Electron Heating in the Trans-Relativistic Perpendicular Shocks of Tilted Accretion Flows

General relativistic magnetohydrodynamic (GRMHD) simulations of black hole tilted disks -- where the angular momentum of the accretion flow at large distances is misaligned with respect to the black hole spin -- commonly display standing shocks, within a few to tens of gravitational radii from the black hole. In GRMHD simulations of geometrically thick, optically thin accretion flows, applicable to low-luminosity sources like Sgr A* and M87*, the shocks have trans-relativistic speed, moderate plasma beta (the ratio of ion thermal pressure to magnetic pressure is $\beta_\mathrm{pi1} \sim 1-8$), and low sonic Mach number (the ratio of shock speed to sound speed is $M_s \sim 1-5$). We study such shocks with two-dimensional particle-in-cell simulations and we quantify the efficiency and mechanisms of electron heating, for the special case of pre-shock magnetic fields perpendicular to the shock direction of propagation. We find that the post-shock electron temperature $T_\mathrm{e2}$ exceeds the adiabatic expectation $T_\mathrm{e2,ad}$ by an amount $T_\mathrm{e2}/T_\mathrm{e2,ad} - 1 \simeq 0.0016 M_s^{3.6}$, nearly independent of the plasma beta and of the pre-shock electron-to-ion temperature ratio $T_\mathrm{e1}/T_\mathrm{i1}$, which we vary from $0.1$ to unity. We investigate the heating physics for $M_s \sim 5-6$ and find that electron super-adiabatic heating is governed by magnetic pumping at $T_\mathrm{e1}/T_\mathrm{i1}=1$, whereas heating by $B$-parallel electric fields (i.e., parallel to the local magnetic field) dominates at $T_\mathrm{e1}/T_\mathrm{i1}=0.1$. Our results provide physically-motivated subgrid prescriptions for electron heating at the collisionless shocks seen in GRMHD simulations of black hole accretion flows.