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 βₚi1 ~ 1-8), and low sonic Mach number (the ratio of shock speed to sound speed is Mₛ ~ 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 Tₑ2 exceeds the adiabatic expectation Tₑ2,ad by an amount Tₑ2/Tₑ2,ad - 1 0.0016 Mₛ3.6, nearly independent of the plasma beta and of the pre-shock electron-to-ion temperature ratio Tₑ1/Tᵢ1, which we vary from $0.1$ to unity. We investigate the heating physics for Mₛ ~ 5-6 and find that electron super-adiabatic heating is governed by magnetic pumping at Tₑ1/Tᵢ1=1, whereas heating by B-parallel electric fields (i.e., parallel to the local magnetic field) dominates at Tₑ1/Tᵢ1=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.