Effect of the Pair-Annihilation on the Break Energy of GRB Spectra

Abstract.

The widely accepted scenario for producing gamma-ray bursts (GRBs) and their afterglow is the dissipation of the kinetic energy of a relativistic flow by relativistic shocks. The rapid temporal variability requires that the GRB itself must arise from internal shocks within the flow, while the afterglow is due to the external shock produced as the flow is decelerated upon collision with the ambient medium. However, burst hardnesses are generally less than 1 MeV [1], and the apparent clustering of the break energies of GRB spectra in the 50keV-1MeV range is reported [2]. Recently Guetta, Spada & Waxman [3] showed by using a Monte-Carlo simulation that the Thomson optical depth due to $e^\pm$ pairs produced by synchrotron photons plays an important role in the internal shock model. They argued that the inclusion of the pair optical depth effects is essential in determining the break energy. In their study Guetta et al. use the standard inelastic collision approximation in which two colliding shells merge into a single uniform shell. However this method does not necessarily reproduce the internat energy in shocked shell and the break energy adequately, especially for a small relative Lorentz factor of two collisional shells. In this paper, we reexamine the pair optical depth effect using a simple analytic model. We show how the photosphere depends on the parameters of the internal shock model.