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, un-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 $\Lambda$ which measures the coupling of the upstream medium. For $\Lambda \lesssim 1.12$, the upstream is collisional, i.e. strongly coupled, and the strong shock front is about $\mathcal{M}_1 \lambda_{\mathrm{mfp},1}$ thick, where $\lambda_{\mathrm{mfp},1}$ and $\mathcal{M}_1$ are the upstream mean-free-path and Mach number respectively. A transition occurs for $\Lambda \sim 1.12$ beyond which the front is $\sim \mathcal{M}_1\lambda_{\mathrm{mfp},1}\ln \Lambda/\Lambda$ thick for $\Lambda\gtrsim 1.12$. Considering $\Lambda$ can reach billions in astrophysical settings, this allows to understand 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 2 extremes.