - kinetic energy flux through shocks Fk = (1/2)r1Vs3 - net thermal energy flux generated at shocks Fth = (3/2) [P2-P1(r2/r1)g] u2 = d(M) Fk - CR energy flux emerged from shocks FCR= F(M) Fk r1 Vs= u1 Egas thermalization efficiency:d(M) CR acceleration efficiency:F(M) ECR EANAM2006@KASI.Daejeon.Korea

  3. Astrophysical Plasma thermal ions & electrons Cosmic ray ions & electrons - Astrophysical plasmas are composed of thermal particles and cosmic ray particles. - turbulent velocities and B fields are ubiquitous in astrophysical plasmas. - Interactions among these components are important in understanding the CR acceleration. EANAM2006@KASI.Daejeon.Korea

  4. Interactions btw particles and fields: examples scattering of particles in turbulent magnetic fields isotropization in local fluid frame transport can be treated as diffusion process downstream upstream streaming CRs upstream of shocks  excite large-amplitude Alfven waves  amplify B field ( Lucek & Bell 2000) EANAM2006@KASI.Daejeon.Korea

  5. Numerical Methods to study the Particle Acceleration • Full plasma simulations: follow the individual particles and B fields, • provide most complete picture, but computationally very expensive • (see the next talk by Hoshino) • Monte Carlo Simulations with a scattering model: steady-state only • particles scattered with a prescribed model assuming a steady-state shock structure • reproduces observed particle spectrum (Ellison, Baring 90s) • Two-Fluid Simulations:   solve for ECR + gasdynamics Eqns • computationally cheap and efficient, but strong dependence on closure • parameters ( ) and injection rate (Drury, Dorfi, KJ 90s) • - Kinetic Simulations :   solve for f(p) + gasdynamics Eqns • 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:CRASH (Cosmic Ray Amr SHock code) • 1D plane-parallel and spherical grid comving with a shock • AMR technique, self-consistent thermal leakage injection model EANAM2006@KASI.Daejeon.Korea

  6. Injection coefficient shock B n QBn Geometry of an oblique shock • In kinetic simulations • Instead of following individual particle trajectories and evolution of fields • diffusion approximation (isotropy in local fluid frame is required) • Diffusion-convection equation for f(p) = isotropic part So complex microphysics of interactions are described by macrophysical models for k(p) & Q x EANAM2006@KASI.Daejeon.Korea

  7. 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 fluid frame •  instead of u • smaller velocity jump and less efficient acceleration Pc streaming CRs upstream EANAM2006@KASI.Daejeon.Korea

  8. Parallel (QBn=0) vs. Perpendicular (QBn=90) shock Slide from Jokipii (2004): KAW3 FULL MHD + CR terms Gasdynamics + CR terms Injection is less efficient, but the acceleration is faster at perpendicular shocks EANAM2006@KASI.Daejeon.Korea

  9. 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 EANAM2006@KASI.Daejeon.Korea

  10. Parallel diffusion coefficient • For completely random field (scattering within one gyroradius, h=1) • “Bohm diffusion coefficient” minimum value This is often considered as a valid assumption because of self-excited Alfven waves in the precursor of strong shocks. • 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 EANAM2006@KASI.Daejeon.Korea

  11. 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 self-generated wave leaking particles Suprathermal particles leak out of thermal pool into CR population. EANAM2006@KASI.Daejeon.Korea

  12. “Transparency function”: probability that particles at a given velocity can leak upstream. e.g. tesc = 1 for CRs tesc = 0 for thermal ptls CRs Smaller eB : stronger turbulence, difficult to cross the shock, less efficient injection eB=0.25 So the injection rate is controlled by the shock Mach number and eB eB=0.3 gas ptls EANAM2006@KASI.Daejeon.Korea

  13. Basic Equations for 1D plane- parallel CR shock ordinary gas dynamic Eq. + Pc terms across the shock outside the shock S = modified entropy = Pg/rg-1 to follow adiabatic compression in the precursor W= wave dissipation heating L= thermal energy loss due to injection EANAM2006@KASI.Daejeon.Korea

  14. CR modified shock:diffusive nature of CR pressure introduces some characteristics different from a gasdynamic shock. • diffusion scales: td (p)= k(p)/us2, ld (p)= k(p)/us •  wide range of scales in the problem: from pinj to pmax •  numerically challenging ! •  not a simple jump across a shock • total transition = precursor + subshock • acceleration time scale: tacc(p) td(p) •  instantaneous acceleration is not valid • so time-dependent calculation is required • particles experience different Du depending on p • due to the precursor velocity gradient + ld (p) •  f(x,p,t): NOT a simple power-law, but a concave curve • should be followed by diffusion-convection equation EANAM2006@KASI.Daejeon.Korea

