Study of Time Evolution of the Bend-over Energy in the Energetic Particle Spectrum at a Parallel Shock

Shock acceleration is considered one of the most important mechanisms for the acceleration of astrophysical energetic particles. In this work, we calculate the trajectories of a large number of test charged particles accurately in a parallel shock with magnetic turbulence. We investigate the time evolution of the accelerated particle energy spectrum in the downstream of the shock, in order to understand the acceleration mechanism of energetic particles. From simulation results we obtain power-law energy spectra with a bend-over energy, E-0, increasing with time. With the particle mean acceleration time and mean momentum change during each cycle of the shock crossing from the diffusive shock acceleration model (following Drury), a time-dependent differential equation for the maximum energy, E-acc, of particles accelerated at the shock can be approximately obtained. We assume the theoretical bend-over energy as E-acc. It is found that the bend-over energy from simulations agrees well with the theoretical bend-over energy using the nonlinear diffusion theory, NLGCE-F, in contrast to that using the classic quasi-linear theory.