On the electron-ion temperature ratio established by collisionless shocks

TL;DR

This paper derives simple relations for electron-ion temperature equilibration in collisionless shocks, matching observations and predicting how the temperature ratio varies with Mach number and plasma conditions.

Contribution

It provides an analytic model for minimal electron-ion temperature ratios in collisionless shocks, incorporating effects of Mach number and magnetic fields, aligning with observational data.

Findings

At low Mach numbers, electrons and ions are nearly equilibrated.

For high Mach numbers, the temperature ratio scales with mass ratio m_e/m_i.

The model suggests observed supernova remnant data may overestimate Mach numbers.

Abstract

Astrophysical shocks are often collisionless shocks. An open question about collisionless shocks is whether electrons and ions each establish their own post-shock temperature, or whether they quickly equilibrate in the shock region. Here we provide simple relations for the minimal amount of equilibration to expect. The basic assumption is that the enthalpy-flux of the electrons is conserved separately, but that all particle species should undergo the same density jump across the the shock. This assumption results in an analytic treatment of electron-ion equilibration that agrees with observations of collisionless shocks: at low Mach numbers ($<2$) the electrons and ions are close to equilibration, whereas for Mach numbers above $M \sim 60$ the electron-ion temperature ratio scales with the particle masses $T_e/T_i = m_e/m_i$. In between these two extremes the electron-ion temperature ratio scales as $T_e/T_i \propto 1/M_s^2$. This relation also hold if adiabatic compression of the electrons is taken into account. For magnetised plasmas the compression is governed by the magnetosonic Mach number, whereas the electron-ion temperatures are governed by the sonic Mach number. The derived equations are in agreement with observational data at low Mach numbers, but for supernova remnants the relation requires that the inferred Mach numbers for the observations are over- estimated, perhaps as a result of upstream heating in the cosmic-ray precursor. In addition to predicting a minimal electron/ion temperature ratio, we also heuristically incorporate ion-electron heat exchange at the shock, quantified with a dimensionless parameter ${\xi}$. Comparing the model to existing observations in the solar system and supernova remnants suggests that the data are best described by ${\xi} \sim 5$ percent. (Abridged abstract.)