Early acceleration of electrons and protons at the nonrelativistic quasiparallel shocks with different obliquity angles

Abstract

The early acceleration of protons and electrons in the nonrelativistic collisionless shocks with three obliquities are investigated through 1D particle-in-cell simulations. In the simulations, the charged particles possessing a velocity of $0.2\, c$ flow towards a reflecting boundary, and the shocks with a sonic Mach number of $13.4$ and a Alf\'{v}en Mach number of $16.5$ in the downstream shock frame are generated. In these quasi-parallel shocks with the obliquity angles $\theta = 15^\circ$, $30^\circ$, and $45^\circ$, some of the protons and the electrons can be injected into the acceleration processes, and their downstream spectra in the momentum space show a power law tail at a time of $1.89\times10^5 \omega_{\rm pe}^{-1}$, where $\omega_{\rm pe}$ is the electron plasma frequency. Moreover, the charged particles reflected at the shock excite magnetic waves upstream of the shock. The shock drift acceleration is more prominent with a larger obliquity angle for the shocks, but the accelerated particles diffuse parallel to the shock propagation direction more easily to participate in the diffusive shock acceleration. At the time still in the early acceleration stage, more energetic protons and electrons appear in the downstream of the shock for $\theta = 15^\circ$ compared with the other two obliquities; moreover, in the upstream region, the spectrum of the accelerated electrons is the hardest for $\theta_{\rm nB} = 45^\circ$ among the three obliquities, whereas the proton spectra for $\theta_{\rm nB} = 15^\circ$ and $45^\circ$ are similar as a result of the competition of the effectiveness of the shock drift acceleration and the diffusive shock acceleration.