Electron Acceleration at Quasi-parallel Non-relativistic Shocks: A 1D Kinetic Survey

We present a survey of 1D kinetic particle-in-cell simulations of quasi-parallel non-relativistic shocks to identify the environments favorable for electron acceleration. We explore an unprecedented range of shock speeds vₛₕ≈ 0.067-0.267\,c, Alfv\'{e}n Mach numbers MA = 5-40, sonic Mach numbers Mₛ = 5-160, as well as the proton-to-electron mass ratios mᵢ/mₑ=16-1836. We find that high Alfv\'{e}n Mach number shocks can channel a large fraction of their kinetic energy into nonthermal particles, self-sustaining magnetic turbulence and acceleration to larger and larger energies. The fraction of injected particles is 0.5% for electrons and ≈ 1% for protons, and the corresponding energy efficiencies are 2% and ≈ 10%, respectively. The extent of the nonthermal tail is sensitive to the Alfv\'{e}n Mach number; when MA 10, the nonthermal electron distribution exhibits minimal growth beyond the average momentum of the downstream thermal protons, independently of the proton-to-electron mass ratio. Acceleration is slow for shocks with low sonic Mach numbers, yet nonthermal electrons still achieve momenta exceeding the downstream thermal proton momentum when the shock Alfv\'{e}n Mach number is large enough. We provide simulation-based parametrizations of the transition from thermal to nonthermal distribution in the downstream (found at a momentum around pi,e/mᵢvₛₕ ≈ 3√mi,e/mᵢ), as well as the ratio of nonthermal electron to proton number density. The results are applicable to many different environments and are important for modeling shock-powered nonthermal radiation.