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

The jet opening angle theta(jet) and the bulk Lorentz factor Gamma(0) are crucial parameters for the computation of the energetics of gamma-ray bursts (GRBs). From the similar to 30 GRBs with measured theta(jet) or Gamma(0) it is known that (i) the real energetic E-gamma, obtained by correcting the isotropic equivalent energy E-iso for the collimation factor similar to theta(2)(jet), is clustered around 10(50)-10(51) erg and it is correlated with the peak energy E-p of the prompt emission and (ii) the comoving frame E-p' and E-gamma'. are clustered around typical values. Current estimates of Gamma(0) and theta(jet) 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 Gamma(0) and theta(jet) can reproduce a set of observational constraints. Assuming that all bursts have the same E-p' and E-gamma' in the comoving frame, we find that Gamma(0) and theta(jet) 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 theta(jet) and Gamma(0) and (b) an intrinsic relation between the peak values of their distributions, i.e. theta(2.5)(jet)Gamma(0) = const. On average, larger values of Gamma(0) (i.e. the 'faster' bursts) correspond to smaller values of theta(jet) (i.e. the 'narrower'). We predict that similar to 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 theta(jet) < 1/Gamma(0). Finally, we estimate that the local rate of GRBs is similar to 0.3 per cent of all local Type Ib/c supernova (SNIb/c) and similar to 4.3 per cent of local hypernovae, i.e. SNIb/c with broad lines.