Ion injection and acceleration at parallel shocks: Comparisons of self-consistent plasma simulations with existing theories

We present a simulation study of particle acceleration by parallel collisionless shocks and compare our results with existing shock acceleration theories, in particular, diffusive and shock drift acceleration. We extend the earlier work in this field by drawing attention to the similarities and discrepancies between the results obtained from the more realistic, self-consistent simulations and these well-established theories. We have simulated several shocks of different Alfvén Mach number which propagate into a medium containing magnetic fluctuations, some of which are initially superposed on the background field, and some of which are generated self-consistently by energetic ions streaming ahead of the shock. Although the simulated characteristics of the energetic ions are in general agreement with the shock acceleration theories, we have found some surprising results. For example, the parallel shock, at times, exhibits features similar to that expected at more perpendicular shocks, although, on average the features are consistent with Fermi acceleration predictions (as expected at a parallel shock). Also, quasi-linear theory, which assumes that the magnetic fluctuations about the mean field are small in amplitude, agrees quite well with the simulation results, where the fluctuations are on the order of ΔB/B ∼ 1.