Bulk Lorentz factors of gamma-ray bursts

Considering the t_p_ of 66 long GRBs and the 85 most constraining upper limits, we estimate {Gamma}_0_ or a lower limit {Gamma}_0_^LL^. Using censored data analysis methods, we reconstruct the most likely distribution of t_p_. All t_p_ are larger than the time T_p,{gamma}_ when the prompt {gamma}-ray emission peaks, and are much larger than the time T_ph_ when the fireball becomes transparent, that is, t_p_>T_p,{gamma}_>T_ph_.

The reconstructed distribution of {Gamma}_0_ has median value ~300 (150) for a uniform (wind) circumburst density profile. In the comoving frame, long GRBs have typical isotropic energy, luminosity, and peak energy <E_iso_>=3(8)x10^50^erg, <L_iso_>=3(15)x10^47^erg/s, and <E_peak_>=1(2)keV in the homogeneous (wind) case. We confirm that the significant correlations between {Gamma}_0_ and the rest frame isotropic energy (E_iso_), luminosity (L_iso_), and peak energy (E_p_) are not due to selection effects.

When combined, they lead to the observed E_p_-E_iso_ and E_p_-L_iso_ correlations. Finally, assuming a typical opening angle of 5 degrees, we derive the distribution of the jet baryon loading which is centered around a few 10^-6^M_{\sun}_.