We recently found that Gamma–Ray Burst energies and luminosities , in their comoving frame , are remarkably similar .
This , coupled with the clustering of energetics once corrected for the collimation factor , suggests the possibility that all bursts , in their comoving frame , have the same peak energy E ^ { \prime } _ { p } ( of the order of a few keV ) and the same energetics of the prompt emission E ^ { \prime } _ { \gamma } ( of the order of 2 \times 10 ^ { 48 } erg ) .
The large diversity of bursts energies is then due to the different bulk Lorentz factor \Gamma _ { 0 } and jet aperture angle \theta _ { jet } .
We investigated , through a population synthesis code , what are the distributions of \Gamma _ { 0 } and \theta _ { jet } compatible with the observations .
Both quantities must have preferred values , with log–normal best fitting distributions and \langle \Gamma _ { 0 } \rangle \sim 275 and \langle \theta _ { jet } \rangle \sim 8.7 ^ { \circ } .
Moreover , the peak values of the \Gamma _ { 0 } and \theta _ { jet } distributions must be related – \theta _ { jet } ^ { 2.5 } \Gamma _ { 0 } =const : the narrower the jet angle , the larger the bulk Lorentz factor .
We predict that \sim 6 % 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 \sim 0.3 % of all local SNIb/c and \sim 2.5 % of local hypernovae , i.e .
SNIb/c with broad absorption lines .