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The strong optical flash observed by ROTSE, as well as the radio flareassociated with GRB 990123 can be attributed to the emission of the fireballejecta, initially heated by the reverse shock. We numerically study theevolution of an adiabatic relativistic fireball interacting with an ambientuniform medium, both in the initial energy transfer stage and in its lateevolution. It is shown that the Blandford-McKee solution adequately describesthe evolution of the shocked shell quite early on and for as long as thefireball material has relativistic temperatures. In the case where the reverseshock is only mildly relativistic the shocked shell becomes cold almostimmediately and the evolution deviates from the Blandford-McKee solution. Wederive analytical expressions for the ejecta evolution in its cold regime. Thissolution gives a good approximation to the numerical results. We estimate theradiation from the fireball ejecta using the numerical hydrodynamic evolutionin both cases: cold and hot shells. Surprisingly, we find that both evolutionsgive rather similar light curves, decaying approximately as $T^{-2}$ in theoptical and peaking at about one day in the radio, even though thehydrodynamics is different.Comment: 11 pages, 13 figures, Submitted to Ap

Optical Flashes and Radio Flares in Gamma‐Ray Burst Afterglow: Numerical Study