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Within a test-particle description of the acceleration process in parallel nonrelativistic shocks, we present an analytic treatment of the electron injection. We estimate the velocity distribution of the injected electrons as the product of the post-shock thermal distribution of electrons times the probability for electrons with a given velocity to be accelerated; the injection efficiency is then evaluated as the integral of this velocity distribution. We estimate the probability of a particle to be injected as that of going back to the upstream region at least once. This is the product of the probability of returning to the shock from downstream times that of recrossing the shock from downstream to upstream. The latter probability is expected to be sensitive to details of the process of electron thermalisation within the (collisionless) shock, a process that is poorly known. In order to include this effect, for our treatment we use results of a numerical, fully kinetic study, by Bykov & Uvarov (1999). According to them, the probability of recrossing depends on physics of thermalisation through a single free parameter (Gamma), which can be expressed as a function of the Mach number of the shock, of the level of electron-ion equilibration, as well as of the spectrum of turbulence. It becomes apparent, from our analysis, that the injection efficiency is related to the post-shock electron temperature, and that it results from the balance between two competing effects: the higher the electron temperature, the higher the fraction of downstream electrons with enough velocity to return to the shock and thus to be ready to cross the shock from downstream to upstream; at the same time, however, the higher the turbulence, which would hinder the crossing.

Influence of thermalisation on electron injection in supernova remnant shocks