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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 warm medium: hot medium: magnetic field SI: n ~ 1 cm-3 n ~ 0.01 cm-3 <B> 3 G <n> ~ 10-6 m-3 T ~ 10-50 K T ~ 104 K T ~ 106-107 K B ~ <B> n-1/2 <B> ~ 0.3 nT 104 K 1 eV Cosmic rays 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 the thermal plasma and the non-thermal energy distribution. In our Galaxy: PCR Pg (= nkT) PB (= B2/8) ~ 10-13 erg/cm3 Particle Flux ( m2 s sr GeV )-1 Cosmic Ray Spectrum 1 particle/m2 s „Knee” 1 particle/m2 yr „Ankle” 1 particle/km2 yr 1 J 6 1018 eV Energy eV CR collisions in ISM For a high energy collision of a CR particle with the interstellar atom (nucleus) we have (n ~ 1/cm3 and the cross section ~ 10-24 cm2) 1 1 13 6 ~ 3 10 s 10 years 24 10 nc 110 3 10 Cosmic ray sources ? Possible SNRs shock waves. CR energy within the galactic volume ECR = V * CR ~ 1068 cm3 * 10-13 erg/cm3 = 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 L71 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 = B0 + 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 ) u1 u2 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 2 1 5 keV A ( u /1000 km/s) for ions * 2 E mv 2 2 2.5 eV (u/1000 km/s) for electrons where A = mi/mH and for u = u1-u2 >> 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 u1 u2 shock compression R = u1/u2 I order acceleration u p ~ p v where u = u1-u2 in the shock rest frame 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 1978a, 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) f 1 f 1 2 f U f f U p 2 p D t 3 p p p plasma advection spatial diffusion adiabatic compression . 1 p p U 3 10 -2 I order: <Δp>/p ~ U/v ~ II order: <Δp>/p ~ (V/v)2 ~ 10 –8 if we consider relativistic particles with v~c momentum diffusion; „II order Fermi acceleration” (p) D 2t 2 V p v 2 2 cf. Schlickeiser 1987 Diffusive acceleration at stationary planar shock propagating along the magnetic field: B || x-axis; „parallel shock” U f f x outside the shock , U u1 or u2 , || x , f f(x,p) f f ui || , i 1, 2 x x x + continuity of particle density and flux at the shock f=f(p) the phase-space Distribution of shock accelerated particles f ( p) Ap p p p' 1 f ( p' )dp' 0 particles injected at the shock 3R R 1 Momentum distribution: n( p ) p , where background particles advected from -∞ INDEPENDENT ON ASSUMPTIONS ABOUT LOCAL CONDITIONS NEAR THE SHOCK R2 2 R 1 Spectral index depends ONLY on the shock compression R 1 2 1 2 M 5 for adiabatic 3 index u1 , M shock Mach vs ,1 , 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 eV -> =2.7 the escape from the Galaxy scales as ~E0.5, 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)