kang

Slide1: 

Particle Acceleration in Astrophysical Shock Waves. (Hydrodynamic approach to Diffusive Shock Acceleration) Hyesung Kang, Pusan National University, KOREA T. W. Jones, University of Minnesota, USA

Slide2: 

SN1006 Coma clusters of galaxies Solar wind & Earth’s bow shock Supernova Remant AGN jets Shocks and Cosmic Ray Particles are ubiquitous in astrophysical environments.

Slide3: 

1. Diffusive particle acceleration at astrophysical shocks: (Axford et al. 1977, Bell 1978, Blandford & Ostriker 1978) Fermi 1st order, self-generation of waves by streaming CRs, thermal leakage injection with x= 10-3 , Galactic CRs by SNRs 2. Hydrodynamic approach (Two fluid) : Drury & Volk 1981 - Gas fluid & CR fluid computationally cheap and efficient, but strong dependence on closure parameters ( k, g c ) and injection rate. key questions: post Pc & modified shock structure (Dorfi 1984, 1985, Kang & Jones 1990)

Slide4: 

U2 U1 Shock front particle downstream upstream shock rest frame Diffusive Shock Acceleration in quasi-parallel shocks “ Fermi first order process” Alfven waves in a converging flow act as converging mirrors  particles are scattered by waves  cross the shock many times energy gain at each crossing Converging mirrors B mean field

Slide5: 

3. Diffusion-Convection Equation approach: f(p) (Bell 1987, Falle & Giddings 1987, Kang & Jones 1991)  Berezkho et al. 1994: piston driven shock in 1D spherical geometry, applied to SNRs, renormalization of space variables with diffusion length i.e. : momentum dependent grid spacing  Kang & Jones 2001: CRASH (Cosmic Ray Amr SHock code) 1D plane-parallel and spherical grid comving with a shock AMR technique, self-consistent thermal leakage injection model

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4. Monte Carlo Simulations with a scattering model: -particles scattered with a prescribed model assuming a steady-state shock structure : scattering mean free path -reproduces observed particle spectrum (Ellison et al. 1990, Baring) at interplanetary shocks (direct measurement) and SNRs (radiation spectrum). Earth’s Bow Shock

Slide7: 

Hybrid Plasma simulations for the proton injection & acceleration at oblique shocks : Giacalone (2005) (electrons: charge-neutralizing fluid) (DB/B)2=1 streaming CRs upstream of shocks  excite large-amplitude Alfven waves  amplify B field ( Lucek & Bell 2000, Bell 2004) 5. Plasma Simulations

Slide8: 

Hydrodynamic Approach to DSA at quasi-parallel shocks Slide from Jokipii (2004): KAW3 FULL MHD + CR terms Gasdynamics + CR terms (Kang & Jones)

Slide9: 

In DSA 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 CRs are scattered and isotropized in the wave frame rather than the fluid frame Bohm diffusion model : For completely random field (scattering within one gyroradius)

Slide10: 

Thermal leakage injection at quasi-parallel shocks: due to small anisotropy in velocity distribution in local fluid frame, suprathermal particles in Maxwellian tail leak upstream of shock injected as CRs accelerated via Feremi process B0 uniform field self-generated wave leaking particles Bw compressed waves hot thermalized plasma unshocked gas Suprathermal particles leak out of thermal pool into CR population.

Slide11: 

“Transparency function”: probability that particles at a given velocity can leak upstream. e.g. tesc = 1 for CRs tesc = 0 for thermal ptls CRs gas ptls e B=0.3 e B=0.25 Smaller e B : stronger turbulence, difficult to cross the shock, less efficient injection

Slide12: 

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 bulk flow frame  instead of u  Particles experience smaller vel jump and so less efficient acceleration upstream

Slide13: 

Basic Equations for 1D plane- parallel CR 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 across the shock outside the shock ordinary gas dynamic Eq. + Pc terms

Slide14: 

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 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

Numerical Tool:CRASH (Cosmic Ray Amr SHock ) Code: 

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

Slide16: 

- 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 diffusion scales “Similarity” in the dynamic evolution of CR shocks - Thermal injection rate: depends on Ms and eB

Slide17: 

Comparison of 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/td=10 PCR,2 approaches time asymptotic values for t/to > 1. At t=0, Ms=20 gasdynamic shock Self similar evolution of CR modified shock

Slide18: 

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 - T0= 106 K and us= (150 km/s) Ms 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: (strong p dependence for sub-relativisitc particles) For a given Ms number, T0 determines the shock speed

Slide19: 

Pcr,2 reaches to an asymptotic value, The shock structure stretches linearly with t.  the shock flow becomes self-similar. To obtain the CR acceleration efficiency, we have carried Diffusive Shock Acceleration simulation of 1D Plane parallel Shock CR modified shocks - presusor + subshock - reduced Pg enhanced compression f(p)p4: concave curvat

Slide20: 

G p : non-linear concave curvature q ~ 4.2 near pinj q ~ 3.6 near pmax f( xs, p)p4 at the shock at t/td = 10

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Pc,2 increases with Ms but asymptotes to 50% of shock ram pressure. = Fraction of injected CRs higher for higher Ms. Time Evolution of

Slide22: 

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 more efficient acceleration pre-existing CRs: higher injection: more CRs - these dependences are weak for strong shocks Pre-existing CRs eB=0.25 eB=0.2 q=0.1

Slide23: 

From DSA kinetic simulations using our CRASH code for parallel shocks: -Thermal leakage injection rate is controlled mainly by Ms and the level of upstream turbulent B fields (eB).  a fraction of x= 10-4 - 10-3 of the incoming particles become CRs. - The CR acceleration rate depends mainly on Ms, because tacc/td = fcn(Ms) in other words Du/u = fcn(Ms) - The shock structure broadens as lshock ~us t/8, linearly with time, independent of the diffusion coefficient. Thus the evolution of CR shocks becomes approximately ``self-similar” in time.  F(Ms) increases with Ms, depends on eB and but it asymptotes to a limiting value of F ~ 0.5 for Ms > 30. DSA with Bohm diffusion is very efficient. SUMMAY

Slide24: 

Complex plasma interactions between particles and fields - Turbulent MHD  properties of waves Plasma instabilities  wave excitations & dissipations resonant & non-reson. scatterings ion-neutral collisions Diffuse Shock Acceleration at astrophysical shocks - wave energy density  Diffusion coefficient: k(p) - wave interaction models : uw, W - thermal leakage model: tesc(p)

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Thank You !