Analytic solution for nonlinear shock acceleration in the Bohm limit

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. The solution of this equation 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 ≡ MΛ1, where Λ1 = ηp1/mc, with a suitably normalized injection rate η, the Mach number M 1, and the cut-off 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 (i) for this efficient solution (or, equivalently, for multiple solutions) to exist, the parameter ζ = η √ p0p1/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 (ii) somewhat surprisingly, there is also an upper limit to this parameter (iii) the total shock compression ratio r increases with M and saturates at a level that scales as r ∝ Λ1 (iv) despite the fact that r can markedly exceed r = 7 (as for a purely thermal ultra-relativistic gas), the downstream power-law spectrum turns out to have the universal index q = 3 /2 over a broad momentum range. This coincides formally with the test particle result for a shock of r = 7 (v) 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. Subject headings: acceleration of particles, cosmic rays, diffusion, hydrodynamics, shock waves, supernova remnants