Relativistic generalization of formation and ion reflection conditions in electrostatic shocks

The theoretical model by Sorasio et al. (2006) for the steady state Mach number of electrostatic shocks formed in the interaction of two plasma slabs of arbitrary density and temperature is generalized for relativistic electron and non-relativistic ion temperatures. We find that the relativistic correction leads to lower Mach numbers, and as a consequence, ions are reflected with lower energies. The steady state bulk velocity of the downstream population is introduced as an additional parameter to describe the transition between the minimum and maximum Mach numbers in dependence of the initial density and temperature ratios. In order to transform the soliton-like solution in the upstream region into a shock, a population of reflected ions is considered and differences to a zero-ion temperature model are discussed.