Stochastic Particle Acceleration at Magnetohydrodynamic Shock Waves

Abstract

The acceleration of energetic particles at a single shock wave in the test-particle approximation is considered. Using the concept of a microscopic treatment a steady-state solution of the cosmic ray transport equation in momentum space describing first-order Fermi acceleration of energetic charged particles at a plane parallel shock and second-order Fermi acceleration in the downstream region or in both the downstream and upstream regions of the shock is derived. The solution depends on the shock compression ratio, the momentum dependence of the spatial diffusion coefficient and the alfvenic Mach number. In the limit of no second-order Fermi acceleration and a constant spatial diffusion coefficient the power law characteristic of first-order Fermi acceleration depending only on the compression ratio obtained previously is recovered. The second-order effects lead to a flattening of the spectrum for any shock other than a strong adiabatic shock and make the spectral index more independent of the shock’s compression ratio.