Analytic model for the electrostatic potential jump across collisionless shocks, with application to Earth's bow shock

Abstract

Existing observational analyses demonstrate that the electrostatic potential jump across collisionless shocks (Δϕ′ in the de Hoffmann–Teller frame) strongly influences the behavior of electrons. The empirical relations inferred from these data provide crucial constraints on shock‐related phenomena, such as foreshock electron beams and radio emission. However, a theoretical interpretation of these relations, especially their parameter dependences, is necessary. We present an analytic model for Δϕ′ which incorporates the full dependences on solar wind and shock conditions. The model involves three assumptions for the spatially varying number density n(x), magnetic field B(x), and perpendicular and parallel electron temperatures (Te⟂(x) and Te∥(x) across the shock: (1) n(x) ∝ B(x) in a piecewise linear fashion; (2) Te⟂(x) ∝ B(x); and (3) ΔTe∥ ∝ ΔTe⟂. Empirical and theoretical arguments for these assumptions exist for moderate to strong shocks, breaking down for weak shocks (assumption 3) and shocks where Te increases by a factor >4 (assumption 2). The model is qualitatively and quantitatively consistent with the known empirical relations between Δϕ′ and the jumps in Te and B across the shock, providing a theoretical basis for them. We use the model to test for a possible correlation between Δϕ′ and the change in the normal component of the ion ram energy, ΔEram. We find that the relation between Δϕ′ and ΔEram is only approximately linear over a range of shockfront locations for constant solar wind conditions, with the relation depending strongly on the angle θbu,1 between the upstream flow speed u1 and magnetic field B1 and on the upstream ratio u1/vA,1 of flow speed to Alfvèn speed. Our model provides a natural interpretation for the spread of observational data around the best linear fit suggested by Hull et al. [2000].