Self-consistent Monte Carlo simulations of proton acceleration in coronal shocks: Effect of anisotropic pitch-angle scattering of particles

Abstract

Context. Solar energetic particles observed in association with coronal mass ejections (CMEs) are produced by the CME-driven shock waves. The acceleration of particles is considered to be due to diffusive shock acceleration (DSA). Aims. We aim at a better understanding of DSA in the case of quasi-parallel shocks, in which self-generated turbulence in the shock vicinity plays a key role. Methods. We have developed and applied a new Monte Carlo simulation code for acceleration of protons in parallel coronal shocks. The code performs a self-consistent calculation of resonant interactions of particles with Alfv\'en waves based on the quasi-linear theory. In contrast to the existing Monte Carlo codes of DSA, the new code features the full quasi-linear resonance condition of particle pitch-angle scattering. This allows us to take anisotropy of particle pitch-angle scattering into account, while the older codes implement an approximate resonance condition leading to isotropic scattering.We performed simulations with the new code and with an old code, applying the same initial and boundary conditions, and have compared the results provided by both codes with each other, and with the predictions of the steady-state theory. Results. We have found that anisotropic pitch-angle scattering leads to less efficient acceleration of particles than isotropic. However, extrapolations to particle injection rates higher than those we were able to use suggest the capability of DSA to produce relativistic particles. The particle and wave distributions in the foreshock as well as their time evolution, provided by our new simulation code, are significantly different from the previous results and from the steady-state theory. Specifically, the mean free path in the simulations with the new code is increasing with energy, in contrast to the theoretical result.