Distribution of Gamma-ray Burst Ejecta Energy with Lorentz Factor

The early X-ray afterglow for a significant number of gamma-ray bursts detected by the Swift satellite is observed to have a phase of very slow flux decline with time (Fν ∝ t−α with 0.2 . α . 0.8) for 102.5 s . t . 104 s, while the subsequent decline is the usual 1 . α3 . 1.5 behavior, that was seen in the pre-Swift era. We show that this behavior is a natural consequence of a small spread in the Lorentz factor of the ejecta, by a factor of ∼ 2 − 4, where the slower ejecta gradually catch-up with the shocked external medium, thus increasing the energy of forward shock and delaying its deceleration. The end of the “shallow” flux decay stage marks the beginning of the Blandford-McKee self similar external shock evolution. This suggests that most of the energy in the relativistic outflow is in material with a Lorentz factor of ∼ 30 − 50. Subject headings: gamma-rays: bursts — shock waves — hydrodynamics