Relativistic Spherical Shocks in Expanding Media

We investigate the propagation of spherically symmetric shocks in relativistic homologously expanding media with density distributions following a power-law profile in their Lorentz factor. That is, $\rho_{ej} \propto t^{-3}\gamma_{e}(R,t)^{-\alpha}$, where $\rho_{ej}$ is the medium proper density, $\gamma_{e}$ is its Lorentz factor, $\alpha>0$ is constant and $t$, $R$ are the time and radius from the center. We find that the shocks behavior can be characterized by their proper velocity, $U'=\Gamma_s'\beta_s'$, where $\Gamma_s'$ is the shock Lorentz factor as measured in the immediate upstream frame and $\beta_s'$ is the corresponding 3-velocity. While generally, we do not expect the shock evolution to be self-similar, for every $\alpha>0$ we find a critical value $U'_c$ for which a self-similar solution with constant $U'$ exists. We then use numerical simulations to investigate the behavior of general shocks. We find that shocks with $U'>U'_c$ have a monotonously growing $U'$, while those with $U'<U'_c$ have a decreasing $U'$ and will eventually die out. Finally, we present an analytic approximation, based on our numerical results, for the evolution of general shocks in the regime where $U'$ is ultra-relativistic.