Bridging the gap between collisional and collisionless shock waves

While the front of a fluid shock is a few mean-free-paths thick, the front of a collisionless shock can be orders of magnitude thinner. By bridging between a collisional and a collisionless formalism, we assess the transition between these two regimes. We consider non-relativistic, non-magnetized, planar shocks in electron-ion plasmas. In addition, our treatment of the collisionless regime is restricted to high-Mach-number electrostatic shocks. We find that the transition can be parameterized by the upstream plasma parameter $\varLambda$ which measures the coupling of the upstream medium. For $\varLambda \lesssim 1.12$, the upstream is collisional, i.e. strongly coupled, and the strong shock front is about $M_1 λ _{mfp,1}$ thick, where $λ _{mfp,1}$ and $M_1$ are the upstream mean free path and Mach number, respectively. A transition occurs for $\varLambda ∼ 1.12$ beyond which the front is $∼ M_1λ _{mfp,1}\ln \varLambda /\varLambda$ thick for $\varLambda \gtrsim 1.12$. Considering that $\varLambda$ can reach billions in astrophysical settings, this allows an understanding of how the front of a collisionless shock can be orders of magnitude smaller than the mean free path, and how physics transitions continuously between these two extremes.