Self-regulation of particle acceleration in quasi-parallel shocks in MHD-PIC simulations

Shock waves in supernova remnants are known to produce non-thermal particles, cosmic rays, via the first order Fermi acceleration mechanism. The most self-consistent way to study this process is to solve kinetic equations using particle-in-cell methods, which are computationally expensive due to the need to resolve microscopic plasma scales. In this work, we study electron acceleration in quasi-parallel shocks using a hybrid code which treats cosmic ray electrons (CRe) as particles, while the thermal electron-ion plasma is described as a fluid by magnetohydrodynamic equations. This MHD-PIC method captures the microphysics of CRe and includes their momentum and energy feedback in the fluid equations. We study the acceleration and feedback mechanisms of CRe on macroscopic scales during the long-term evolution of the system. Instead of utilizing constant injection parameters, we implement injection prescriptions based on a self-consistent reflectivity of the shock which we determine by tracing test particles through MHD turbulence. This approach allows us to study the long-term evolution of the upstream turbulence and the particle acceleration process. We show how the injection efficiency of quasi-parallel shocks is self-regulated by accelerated particles.