Bulk Lorentz factors of gamma-ray bursts

Knowledge of the bulk Lorentz factor Gamma(0) of gamma-ray bursts (GRBs) allows us to compute their comoving frame properties shedding light on their physics. Upon collisions with the circumburst matter, the fireball of a GRB starts to decelerate, producing a peak or a break (depending on the circumburst density profile) in the light curve of the afterglow. Considering all bursts with known redshift and with an early coverage of their emission, we find 67 GRBs (including one short event) with a peak in their optical or GeV light curves at a time t(p). For another 106 GRBs we set an upper limit t(p)(UL). The measure of t(p) provides the bulk Lorentz factor Gamma(0) of the fireball before deceleration. We show that t(p) is due to the dynamics of the fireball deceleration and not to the passage of a characteristic frequency of the synchrotron spectrum across the optical band. Considering the t(p) of 66 long GRBs and the 85 most constraining upper limits, we estimate Gamma(0) or a lower limit Gamma(LL)(0). Using censored data analysis methods, we reconstruct the most likely distribution of t(p). All t(p) are larger than the time T-p,T-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 similar to 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) x 10(50) erg, < L-iso > = 3(15) x 10(47) erg s(-1), 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 e ff ects. 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-circle dot.