Optical Flashes and Radio Flares in GRBs afterglow: Numerical Study

The strong optical flash observed by ROTSE, as well as the radio flare associated with GRB 990123 can be attributed to the emission of the fireball ejecta, initially heated by the reverse shock. We numerically study the evolution of an adiabatic relativistic fireball interacting with an ambient uniform medium, both in the initial energy transfer stage and in its late evolution. It is shown that the Blandford-McKee solution adequately describes the evolution of the shocked shell quite early on and for as long as the fireball material has relativistic temperatures. In the case where the reverse shock is only mildly relativistic the shocked shell becomes cold almost immediately and the evolution deviates from the Blandford-McKee solution. We derive analytical expressions for the ejecta evolution in its cold regime. This solution gives a good approximation to the numerical results. We estimate the radiation from the fireball ejecta using the numerical hydrodynamic evolution in both cases: cold and hot shells. Surprisingly, we find that both evolutions give rather similar light curves, decaying approximately as $T^{-2}$ in the optical and peaking at about one day in the radio, even though the hydrodynamics is different.