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 invalid the first assumption. In addition, an external magnetic field can sustain stable pressure anisotropies, invaliding the second assumption. It is therefore unclear whether the density jump of a collisionless shock fulfils the MHD jump or not. 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 $Δ_{RH}$, the relative departure from the Rankine-Hugoniot jump. For small departures we propose $Δ_{RH} = + \mathcal{O}(10^{-1-3.7κ})t^κ- σ\mathcal{O}(1)$ where $t$ is the timescale of the simulation, $σ$ the magnetization parameter and $κ$ a constant of order unity. The first term stems from the energy leakage into accelerated particle. The second term stems from the downstream anisotropy triggered by the field (assuming an isotropic upstream). This relation allows to assess to which extent a collisionless shock fulfils 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 $σ$'s so that we find here a departure window. The results obtained have to be checked against full 3D simulations.

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