2005AJ....130..418C

Assuming that the observed gamma-ray burst (GRB) rate as a function of redshift is proportional to the corrected star formation rate, we derive the empirical distribution of the viewing angles of long BATSE GRBs, P^em(θ), and the distribution of these bursts in the (θ, z)-plane, P^em(θ,z), by using a tight correlation between the collimation-corrected gamma-ray energy (E_γ) and the peak energy of the νF_ν spectrum measured in the rest frame (E'_p). Our results show that P^em(θ) is well fitted by a lognormal distribution centered at logθ/1rad=-0.76 with a width of σ_logθ=0.57. We test different universal structured jet models by comparing model predictions, P^th(θ) and P^th(θ,z), with our empirical results. To make the comparisons reasonable, an ``effective'' threshold, which corresponds to the sample selection criteria of the long GRB sample, is used. We find that (1) P^th(θ) predicted by a power-law jet model is consistent with P^em(θ), but P^th(θ,z) predicted by this model is significantly different from P^em(θ,z); (2) P^th(θ,z) predicted by a single-Gaussian jet model is more consistent with P^em(θ,z) than that predicted by the power-law jet model, but P^th(θ) predicted by this model rapidly drops at θ>0.3 rad and greatly deviates from P^em(θ); and (3) both the P^th(θ) and P^th(θ,z) predicted by a two-Gaussian jet model are roughly consistent with our empirical results. A brief discussion shows that cosmological effects on the E_γ-E'_p relation do not significantly affect our results, but sample selection effects on this relationship might significantly influence our results.