The faster the narrower: characteristic bulk velocities and jet opening angles of gamma-ray bursts

The jet opening angle θjet and the bulk Lorentz factor Γ0 are crucial parameters for the computation of the energetics of gamma-ray bursts (GRBs). From the ∼30 GRBs with measured θjet or Γ0 it is known that (i) the real energetic E γ, obtained by correcting the isotropic equivalent energy E iso for the collimation factor ∼ θ2 jet, is clustered around 1050-1051 erg and it is correlated with the peak energy E p of the prompt emission and (ii) the comoving frame E′p and E′γ are clustered around typical values. Current estimates of Γ0 and θjet are based on incomplete data samples and their observed distributions could be subject to biases. Through a population synthesis code we investigate whether different assumed intrinsic distributions of Γ0 and θjet can reproduce a set of observational constraints. Assuming that all bursts have the same E′p and E′γ in the comoving frame, we find that Γ0 and θjet cannot be distributed as single power laws. The best agreement between our simulation and the available data is obtained assuming (a) log-normal distributions for θjet and Γ0 and (b) an intrinsic relation between the peak values of their distributions, i.e. θjet 2.5Γ0 = const. On average, larger values of Γ0 (i.e. the 'faster' bursts) correspond to smaller values of θjet (i.e. the 'narrower'). We predict that ∼6 per cent of the bursts that point to us should not show any jet break in their afterglow light curve since they have sin θjet < 1/Γ0. Finally, we estimate that the local rate of GRBs is ∼0.3 per cent of all local Type Ib/c supernova (SNIb/c) and ∼4.3 per cent of local hypernovae, i.e. SNIb/c with broad lines.