  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.
