Analytic solution for nonlinear shock acceleration in the Bohm limit

Details Analytic solution for nonlinear shock acceleration in the Bohm limit Analytic solution for nonlinear shock acceleration in the Bohm limit

1997-07-14

arxiv.org/abs/astro-ph/9707152v1

Astrophys.J. 485 (1997) 638

Astronomy and Astrophysics

Astrophysics

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

Scientific paper

10.1086/304471

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 $\Lambda = M^{-3/4}\Lambda_1 $, where $ \Lambda_1 =\eta p_1/mc $, with a suitably normalized injection rate $ \eta $, 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 $ \zeta =\eta\sqrt{p_0 p_1}/mc $ must exceed a critical value $\zeta_{cr} \sim 1$ ($p_0$ is the injection momentum), (ii) the total shock compression ratio r increases with M and saturates at a level that scales as $ r \propto \Lambda_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.

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