Diffusive shock acceleration an introduction Interstellar medium Rarefied

Diffusive shock acceleration: an introduction

Interstellar medium Rarefied ( thermal) plasma filling the galactic space <n> ~ 1 cm-3 (CGS units are simple) molecular clouds: n ~ 100 -1000 cm-3 T ~ 10 -50 K warm medium: hot medium: T ~ 104 K T ~ 106 -107 K magnetic field SI: n ~ 1 cm-3 n ~ 0. 01 cm-3 <B> 3 G <n> ~ 10 -6 m-3 B ~ <B> n-1/2 <B> ~ 0. 3 n. T 104 K 1 e. V

Cosmic rays are energetic particles. Primary: - protons and heavier nuclei - electrons (and positrons) Secondary CR include also: - antiprotons, positrons, neutrinos, gamma rays with energies much above thermal plasma and the non-thermal energy distribution. In our Galaxy: PCR Pg (= nk. T) PB (= B 2/8 ) ~ 10 -13 erg/cm 3

Particle Flux ( m 2 s sr Ge. V )-1 Cosmic Ray Spectrum 1 particle/m 2 s „Knee” 1 particle/m 2 yr „Ankle” 1 particle/km 2 yr 1 J 6 1018 e. V Energy e. V

CR collisions in ISM For a high energy collision of a CR particle with the interstellar atom (nucleus) we have (n ~ 1/cm 3 and the cross section ~ 10 -24 cm 2)

Cosmic ray sources ? Possible SNRs shock waves. CR energy within the galactic volume ECR = V * CR ~ 1068 cm 3 * 10 -13 erg/cm 3 = 1055 erg Mean CR residence time CR = 2 *107 yr CR production required for a steady-state ECR / CR ~ 1040 erg/s 1 SN / 100 yrs injects ~1051 erg /3*109 s 3*1041 erg/s 10% efficiency is enough

Tycho X-ray picture from Chandra

X-ray H-alpha Supernova remnant Dem L 71

Particle acceleration in the interstellar medium Inhomogeneities of the magnetized plasma flow lead to energy changes of energetic charged particles due to electric fields δE = δu/c ✕ B B = B 0 + δB u - compressive discontinuities: shock waves - tangential discontinuities and velocity shear layers - MHD turbulence B

Cas A 1 -D shock model for „small” CR energies from Chandra

Schematic view of the collisionless shock wave ( some elements in the shock front rest frame, other in local plasma rest frames ) u 1 u 2 δE ≠ 0 thermal plasma v~10 km/s v~1000 km/s CR B upstream d shock transition layer downstream

Particle energies downstream of the shock evaluated from upstream-downstream Lorentz transformation where A = mi/m. H and for u = u 1 -u 2 >> vs, 1 upstream sound speed Cosmic rays (suprathermal particles) rg, CR >> rg(E*i) ~ 10 9 -10 cm ~ d E >> E*i (for B ~ a few μG) how to get particles with E>>E*i - particle injection problem

Modelling the injection process by PIC simulations. For electrons, see e. g. , Hoshino & Shimada (2002) shock detailes vx, i/ush vx, e/ush |ve|/ush Ey Bz/Bo Ex x/(c/ωpe)

suprathermal electrons Maxwellian I-st order Fermi acceleration

Diffusive shock acceleration: rg >> d u 1 u 2 shock compression R = u 1/u 2 I order acceleration where u = u 1 -u 2 in the shock rest frame Compressive discontinuity of the plasma flow leads to acceleration of particles reflecting at both sides of the discontinuity: diffusive shock acceleration (I-st order Fermi)

To characterize the accelerated particle spectrum one needs information about: 1. „low energy” normalization (injection efficiency) 2. spectral shape (spectral index for the power-law distribution) 3. upper energy limit (or acceleration time scale)

CR scattering at magnetic field perturbations (MHD waves) Development of the shock diffusive acceleration theory Basic theory: Krymsky 1977 Axford, Leer and Skadron 1977 Bell 1978 a, b Blandford & Ostriker 1978 Acceleration time scale, e. g. : Lagage & Cesarsky 1983 - parallel shocks Ostrowski 1988 - oblique shocks Non-linear modifications (Drury, Völk, Ellison, and others) Drury 1983 (review of the early work)

Energetic particles accelerated at the shock wave: kinetic equation for isotropic part of the dist. function f(t, x, p) plasma advection spatial diffusion . adiabatic compression momentum diffusion; „II order Fermi acceleration” I order: <Δp>/p ~ U/v ~ 10 -2 II order: <Δp>/p ~ (V/v)2 ~ 10 – 8 if we consider relativistic particles with v~c cf. Schlickeiser 1987

Diffusive acceleration at stationary planar shock propagating along the magnetic field: B || x-axis; „parallel shock” outside the shock + continuity of particle density and flux at the shock f=f(p)

the phase-space Distribution of shock accelerated particles injected at the shock background particles advected from -∞ INDEPENDENT ON ASSUMPTIONS ABOUT LOCAL CONDITIONS NEAR THE SHOCK Momentum distribution:

Spectral index depends ONLY on the shock compression adiabatic index shock Mach number For a strong shock (M>>1): R = 4 and α = 4. 0 (σ = 2. 0) (for CR dominated shock: γ ≈ 4/3 R ≈ 7. 0 and γ ≈ 3. 5) Spectral shape nearly parameter free, with the index α very close to the values observed or anticipated in real sources. Diffusive shock acceleration theory in its simplest test particle non-relativistic version became a basis of most studies considering energetic particle populations in astrophysical sources.

Spectral index the observed spectrum below 1015 e. V -> =2. 7 the escape from the Galaxy scales as ~E 0. 5, thus the injection spectral index i=2. 2 It is very close to the above value DSA=2. 0 for M>>1 In real shocks with finite M the above value of i very well fits the modelled effective spectral index (like by Berezkho & Voelk for SNRs)