Time sequence of energetic particle spectra in quasiparallel shocks in large simulation systems

We have performed collisionless shock simulations using a one-dimensional hybrid particle-in-cell method to investigate the energy spectra of the differential intensity around the quasiparallel shocks. The system size is sufficiently large (200 000 ion inertia length) in order to eliminate the unphysical effect caused by the upstream boundary. The obtained spectrum of the differential intensity have the shape of the power-law with exponentially falling off in higher energy as predicted in previous simulations, however, the power-law indices and e-folding energy do not depend on the shock parameters, the shock Mach number MA (7.1–11.7) and the shock angle θBN (10°–40°). The power-law index is ∼1.0. This number is close to the prediction by the standard diffusive shock acceleration theory. However, the reason for the agreements between the values is not clear because an additional acceleration process is also observed in the present runs; this process was reported in [Sugiyama and Terasawa, Adv. Space Res. 24, 73 (1999)] and [Sugiyama et al., J. Geophys. Res. 106, 21657 (2001)]. The e-folding energy linearly increases in time for all runs. One of the reasons for these independent profiles on shock parameters is that the pitch angle distribution in the upstream region shows similar profiles in each run.