Steady-state structure of relativistic collisionless shocks

Abstract

We explore analytically the structure of relativistic shock and solitary wave solutions in collisionless plasmas. In the wave frame of reference, a cold plasma is flowing from one end and impacting on a low velocity plasma. First we show that under astrophysical conditions, a cold electron-positron plasma is unstable with respect to a two-stream instability in the interface between these regions. The instability heats the inflowing cold plasma rapidly, on a timescale comparable to the inverse of its plasma frequency. We then derive time-independent equations to describe the resulting hot state of the pair plasma, and describe the conditions under which the spatially uniform solution is the unique stable solution for the post shock conditions. We also examine plasmas composed of cold protons and hot electrons, and show that the spatially uniform solution is the unique stable solution there as well. We state the shock jump conditions which connect a cold, electron-proton plasma to a hot electron-proton plasma. The generic feature evident in all of these models is that the plasma's initial, directed kinetic energy gets almost completely converted into heat. The magnetic field plays the role of catalyst which can induce the plasma instability, but our solutions indicate that the macroscopic field only gets amplified by a factor of approximately three in the frame of the shock.