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The jet opening angle theta_jet and the bulk Lorentz factor Gamma_0 arecrucial parameters for the computation of the energetics of Gamma Ray Bursts(GRBs). From the ~30 GRBs with measured theta_jet or Gamma_0 it is known that:(i) the real energetic E_gamma, obtained by correcting the isotropic equivalentenergy E_iso for the collimation factor ~theta_jet^2, is clustered around10^50-10^51 erg and it is correlated with the peak energy E_p of the promptemission and (ii) the comoving frame E'_p and E'_gamma are clustered aroundtypical values. Current estimates of Gamma_0 and theta_jet are based onincomplete data samples and their observed distributions could be subject tobiases. Through a population synthesis code we investigate whether differentassumed intrinsic distributions of Gamma_0 and theta_jet can reproduce a set ofobservational constraints. Assuming that all bursts have the same E'_p andE'_gamma in the comoving frame, we find that Gamma_0 and theta_jet cannot bedistributed as single power-laws. The best agreement between our simulation andthe available data is obtained assuming (a) log-normal distributions fortheta_jet and Gamma_0 and (b) an intrinsic relation between the peak values oftheir distributions, i.e theta_jet^2.5*Gamma_0=const. On average, larger valuesof Gamma_0 (i.e. the "faster" bursts) correspond to smaller values of theta_jet(i.e. the "narrower"). We predict that ~6% of the bursts that point to usshould not show any jet break in their afterglow light curve since they havesin(theta_jet)<1/Gamma_0. Finally, we estimate that the local rate of GRBs is~0.3% of all local SNIb/c and ~4.3% of local hypernovae, i.e. SNIb/c withbroad-lines.

The faster the narrower: characteristic bulk velocities and jet opening angles of Gamma Ray Bursts