On the electron-ion temperature ratio established by collisionless shocks

¹ Anton Pannekoek Institute/GRAPPA, University of Amsterdam, PO Box 94249, 1090 GE Amsterdam, The Netherlands
e-mail: j.vink@uva.nl
² Anton Pannekoek Institute, University of Amsterdam, PO Box 94249, 1090 GE Amsterdam, The Netherlands
³ A.F. Ioffe Physical-Technical Institute, 194021 St. Petersburg, and St. Petersburg State Politechnical University, Russia
⁴ APC, Univ. Paris Diderot, CNRS/IN2P3, CEA/Irfu, Obs. de Paris, Sorbonne Paris Cité, 75013 Paris, France

Received: 15 July 2014
Accepted: 4 May 2015

Abstract

Astrophysical shocks are often collisionless shocks, in which the changes in plasma flow and temperatures across the shock are established not through Coulomb interactions, but through electric and magnetic fields. An open question about collisionless shocks is whether electrons and ions each establish their own post-shock temperature (non-equilibration of temperatures), or whether they quickly equilibrate in the shock region. Here we provide a simple, thermodynamic, relation for the minimum electron-ion temperature ratios that should be expected as a function of Mach number. 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 shock, in order for the plasma to remain charge neutral. The only form of additional electron heating that we allow for is adiabatic heating, caused by the compression of the electron gas. These assumptions result in an analytic treatment of expected electron-ion temperature ratio that agrees with observations of collisionless shocks: at low sonic Mach numbers, M_s ≲ 2, the electron-ion temperature ratio is close to unity, whereas for Mach numbers above M_s ≈ 60 the electron-ion temperature ratio asymptotically approaches a temperature ratio of T_e/T_i = m_e/ ⟨ m_i ⟩. In the intermediate Mach number range the electron-ion temperature ratio scales as T_e/T_i ∝ M_s^-2. In addition, we calculate the electron-ion temperature ratios under the assumption of adiabatic heating of the electrons only, which results in a higher electron-ion temperature ratio, but preserves the T_e/T_i ∝ M_s^-2 scaling. We also show that for magnetised shocks the electron-ion temperature ratio approaches the asymptotic value T_e/T_i = m_e/ ⟨ m_i ⟩ for lower magnetosonic Mach numbers (M_ms), mainly because for a strongly magnetised shock the sonic Mach number is larger than the magnetosonic Mach number (M_ms ≤ M_s). The predicted scaling of the electron-ion temperature ratio is in agreement with observational data for magnetosonic Mach numbers between 2 and 10, but for supernova remnants the relation requires that the inferred Mach numbers for the observations are overestimated, 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 ξ, which is the fraction of the enthalpy-flux difference between electrons and ions that is used for equilibrating the electron and ion temperatures. Comparing the model to existing observations in the solar system and supernova remnants suggests that the data are best described by ξ ≳ 5%, but also provides a hint that the Mach number of some supernova remnant shocks may have been overestimated; perhaps as a result of heating in the cosmic-ray precursor.