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 v ₒ₇ 0. 067-0. 267\, c, Alfv\'en Mach numbers M ₀ = 5-40, sonic Mach numbers M ₒ = 5-160, as well as the proton-to-electron mass ratios m ₈/m ₄=16-1836. We find that high Alfv\'en 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\'en Mach number; when M ₀ 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\'en 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 p ₈, ₄/m ₈v ₒ₇ 3m ₈, ₄/₌_ ₈), 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.