Comments: 13 pages, 8 figures, Accepted for publication in MNRAS, Comments and questions are welcome!
Subjects:High Energy Astrophysical Phenomena (astro-ph.HE)
We extend the standard model of forward-reverse shock (FS-RS) for gamma-ray
burst (GRB) afterglow to more general cases. On one hand, we derive the
analytical solution to the hydrodynamics of the shocks in two limiting cases,
i.e., an ultra-relativistic reverse shock case and a Newtonian reverse shock
case. Based on the asymptotic solutions in these two limiting cases, we
constitute a semi-analytical solution for the hydrodynamics of the shocks in
the generic case, covering the mildly-relativistic reverse shock case. On the
other hand, we derive the evolution of the system taking into account the
condition of energy conservation which is not satisfied in the standard FS-RS
model. A generic solution of semi-analytical expressions is also given. In both
the extended standard FS-RS model (satisfying pressure balance condition) and
the model satisfying energy conservation, we find that the results in the
ultra-relativistic reverse shock case and in the early stage of the Newtonian
reverse shock case are different from those in the standard FS-RS model by only
a factor that close to one while the same initial conditions adopted. However,
the asymptotic solutions in the limiting cases are not good approximations to
those in the intermediate case. Our semi-analytical results agree well with the
numerical results for a large range of model parameters, and hence can be
easily employed to diagnose the physical quantities of the GRB shell and
circumburst environment.
