  During the gravitational core collapse of a massive progenitor star which may
give rise to at least a class of gamma-ray bursts (GRBs) associated with
supernovae, a stellar core rapidly passes through a short yet important phase
of neutronization, producing a huge amount of energetic neutrinos and photons
which contribute to the total pressure within the progenitor core. The
collection of neutrinos, photons and gas materials together may be approximated
as a fluid with a polytropic index $\gamma=4/3$ under the action of
self-gravity. With a substantial generalization and using analytical and
numerical methods (Lou & Cao 2008), we recently constructed and examined
various self-similar solutions to describe collapses, rebound shocks and flows
systematically in a $\gamma=4/3$ polytropic gas mixture with spherical
symmetry, and compare our results with those of Goldreich & Weber (1980). It is
also possible to construct central void solutions without or with shocks.
Various features and characteristics of this nonlinear relativistically hot gas
dynamics, including asymptotic and exact solutions, are presented. This more
general polytropic model analysis provides the dynamic basis of understanding
the evolution of rebound shocks in supernovae (SNs) and the results may be also
utilized to benchmark hydrodynamic simulations.
