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,, where is the medium proper density, is its Lorentz factor, α > 0 is constant, and t, r are the time and radius from the centre. We find that the shocks behaviour can be characterized by their proper velocity,, where is the shock Lorentz factor as measured in the immediate upstream frame and is the corresponding three velocity. While generally, we do not expect the shock evolution to be self-similar, for every α > 0 we find a critical value for which a self-similar solution with constant U′ exists. We then use numerical simulations to investigate the behaviour of general shocks. We find that shocks with have a monotonously growing U′, while those with 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 ultrarelativistic.