Electron Heating in 2D Particle-in-Cell Simulations of Quasi-Perpendicular Low-Beta Shocks

We measure the thermal electron energization in 1D and 2D particle-in-cell (PIC) simulations of quasi-perpendicular, low-beta (${\beta }_p=0.25$) collisionless ion-electron shocks with mass ratio $m_i\mathrm{/}{m}_e=200$, fast Mach number ${\mathcal{M}}_{ms}=1$-$4$, and upstream magnetic field angle ${\theta }_{Bn}=55$-${85}^{\circ }$ from shock normal $\widehat{\mathit{n}}$. It is known that shock electron heating is described by an ambipolar, $\mathit{B}$-parallel electric potential jump, $\mathrm{\Delta }{\phi }_{\parallel }$, that scales roughly linearly with the electron temperature jump. Our simulations have $\mathrm{\Delta }{\phi }_{\parallel }\mathrm{/}(0.5m_i{u_{\mathrm{s}\mathrm{h}}}^2)\sim 0.1$-$0.2$ in units of the pre-shock ions’ bulk kinetic energy, in agreement with prior measurements and simulations. Different ways to measure ${\phi }_{\parallel }$, including the use of de Hoffmann-Teller frame fields, agree to tens-of-percent accuracy. Neglecting off-diagonal electron pressure tensor terms can lead to a systematic underestimate of ${\phi }_{\parallel }$ in our low-${\beta }_p$ shocks. We further focus on two ${\theta }_{Bn}={65}^{\circ }$ shocks: a ${\mathcal{M}}_s=4$ (${\mathcal{M}}_A=1.8$) case with a long, $30d_i$ precursor of whistler waves along $\widehat{\mathit{n}}$, and a ${\mathcal{M}}_s=7$ (${\mathcal{M}}_A=3.2$) case with a shorter, $5d_i$ precursor of whistlers oblique to both $\widehat{\mathit{n}}$ and $\mathit{B}$; $d_i$ is the ion skin depth. Within the precursors, ${\phi }_{\parallel }$ has a secular rise towards the shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of the ${\mathcal{M}}_s=4$, ${\theta }_{Bn}={65}^{\circ }$ case, ${\phi }_{\parallel }$ shows a weak dependence on the electron plasma-to-cyclotron frequency ratio ${\omega }_{pe}\mathrm{/}{\mathrm{\Omega }}_{ce}$, and ${\phi }_{\parallel }$ decreases by a factor of 2 as $m_i\mathrm{/}{m}_e$ is raised to the true proton-electron value of 1836.