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Shock acceleration is considered one of the most important mechanisms for theacceleration of astrophysical energetic particles. In this work, we calculatethe trajectories of a large number of test charged particles accurately in aparallel shock with magnetic turbulence. We investigate the time evolution ofthe accelerated-particle energy spectrum in the downstream of the shock inorder to understand the acceleration mechanism of energetic particles. Fromsimulation results we obtain power-law energy spectra with a bend-over energy,$E_0$, increasing with time. With the particle mean acceleration time and meanmomentum change during each cycle of the shock crossing from diffusive shockacceleration model (following Drury), a time-dependent differential equationfor the maximum energy, $E_{acc}$, of particles accelerated at the shock, canbe approximately obtained. We assume the theoretical bend-over energy as$E_{acc}$. It is found that the bend-over energy from simulations agrees wellwith the theoretical bend-over energy using the non-linear diffusion theory,NLGCE-F, in contrast to that using the classic quasi-linear theory (QLT).

Study of time evolution of the bend-over energy in the energetic particle spectrum at a parallel shock