Monte Carlo shock-like solutions to the boltzmann equation with collective scattering

We present the results of Monte Carlo simulations of steady state shocks generated by a collision operator that isotropizes the particles by means of elastic scattering in some locally defined frame of reference. The simulations include both the back-reaction of accelerated particles on the inflowing plasma and the free escape of high-energy particles from finite shocks. Energetic particles are found to be naturally extracted out of the background plasma by the shock process with an efficiency in good quantitative agreement with (a) an earlier analytic approximation (Eichler 1983, 1984) and (b) observations (Gosling et al. 1981) of the entire particle spectrum at a quasi-parallel interplanetary shock. We use the analytic approximation, which allows a self-consistent determination of the effective adiabatic index of the shocked gas, to calculate the overall acceleration efficiency and particle spectrum for cases where ultrarelativistic energies are obtained. We find that shocks of the strength necessary to produce galactic cosmic rays put approximately 15% of the shock energy into relativistic particles