Density jump as a function of the field for parallel relativistic collisionless shocks

Collisionless shocks are frequently analysed using the magnetohydrodynamic (MHD) formalism, even though the required collisionality hypothesis is not fulfilled. In a previous work (Bret & Narayan, 2018 J. Plasma Phys. vol. 84 , p. 905840604), we presented a model of collisionless shock displaying an important departure from the expected MHD behaviour, in the case of a strong flow aligned magnetic field. This model was non-relativistic. Here, it is extended to the relativistic regime, considering zero upstream pressure and upstream Lorentz factor $\gg 1$ . The result agrees satisfactorily with Particle-in-Cell simulations and shows a similar, and important, departure from the MHD prediction. In the strong field regime, the density jump $r$ , seen in the downstream frame, behaves like $r \sim 2 + 1/\gamma _{\mathrm{up}}$ , while MHD predicts 4 ( $\gamma _{\mathrm{up}}$ is the Lorentz factor of the upstream measured in the downstream frame). Only pair plasmas are considered.