Constraints on the bulk Lorentz factor in the internal shock scenario for gamma-ray bursts

We investigate, independently of specific emission models, the constraints on the value of the bulk Lorentz factor Gamma of a fireball. We assume that the burst emission comes from internal shocks in a region transparent to Thomson scattering, and before deceleration caused by the swept-up external matter is effective. We consider the role of Compton drag in decelerating fast-moving shells before they interact with slower ones, thus limiting the possible differences in the bulk Lorentz factor of shells. Tighter constraints on the possible range of Gamma are derived by requiring that the internal shocks transform more than a few per cent of the bulk energy into radiation. Efficient bursts may require a hierarchical scenario, where a shell undergoes multiple interactions with other shells. We conclude that fireballs with average Lorentz factors larger than 1000 are unlikely to give rise to the observed bursts.