We study the early afterglows of gamma-ray bursts produced by geometrically thick fireballs, following the development of the external shock as energy is continually supplied to the shocked material. We study the dependence of the early afterglow slope on the luminosity history of the central engine. The resulting light curves are modeled with power-law functions, and the importance of a correct choice of the reference time t0 is investigated. We find that deviations from a simple power law are observed only if a large majority of the energy is released at late times. The light curve in this case can be described as a simple power law if the reference time is set to be close to the end of the burst. We applied our analysis to the cases of GRB 050219a and GRB 050315. We show that the early steep decay of the afterglow cannot result from the interaction of the fireball with the ambient medium. We conclude that the early X-ray afterglow emission is associated with the prompt phase, and we derive limits on the radius at which the prompt radiation is produced.
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