We propose a method to estimate the pseudo jet opening half-angle of GRBs using the spectral peak energy ( { E _ { p } } ) –peak luminosity relation ( so called Yonetoku relation ) as well as the { E _ { p } } –collimation-corrected \gamma -ray energy relation ( so called Ghirlanda relation ) .
For bursts with known jet break times and redshifts , we compared the pseudo jet opening half-angle with the standard one and found that the differences are within a factor 2 .
We apply the method to 689 long GRBS .
We found that the distribution function of the pseudo jet opening half-angle obeys f ( \theta _ { j } ) \propto \theta _ { j } ^ { -2.2 \pm 0.2 } with possible cutoffs for \theta _ { j } < 0.04 and \theta _ { j } > 0.3 although the log-normal fit is also possible . \theta ^ { -2 } distribution is compatible with the structured jet model .
From the distribution function we found that the beaming correction for the rate of GRBs is \sim 340 , which means \sim 10 ^ { -5 } yr ^ { -1 } galaxy ^ { -1 } or only one in 10 ^ { 2 } type Ib/c supernovae .
We also found the evolution of the distribution function as a function of the redshift .