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 $v_{\rm sh}\approx 0.067-0.267\,c$, Alfv\'{e}n Mach numbers $\mathcal{M}_{\rm A} = 5-40$, sonic Mach numbers $\mathcal{M}_{\rm s} = 5-160$, as well as the proton-to-electron mass ratios $m_{\rm i}/m_{\rm e}=16-1836$. 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 $\lesssim 0.5\%$ for electrons and $\approx 1\%$ for protons, and the corresponding energy efficiencies are $\lesssim 2\%$ and $\approx 10\%$, respectively. The extent of the nonthermal tail is sensitive to the Alfv\'{e}n Mach number; when $\mathcal{M}_{\rm A}\lesssim 10$, 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 $p_{\rm i,e}/m_{\rm i}v_{\rm sh} \approx 3\sqrt{m_{\rm i,e}/m_{\rm i}}$), 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.