  15. “Similarity” in the dynamic evolution of CR shocks - Thermal injection rate:depends on Ms and eB - For given shock parameters: Ms, us the CR acceleration depends on the shock Mach number only. So, for example, the evolution of CR modified shocks is “approximately” similar for two shocks with the same Ms but with different us, if presented in units of EANAM2006@KASI.Daejeon.Korea

  16. Three runs with k(p) = 0.1 p2/(p2+1)1/2 k(p) = 10-4 p k(p) = 10-6 p at a same time t/to=10 PCR,2 approaches time asymptotic values for t/to > 1. At t=0, Ms=20 gasdynamic shock EANAM2006@KASI.Daejeon.Korea

  17. 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 (pinj/mc=10-2) to outer scales for the highest p (~106) 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 Nrf=100 Kang et al. 2001 EANAM2006@KASI.Daejeon.Korea

  18. “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 analytic shock jump condition • Need numerical simulations to calculate the CR acceleration efficiency precursor Kang, Jones & Gieseler 2002 EANAM2006@KASI.Daejeon.Korea

  19. Evolution from a gasdynamic shock to a CR modified shock. 1) initial states : a gasdynamic shock at x=0 at t=0 - T0= 104 K and us= (15 km/s) Ms , 10<Ms<80 - T0= 106 K and us= (150 km/s) Ms, 2<Ms<30 2) Thermal leakage injection : - more turbulent B smaller eB smaller injection - pure injection model : f0 = 0 - power-law pre-existing CRs: fup(p) = f0 (p/pinj) -5 3) B field strength : 4) Bohm type diffusion: EANAM2006@KASI.Daejeon.Korea

  20. Pcr,2 reaches to an asymptotic value, The shock structure stretches linearly with t.  the shock flow becomes self-similar. CR energy/shock kin. E. EANAM2006@KASI.Daejeon.Korea

  21. Pc,2 increases with Ms but asymptotes to 50% of shock ram pressure. Fraction of injected CR particles is higher for higher Ms. EANAM2006@KASI.Daejeon.Korea

  22. f( xs, p)p4 at the shock at t/to = 10 G p : non-linear concave curvature q ~ 4.2 near pinj q ~ 3.6 near pmax EANAM2006@KASI.Daejeon.Korea

  23. CR acceleration efficiency F vs. Ms for plane-parallel shocks • The CR acceleration efficiency is determined mainly by Ms . It increases with Ms, but it asymptotes to a limiting value of F ~ 0.5 for Ms > 30. • larger q ( larger uA/cs): • less efficient acceleration • due to the wave drift in precursor • - larger eB, weaker turbulence: • more efficient injection, • and less efficient acceleration • pre-existing CRs: higher injection: • more CRs • - these dependences are weak for • strong shocks eB=0.2 Pre-exist Pc eB=0.25 EANAM2006@KASI.Daejeon.Korea

  24. SUMMAY From DSA simulations using our CRASH code for parallel shocks: -Thermal leakage injection rate is controlled mainly by Ms and the level of downstream turbulent B fields. (a fraction of x= 10-4 - 10-3 of the incoming particles become CRs.) - The CR acceleration rate depends on Ms, because tacc/td = fcn(Ms) in other words Du/u = fcn(Ms) - The postshock CR pressure reaches a stable value after a balance between fresh injection/acceleration and advection/diffusion of the CR particles away from the shock is established. -The shock structure broadens as lshock ~us t/8, linearly with time, independent of the diffusion coefficient.  So the evolution of CR shocks becomes approximately ``self-similar” in time.  It makes sense to define the CR energy ratio F for the acceleration efficiency - F(Ms) increases with Ms, depends on eB, EB/Eth (wave drift speed), but it asymptotes to a limiting value of F ~ 0.5 for Ms > 30. EANAM2006@KASI.Daejeon.Korea

  25. Thank You ! EANAM2006@KASI.Daejeon.Korea

  26. Plasma simulations at oblique shocks : Giacalone (2005a) (DB/B)2=1 parallel perp. Injection rate weakly depends on QBn for fully turbulent fields. ~ 10 % reduction at perpendicular shocks The perpendicular shock accelerates particles to higher energies compared to the parallel shock at the same simulation time . EANAM2006@KASI.Daejeon.Korea

  27. gas CR • Observational example: • particle spectra in the Solar wind • (Mewaldt et al 2001) • -Thermal+ CR populations • suprathermal particles • leak out of thermal pool • into CR population EANAM2006@KASI.Daejeon.Korea

  28. 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 EANAM2006@KASI.Daejeon.Korea

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