On the Energy of Gamma-Ray Bursts

We show that gamma-ray burst (GRB) afterglow observations strongly suggest, within the fireball model framework, that radiating electrons are shock accelerated to a power-law energy distribution, with universal index p \approx 2.2, and that the fraction of shock energy carried by electrons, \xi_e, is universal and close to equipartition, \xi_e ~ 1/3. For universal p and \xi_e, a single measurement of the X-ray afterglow flux on the time scale of a day provides a robust estimate of the fireball energy per unit solid angle, \epsilon, averaged over a conical section of the fireball of opening angle \theta ~ 0.1. Applying our analysis to BeppoSAX afterglow data we find that: (i) Fireball energies are in the range of 4\pi\epsilon=10^{51.5} to 10^{53.5} erg; (ii) The ratio of observed $\gamma$-ray to total fireball energy per unit solid angle, \epsilon_\gamma / \epsilon, is of order unity, satisfying abs[log10(\epsilon_\gamma/\epsilon)] =0.1. Our results imply that if typical opening angles are \theta ~ 0.1, a value consistent with our analysis, the total energy associated with a GRB event is in the range of 10^{50} erg to 10^{51.5} erg.