Thermal electron heating rate: A derivation

Abstract

The thermal electron heating rate, Qe, is an important heat source term in the ionospheric electron energy balance equation, representing heating by photoelectrons or by precipitating higher energy electrons. A formula for the thermal electron heating rate is derived from the kinetic equation by using the electron‐electron collision operator as given by the unified theory of Kihara and Aono. This collision operator includes collective interactions to produce a finite collision operator with an exact Coulomb logarithm term. The derived heating rate Qe is the sum of three terms, Qe = Qp + S + Qint, which are, respectively, (1) primary electron production term giving the heating from newly created electrons that have not yet suffered collisions with the ambient electrons, (2) a heating term evaluated on the energy surface meυ²/2 = ET at the transition between Maxwellian and tail electrons at ET, and (3) the integral term representing heating of Maxwellian electrons by energetic tail electrons at all energies > ET. Published ionospheric electron temperature studies have used only the integral term Qint with differing lower integration limits. There can be a significant numerical difference between Qe and Qint. Use of the incomplete heating rate could lead to erroneous conclusions regarding electron heat balance, since Qe is greater than Qint by as much as a factor of 2. The sensitivity of the heating rate to the method of calculating the energetic (tail) electron distribution function by using either a linear or a quadratic collision operator is demonstrated. Choice of the transition energy, ET, between the thermal (Maxwellian) population and the energetic tail electrons significantly affects the magnitude of the individual heating rate terms. The net heating rate Qe is less sensitive to the value of ET than are the individual terms.