Stochastic particle acceleration at parallel astrophysical shock waves

Details Stochastic particle acceleration at parallel astrophysical shock waves Stochastic particle acceleration at parallel astrophysical shock waves

Sep 1993

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993a%26a...276..614s&link_type=abstract

Astronomy and Astrophysics, Vol. 276, p. 614 (1993)

Astronomy and Astrophysics

Astrophysics

22

Particle Acceleration - Shock Waves - Alfvén Waves - Cosmic Rays

Scientific paper

We present exact analytical solutions to the full cosmic ray transport equation describing the diffusive acceleration of cosmic ray protons at parallel nonrelativistic shock waves including momentum diffusion in the downstream region of the shock. The downstream region is of finite extent to avoid the unphysical situation of background waves acting as an infinite source of energy. The spatial diffusion coefficient of particles is assumed to be independent of particle energy. The solutions allow for (i) any momentum dependence of the particle injection at the shock, (ii) any spatial dependence of the upstream spatial diffusion coefficient, and (iii) any spatial dependence of the downstream spatial diffusion coefficient. We assume that the incoming waves upstream are right- and left-handed circularly polarised backward moving Alfvén waves, as suggested by the streaming instability of the upstream precursor cosmic ray distribution. Interaction of these waves with the fast super-Alfvénic adiabatic shock makes momentum diffusion in the downstream region unavoidable. The downstream solution is an infinite sum of power laws whose spectral indices follow from a transcendental eigenvalue equation. Each individual power law component is weighted by expansion coefficients that depend on the actual downstream position. Quite generally the shape of the downstream distribution function is concavely curved, but approaches a single power law at large momenta. In the formal limit of vanishing momentum diffusion and infinite extent of the downstream region our general solution approaches the classical result of Axford et al. (1977), Krymsky (1977), Bell (1978), and Blandford & Ostriker (1978). For efficient momentum diffusion of particles in the downstream region of the shock, the particle spectra become flatter than in the original treatment of diffusive shock acceleration, thereby removing some of the noted discrepancies of the original theory with the explanation of fiat particle spectra in shell-type supernova remnants and bright spiral galaxies, and with the independence of spectral indices from evolutionary effects in these sources.

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