Diffusive shock acceleration: A method based on integral equation

The structure of a strong shock, significantly modified by cosmic rays (CRs) is studied on the level of a kinetic description, assuming Bohm-type diffusion. The original problem that is commonly formulated in terms of the diffusion-convection equation for the distribution function of CRs, coupled with the thermal plasma through the momentum flux continuity equation, is reduced to a nonlinear integral equation in one variable. The solution of this equation provides selfconsistently both the particle spectrum and the flow profile. We particularly focus on the solution (out of the three possible) with the highest conversion efficiency of the flow energy to the CR energy. It was found that (1) for this or, equivalently, for multiple solutions to exist, the suitably normalized injection rate ν must exceed the value of ∼ p0/p1 1 where p1 and p0 are the cut-off and injection momenta, respectively and, the Mach number M must also be rather large. (2) The total shock compression ratio r increases with M and saturates at a level that scales as r πo α p1/p0. (3) Completely smooth shock transitions do not appear in the steady state kinetic description. The flow deceleration in the CR precursor is equal to the velocity jump at the gaseous subshock multiplied by a factor ∼ vp3/po. © 2001 COSPAR. Published by Elsevier Science Ltd. All rights reserved.