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Understanding the heating of electrons to quasi-thermal energies atcollisionless shocks has broad implications for plasma astrophysics. Itdirectly impacts the interpretation of X-ray spectra from shocks, is importantfor understanding how energy is partitioned between the thermal and cosmic raypopulations, and provides insight into the structure of the shock itself. InGhavamian, Laming & Rakowski (2007) we presented observational evidence for aninverse square relation between the electron-to-proton temperature ratio andthe shock speed at the outer blast waves of supernova remnants in partiallyneutral interstellar gas. There we outlined how lower hybrid waves generated inthe cosmic ray precursor could produce such a relationship by heating theelectrons to a common temperature independent of both shock speed and thestrength of the ambient magnetic field. Here we explore the mechanism of lowerhybrid wave heating of electrons in more detail. Specifically we examine thegrowth rate of the lower hybrid waves for both the kinetic (resonant) andreactive cases. We find that only the kinetic case exhibits a growing mode. Atlow Alfv\'en Mach numbers (~15) the growth of lower hybrid waves can be fasterthan the magnetic field amplification by modified Alfv\'en waves.

The Heating of Thermal Electrons in Fast Collisionless Shocks: The Integral Role of Cosmic Rays