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 simulations of quasi-perpendicular, low-beta ( β p = 0.25) collisionless ion–electron shocks with mass ratio m i / m e = 200, fast Mach number <?CDATA M ₘₛ=1?> M ms = 1 –4, and upstream magnetic field angle θ Bn = 55°–85° from the shock normal <?CDATA n?> n ˆ . It is known that shock electron heating is described by an ambipolar, B -parallel electric potential jump, Δ ϕ ∥ , that scales roughly linearly with the electron temperature jump. Our simulations have <?CDATA Δ φ ∥/(0.5m_iuₛₕ²)~ 0.1?> Δ ϕ ∥ / ( 0.5 m i u sh 2 ) ∼ 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 ϕ ∥ , 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 ϕ ∥ in our low- β p shocks. We further focus on two θ Bn = 65° shocks: a <?CDATA M _s\,=\,4?> M s = 4 ( <?CDATA M _A\,=\,1.8?> M A = 1.8 ) case with a long, 30 d i precursor of whistler waves along <?CDATA n?> n ˆ , and a <?CDATA M _s\,=\,7?> M s = 7 ( <?CDATA M _A\,=\,3.2?> M A = 3.2 ) case with a shorter, 5 d i precursor of whistlers oblique to both <?CDATA n?> n ˆ and B ; d i is the ion skin depth. Within the precursors, ϕ ∥ has a secular rise toward the shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of the <?CDATA M _s\,=\,4?> M s = 4 , θ Bn = 65° case, ϕ ∥ shows a weak dependence on the electron plasma-to-cyclotron frequency ratio ω pe /Ω ce , and ϕ ∥ decreases by a factor of 2 as m i / m e is raised to the true proton–electron value of 1836.