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Analytic Solution for Nonlinear Shock Acceleration in the Bohm Limit

The self-consistent 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. The solution of this equation provides self-consistently 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-3/4Λ1, where Λ1 = ηp1/mc, with a suitably normalized injection rate η, the Mach number M ≫ 1, and the cutoff momentum p1. We are able to confirm in principle the often-quoted hydrodynamic prediction of three different solutions. We particularly focus on the most efficient of these solutions, in which almost all the energy of the flow is converted into a few energetic particles. It was found that (1) for this efficient solution (or, equivalently, for multiple solutions) to exist, the parameter ζ = η(p0p1)1/2/mc must exceed a critical value ζcr ~ 1 (p0 is some point in momentum space separating accelerated particles from the thermal plasma), and M must also be rather large. (2) Somewhat surprisingly, there is also an upper limit to this parameter. (3) The total shock compression ratio r increases with M and saturates at a level that scales as r ∝ Λ1. (4) Despite the fact that r can markedly exceed r = 7 (as for a purely thermal ultrarelativistic gas), the downstream power-law spectrum turns out to have the universal index q = 3½ over a broad momentum range. This coincides formally with the test particle result for a shock of r = 7. (5) Completely smooth shock transitions do not appear in the steady state kinetic description. A finite subshock always remains. It is even very strong, rs ≃ 4 for Λ ≪ 1, and it can be reduced noticeably if Λ ≳ 1.