Empirical Constraints on Cosmological Gamma-Ray Bursts

We place empirical constraints on the physical properties of $\gamma -$ray burst events at cosmological distances. In particular we derive probability distributions for the radiation energy $E_{\gamma }$, the minimum Lorentz factor ${\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}}$, the maximum baryonic mass $M_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and the upper bound on the surrounding gas density $n_{\mathrm{m}\mathrm{a}\mathrm{x}}$ in the events, based on 169 bursts from the first BATSE catalog. Using peak flux as a distance indicator we probe bursts where the constraints are stronger than average. The resulting variance and skewness of the cosmological probability distributions are calculated in addition to their mean values: $\langle E_\gamma\rangle=4\times 10^{51} h^{-2} {\rm erg}$,$\langle \gamma_{\rm min}\rangle= 5\times 10^2$,$\langle M_{\rm max}\rangle=10^{-5}\xi^{-1} M_\odot$,and$\langle (\gamma/\gamma_{\rm min})^{-5} n_{\rm max}\rangle\lsim 10^{2} \xi^{-1}{\rm cm^{-3}}$,where$\xi$isthefractionofthetotalenergywhichisconvertedto$\gamma-$rays. The distribution of burst energies ends at about ${10}^{53}\mathrm{e}\mathrm{r}\mathrm{g}$, close to the binding energy of a neutron star.