Rate of energy gain and maximum energy in diffusive shock acceleration

The problem of diffusive shock acceleration of fast charged particles is reexamined with emphasis on the rate of energy gain, and the maximum energy which can be attained in a given circumstance. The direction of the average magnetic field at the shock is shown to have a large effect. If the perpendicular diffusion coefficient is much smaller than the parallel coefficient, particles can gain much more energy if the shock is quasi-perpendicular than if it is quasi-parallel. The maximum energy attainable can be substantially higher (by a factor of 100 or more) than previous discussions would predict, in cases where the shock is quasi-perpendicular. The energy gain increases as kappa-perpendicular decreases. The principal limitation comes from the requirement that diffusion be a valid approximation to the particle motion, and that the particle be able to diffuse fast enough to encounter the shock many times.