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 slowflux decline with time (F-v proportional to t(-alpha) with 0.2 less than or similar to alpha less than or similar to 0.8) for 10(2.5) less than or similar to t less than or similar to 10(4) s, while the subsequent decline is the usual 1 less than or similar to alpha(3) less than or similar to 1.5 behaviour, which was seen in the pre-Swift era. We show that this behaviour is a natural consequence of a small spread in the Lorentz factor of the ejecta, by a factor of similar to 2-4, where the slower ejecta gradually catch up with the shocked external medium, thus increasing the energy of the 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 similar to 30-50.