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 Γ 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 Γ 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.