Possible observational evidence for the θ−2 angular distribution of the opening half-angle of gamma-ray burst jets

We propose a method to estimate the jet opening half-angle of gamma-ray bursts (GRBs) using the spectral peak energy (E p )-peak luminosity relation (the so-called Yonetoku relation) as well as the E n -collimation-corrected gamma-ray energy relation (the so-called Ghirlanda relation). For bursts with known jet break times and redshifts, we have compared the derived jet opening half-angle with that measured and found that the differences are within a factor of 2. We apply the method to 689 long GRBs. We have found that the distribution function of the derived jet opening half-angle obeys f(θ j ) θ -2.2±0.2 j , although a lognormal fit is also possible. θ -2 distribution is compatible with the structured jet model. From the distribution function we have found that the beaming correction for the rate of GRBs is ∼340, which means ∼10 -5 yr -1 galaxy -1 or only one in 10 2 Type Ib/c supernovae. We have also found the evolution of the distribution function as a function of the redshift.