Cosmic-ray Acceleration and Current-Driven Instabilities

Cosmic-ray Acceleration and Current-Driven Instabilities B. Reville Max-Planck-Institut für Kernphysik, Heidelberg Sep 17 2009, KITP J.G. Kirk, P. Duffy, S.O Sullivan, Y. Ohira, F. Takahara

Outline Analysis of yesterday s football match Particle acceleration at non-relativistic shocks Field amplification - streaming instabilities Transport properties of test particles in amplified field Cosmic ray modified shocks & Free escape boundaries

DSA - The diffusion approximation Transport equation f t + ( uf κ(x, p) f ) = ( x x p 3 p 3 f u ) x Steady-state test particle solution: where q = 3r/r 1 p f + (p) = ap q dp + q 0 p ( ) p q f (p ) p N(E) E 2 for strong shocks - consistent with GCR spectrum allowing for energy-dependent propagation N P 1 f e κ/u P > P1 Shape of the spectrum is independent of κ. x

Wave modification supernova shock precursors steady-state solution: ( ) δb 2 P cr0 = M a B ρu 2 CRs stream at shock speed Streaming instability (resonant Alfvén waves neglecting dissipation) U a t + u U a x = v P cr a x (Lucek & Bell 00, Bell & Lucek 01) non-linear McKenzie & Völk 81 ( ) δb 2 M a (1 y 2 ) = B 2y( y 1/2M a ) see also Drury 83

Observational evidence of MFA?? (Long et al. 2003) L min κ/u 10 19 γ 1/2 kev B 3/2 3µG u 1 10 8 cm Chandra resolution at 2 kpc 3 10 16 cm

Cosmic-ray current driven instability (Bell 2004) 3 component plasma - thermal electrons, protons + non-thermal protons (CRs) non-thermal protons provide current - J cr en cr v sh zero net current and charge qs n s v s = 0, qs n s = 0 B = α J α = J MHD + J cr ω Linearizing the MHD/kinetic equations results: ω 2 = v 2 A k 2 ± B 0J cr ρ 0 c k[1 S(k)] log 10 k

Resonant Mode kr g0 < 1- linear theory P cr ρv 2 s v s c M2 A 1 waves are unmodified, i.e. ω r = v A k growth rate given by ion-cyclotron resonance s(k) 1 + i 3π 16 kr g0 9π 3 e Γ = 16 ρ 1/2 0 c n cr (p > p res ) [< v > 13 ] σa where p res = eb/ck, σ = ln f ln p Kulsrud & Pearce, Melrose & Wentzel, Cesarsky etc.

Non-resonant mode - Growth and saturation Condition for non-res instability ζm 2 A 1 Maximum growth rate Γ max = 1 2 ζm v s A where ζ = n cr p res r g n 0 m p v s saturation when currents associated with waves: k δb 4πn cr eβ sh k 1/r g - saturated field energy B 2 w 8π U cr β sh = ηρu 2 sh β sh fast, efficiently accelerating shocks ideal for magnetic field amplification What do the numerical simulations tell us?

Numerical simulations quasi-linear steady-state solution CRs stream at shock speed simulation box L box κ(p min )/u Box is periodic j cr (x, t) = const. although see Lucek & Bell 2001, Zirakashvili et al 2008, Niemiec et al. 2008, Riquelme & Spitkovsky 2009 j cr B 0 j cr E = u (j cr δb) once fluid is set in motion, energy pumped into system increases

Numerical simulations-mhd (Bell 2004)

Numerical simulations - Non-linear structure Bell 2004 Zirakashvili et al. 2008 BR et al. 2008 Structures are approaching scale of simulation box However, no bulk acceleration of fluid observed

Particle in Cell simulations v d,cr = 0.1c 0.1 0.08 Electron drift velocity Proton drift velocity Relative drift velocity Alfven velocity Velocity [c] 0.06 0.04 0.02 0 0 5 10 15 20 25 30 35 40 Time [τ grow ] Ohira, BR, Kirk, Takahara 2009 (see also: Niemiec et al. 2008, Riquelme & Spitkovsky 2009)

Test particle transport in amplified field Take a snapshot of field in early stages of non-linear development, kl box 1 determine statistical properties of field lines and diffusion integrate particle trajectories dp = qv B dt dx = v dt for many particles, random initial positions and directions

Diffusion coefficients Comparison of diffusion coefficients to Bohm limit in the pre-amplified field B 0 = 3µG, B rms 15µG κ /κ Bohm, κ /κ Bohm 10 1 0.1 0.01 1000 10000 100000 γ BR, O Sullivan, Duffy, Kirk (2008) MNRAS

Cosmic-ray modified shocks - Non-linear DSA When CR pressure becomes large we must go beyond test-particle limit u P CR x=0 Cosmic ray pressure gradient decelerates incoming plasma Total shock compression ratio ρ ds /ρ 0 > 4 sub-shock compression ration< 4 larger effective compression ratio, hardens spectra at higher energies

Cosmic-ray modified shocks - Free escape boundary Scattering waves are self-generated At a certain distance upstream, growth of hydromagnetic waves too slow to confine particles - CRs decouple from plasma magnetic field growth (amplification) driven in transition region

Steady-state solutions coupled hydrodynamic - kinetic equations mass & momentum conservation ρ(x)u(x) = ρ 0 u 0, P cr (x) + ρ(x)u(x) 2 + P g (x) = ρ 0 u 2 0 + P g,0 time dependent transport equation f t + u f x ( κ f ) = 1 du x x 3 dx p f p + Q 0(x, p) Additional boundary condition f (L esc ) = 0 See BR, Kirk & Duffy 2009 for details

Steady-state solutions - method of solution what is the location of escape boundary - transition zone from weak to strong turbulence diffusive current at escape boundary drives growth of waves j cr ( L esc ) = 4πe p 0 κ f x p2 dp, L esc = κ(p ) u 0 nonresonant mode growth dominates provided: ζma 2 j cr p eρ 0 u0 2 MA 2 > 1

Injection efficiency ν R diagram 1 10 ν M=10 50 100 1 R=9.5 R=2.2 R=1.1 500 p 4 f 0 0.1 0.1 0.01 0.01 1 10 100 u 0 = 5000km s 1, n 0 = 1 cm 3, p = 10 3 ν = 4π 3 R mc 2 ρ 0 u0 2 p0 4 f 0(p 0 ), 0.001 0.1 1 10 100 1000 10000 100000 1e+06 p/mc L esc determines max. energy How to determine L esc in self-consistant manner?

Maximum momentum Calculate CR flux (and p ) from numerical SS solution Transition zone determined from condition L adv u sh /Γ max L esc

Maximum momentum L adv L esc 4 3.5 3 10 5 10 4 10 3 10 2 2.5 2 1.5 1 0.5 0 0.01 0.1 ν BR, Kirk & Duffy, ApJ Maximum energy can be calculated as a function of injection parameter

Summary Magnetic field amplification a natural consequence of efficient DSA Non-resonant instability first investigated by Bell likely mechanism Non-linear development and saturation still uncertain, bigger & better simulations required time asymptotic solutions to modified shock problem and boundary conditions investigated particle accelerating shocks appear to be self-organising /self-regulating systems