Nov. 1-3, 2006 Self-Similar Evolution of Cosmic Ray Modified Shocks Hyesung Kang Pusan National University, KOREA

Nov. 1-3, CR energy flux emerged from shocks F CR =  (M) F k Thermal E CR E thermalization efficiency:  (M) CR acceleration efficiency:  (M)  1 V s = u 1 E gas - kinetic energy flux through shocks F k = (1/2)   V s 3 - net thermal energy flux generated at shocks F th = (3/2) [P 2 -P 1        u 2 =  (M) F k E CR

Nov. 1-3, 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. Astrophysical Plasma thermal ions & electrons Cosmic ray ions & electrons

Nov. 1-3, 2006 scattering of particles in turbulent magnetic fields  isotropization in local fluid frame  transport can be treated as diffusion process streaming CRs upstream of shocks  excite large-amplitude Alfven waves  amplify B field ( Lucek & Bell 2000) upstream downstream Interactions btw particles and fields: examples

Nov. 1-3, 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 E CR + 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 Numerical Methods to study the Particle Acceleration

Nov. 1-3, 2006 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 B n  Bn Geometry of an oblique shock shock Injection coefficient x So complex microphysics of interactions are described by macrophysical models for  p  & Q

Nov. 1-3, 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 fluid frame  instead of u  smaller velocity jump and less efficient acceleration generate waves U1U1U1U1 upstream PcPc

Nov. 1-3, 2006  Bn  Bn Parallel (  Bn =0) vs. Perpendicular (  Bn =90) shock Injection is less efficient, but the acceleration is faster at perpendicular shocks Slide from Jokipii (2004): KAW3 FULL MHD + CR terms Gasdynamics + CR terms

Nov. 1-3, 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

Nov. 1-3, 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 This is often considered as a valid assumption because of self-excited Alfven waves in the precursor of strong shocks.

Nov. 1-3, 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 Suprathermal particles leak out of thermal pool into CR population.

Nov. 1-3, 2006 “Transparency function”: probability that particles at a given velocity can leak upstream. e.g.  esc = 1 for CRs  esc = 0 for thermal ptls CRs gas ptls  B =0.3  B =0.25 Smaller  B : stronger turbulence, difficult to cross the shock, less efficient injection So the injection rate is controlled by the shock Mach number and  B

Nov. 1-3, 2006 Basic Equations for 1D plane- parallel CR shock S = modified entropy = P g /   to follow adiabatic compression in the precursor W= wave dissipation heating L= thermal energy loss due to injection across the shock outside the shock ordinary gas dynamic Eq. + P c terms

Nov. 1-3, 2006 CR modified shock: diffusive nature of CR pressure introduces some characteristics different from a gasdynamic shock. - diffusion scales: t d (p)=  (p)/u s 2, l d (p)=  (p)/u s  wide range of scales in the problem: from p inj to p max  numerically challenging !  not a simple jump across a shock total transition = precursor + subshock - acceleration time scale: t acc (p) t d (p)  instantaneous acceleration is not valid so time-dependent calculation is required - particles experience different  u depending on p due to the precursor velocity gradient + l d (p)  f(x,p,t): NOT a simple power-law, but a concave curve should be followed by diffusion-convection equation

Nov. 1-3, For given shock parameters: M s, u s 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 M s but with different u s, if presented in units of “Similarity” in the dynamic evolution of CR shocks - Thermal injection rate: depends on M s and  B

Nov. 1-3, 2006 Three runs with  (p) = 0.1 p 2 /(p 2 +1) 1/2  (p) = p  (p) = p at a same time t/t o =10 P CR,2 approaches time asymptotic values for t/t o > 1. At t=0, M s =20 gasdynamic shock

Nov. 1-3, 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 (p inj /mc=10 -2 ) to outer scales for the highest p (~10 6 ) 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

Nov. 1-3, 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 analytic shock jump condition  Need numerical simulations to calculate the CR acceleration efficiency preshockpostshock

Nov. 1-3, 2006 Evolution from a gasdynamic shock to a CR modified shock. 1) initial states : a gasdynamic shock at x=0 at t=0 - T 0 = 10 4 K and u s = (15 km/s) M s, 10<M s <80 - T 0 = 10 6 K and u s = (150 km/s) M s, 2<M s <30 2) Thermal leakage injection : - more turbulent B  smaller  B  smaller injection - pure injection model : f 0 = 0 - power-law pre-existing CRs: f up (p) = f 0 (p/p inj ) -5 3) B field strength : 4) Bohm type diffusion:

Nov. 1-3, 2006 P cr,2 reaches to an asymptotic value, The shock structure stretches linearly with t.  the shock flow becomes self-similar. CR energy/shock kin. E.

Nov. 1-3, 2006 P c,2 increases with M s but asymptotes to 50% of shock ram pressure. Fraction of injected CR particles is higher for higher M s.

Nov. 1-3, 2006 G p : non-linear concave curvature q ~ 4.2 near p inj q ~ 3.6 near p max f( x s, p)p 4 at the shock at t/t o = 10

Nov. 1-3, 2006 CR acceleration efficiency  vs. Ms for plane-parallel shocks -The CR acceleration efficiency is determined mainly by M s. It increases with M s, but it asymptotes to a limiting value of  ~ 0.5 for M s > 30. -larger  ( larger  A /c s ): less efficient acceleration due to the wave drift in precursor - larger    weaker turbulence: more efficient injection, and less efficient acceleration - pre-existing CRs: higher injection: more CRs - these dependences are weak for strong shocks Pre-exist Pc  B =0.25  B =0.2

Nov. 1-3, 2006 From DSA simulations using our CRASH code for parallel shocks: -Thermal leakage injection rate is controlled mainly by M s and the level of downstream turbulent B fields. (a fraction of  = of the incoming particles become CRs.) - The CR acceleration rate depends on M s, because  acc /t d = fcn(M s ) in other words  u/u = fcn(M s ) - 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 l shock ~u s 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  for the acceleration efficiency -  M s  increases with M s, depends on  B, E B /E th (wave drift speed), but it asymptotes to a limiting value of  ~ 0.5 for M s > 30. SUMMAY

Nov. 1-3, 2006 Thank You !

Nov. 1-3, 2006 Plasma simulations at oblique shocks : Giacalone (2005a) Injection rate weakly depends on  Bn for fully turbulent fields. ~ 10 % reduction at perpendicular shocks (  B/B) 2 =1 The perpendicular shock accelerates particles to higher energies compared to the parallel shock at the same simulation time. parallel perp.

Nov. 1-3, 2006 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 CRgas

Nov. 1-3, 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