Density jump as a function of magnetic field strength for parallel collisionless shocks with anisotropic upstream pressure

The properties of collisionless shocks are frequently assessed in the magnetohydrodynamics (MHD) model. Yet, in a collisionless plasma, an ambient magnetic field can sustain a stable anisotropy in the upstream or the downstream, resulting in a departure from the MHD predicted behavior. We present a model allowing to derive the downstream anisotropy, hence the shock density jump, in terms of the upstream quantities. For simplicity, the case of a parallel shock in pair plasma is considered. Contrary to previous works where the upstream was assumed isotropic, here the upstream anisotropy $A=T_\perp/T_\parallel$ is a free parameter. The strong sonic shock regime is formally identical to the isotropic upstream case. Yet, for intermediate sonic Mach numbers, a variety of behaviors appear as a result of the anisotropy of the upstream.