Reverse shock forming condition for magnetized relativistic outflows: reconciling theories and simulations

Abstract: Reverse shock (RS) emission can be used to probe the properties of the relativistic ejecta, especially the degree of magnetization $\sigma$, in gamma-ray burst (GRB) afterglows. However, there has been confusion in the literature regarding the physical condition for the RS formation, and the role of magnetic fields in the RS dynamics in the Poynting-flux-dominated regime is not fully understood. Exploiting the shock jump conditions, we characterize the properties of a magnetized RS. We compare the RS dynamics and forming conditions from different theories and numerical simulations, and reconcile the discrepancies among them. The strict RS forming condition is found to be $\sigma < \sigma_\mathrm{cr}=(8/3)\gamma_4^2(n_1/n_4)$, where $n_4$ and $n_1$ are the rest-frame number densities of the ejecta and the ambient medium, respectively, $\gamma_4$ is the bulk Lorentz factor, and $\sigma_\mathrm{cr}$ is the critical magnetization. Contrary to previous claims, we prove that this condition agrees with other theoretical and simulated results, which can be further applied to the setup and consistency check of future numerical simulations. Using this condition, we propose a characteristic radius for RS formation, and categorize the magnetized shell into three regimes: 'thick shell' (relativistic RS), 'thin shell' (trans-relativistic RS), and 'no RS' regimes. The critical magnetization $\sigma_\mathrm{cr}$ is generally below unity for thin shells, but can potentially reaches $\sim 100-1000$ in the 'thick shell' regime. Our results could be applied to the dynamical evolution of Poynting-flux-dominated ejecta, with potential applications to self-consistent lightcurve modelling of magnetized relativistic outflows.