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 Aux 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 Lambda = M(-3/4)Lambda(1), where Lambda(1) = eta p(1)/mc, with a suitably normalized injection rate eta, the Mach number M much greater than 1, and the cutoff momentum p(1). We are able to confirm in principle the often-quoted hydrodynamic prediction of three different solutions. We particularly focus on the mast 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 zeta = eta(p(0)p(1))(1/2)/mc must exceed a critical value zeta(er)similar to 1 (p(0) is some Feint 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 proportional to Lambda(1). (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 = 3 1/2 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, r(s) similar or equal to 4 for Lambda much less than 1, and it can be reduced noticeably if Lambda greater than or similar to 1.