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Diffusive shock acceleration: an introduction

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: n ~ 1 cm -3 T ~ 10 4 K

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Diffusive shock acceleration: an introduction

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  1. Diffusive shock acceleration: an introduction

  2. 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: n ~ 1 cm-3 T ~ 104 K hot medium: n ~ 0.01 cm-3 T ~ 106-107 K magnetic field <B>  3 G B ~ <B> n-1/2 SI: <n> ~ 10-6 m-3 <B> ~ 0.3 nT 104 K  1 eV

  3. 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

  4. Cosmic Ray Spectrum 1 particle/m2 s Particle Flux ( m2 s sr GeV )-1 „Knee” 1 particle/m2 yr „Ankle” 1 particle/km2 yr 1 J  61018 eV Energy eV

  5. 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)

  6. 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

  7. Tycho X-ray picture from Chandra

  8. X-ray H-alpha Supernova remnant Dem L71

  9. 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 • compressive discontinuities: shock waves • tangential discontinuities and velocity shear layers - MHD turbulence B = B0 + δB u B

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

  11. 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 d shock transition layer upstream downstream

  12. Particle energies downstream of the shock evaluated from upstream-downstream Lorentz transformation for where A = mi/mH and u = u1-u2 >> vs,1 upstream sound speed Cosmic rays (suprathermal particles) E >> E*i rg,CR >> rg(E*i) ~ 10 9-10cm ~ d (for B ~ a few μG) how to get particles with E>>E*i - particle injection problem

  13. 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)

  14. suprathermal electrons Maxwellian I-st order Fermi acceleration

  15. Diffusive shock acceleration: rg >> d u2 u1 shock compression R = u1/u2 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) I order acceleration where u = u1-u2 in the shock rest frame

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

  17. 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 theearly work)

  18. Energetic particles accelerated at theshock 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

  19. 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)

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

  21. 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.

  22. 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)

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