Analytic solution for nonlinear shock acceleration in the Bohm limit

M. A. Malkov

doi:10.1086/304471

paper

Analytic solution for nonlinear shock acceleration in the Bohm limit

arXiv:astro-ph/9707152 · doi:10.1086/304471

Abstract

The selfconsistent steady state solution for a strong shock, significantly modified by accelerated particles is obtained 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 energetic particles, coupled with the thermal plasma through the momentum flux continuity equation, is reduced to a nonlinear integral equation in one variable. Its solution provides selfconsistently both the particle spectrum and the structure of the hydrodynamic flow. A critical system parameter governing the acceleration process is found to be $Λ = M^{- 3/4} Λ_{1}$ , where $Λ_{1} = η p_{1} / m c$ , with a suitably normalized injection rate $η$ , the Mach number M >> 1, and the cut-off momentum $p_{1}$ . We particularly focus on an efficient solution, in which almost all the energy of the flow is converted into a few energetic particles. It was found that (i) for this efficient solution (or, equivalently, for multiple solutions) to exist, the parameter $ζ = η p_{0} p_{1} / m c$ must exceed a critical value $ζ_{cr} \sim 1$ ( $p_{0}$ is the injection momentum), (ii) the total shock compression ratio r increases with M and saturates at a level that scales as $ r \propto Λ_1 (iii) the downstream power-law spectrum has the universal index q=3.5 over a broad momentum range. (iv) completely smooth shock transitions do not appear in the steady state kinetic description.

39 pages, 3 PostScript figures, uses aasms4.sty, to appear in Aug. 20, 1997 issue ApJ, vol. 485

Analytic solution for nonlinear shock acceleration in the Bohm limit

Abstract

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