Can We Trust MHD Jump Conditions for Collisionless Shocks?

When applied to compute the density jump of a shock, the standard magnetohydrodynamic (MHD) formalism assumes (1) that all the upstream material passes downstream, together with the momentum and energy it carries, and (2) that pressures are isotropic. In a collisionless shock, shock-accelerated particles going back and forth around the front can invalidate the first assumption. In addition, an external magnetic field can sustain stable pressure anisotropies, invalidating the second assumption. It is therefore unclear whether or not the density jump of a collisionless shock fulfills the MHD jump. Here we try to clarify this issue. A literature review is conducted on 68 articles dealing with Particle-In-Cell simulations of collisionless shocks. We analyze the factors triggering departure from the MHD density jump and quantify their influence on Delta(RH), the relative departure from the Rankine-Hugoniot (RH) jump. For small departures we propose Delta(RH) = +O(10(-1-3.7 kappa))t(kappa) - sigma O(1) where t is the timescale of the simulation, sigma is the magnetization parameter and kappa is a constant of order unity. The first term stems from the energy leakage into the accelerated particle. The second term stems from the downstream anisotropy triggered by the field (assuming an isotropic upstream). This relation allows us to assess to what extent a collisionless shock fulfills the RH density jump. In the strong field limit and for parallel shocks, the departure caused by the field saturates at a finite, negative value. For perpendicular shocks, the departure goes to zero at small and high sigma's so that we find here a departure window. The results obtained have to be checked against full 3D simulations.