Photon-photon opacity constraints for relativistically expanding gamma-ray burst sources

Five bright gamma-ray bursts (GRBs) detected by the Burst and Transient Source Experiment (BATSE) have also been detected at higher energies by EGRET. Four are consistent with power-law spectra extending to energies as high as, in the case of GRB930131, 1 GeV. The fifth, and most recent, GRB940217, has a more complex spectrum, with one photon detected at 18 GeV, the most energetic GRB photon detection to date. The optical depth to photon-photon pair production in these sources is extremely large for distances more than about 10pc away if the radiation is emitted isotropically in the observer's frame. This optical depth can be dramatically reduced if the source is moving with a relativstic bulk Lorentz factor Gamma, and recent calculations for this situation have been limited to cases of a beam with opening angle 1 Gamma, or expansions of infinitely thin spherical shells. This paper presents our extension of the pair production otpical depth calculation in relativistically expanding sources to more general geometries, including shells of finite thickness and arbitrary opening angle. We find that the minimum bulk Lorentz factors for the Energy Gamma Ray Experiment Telescope (EGRET) sources to be optically thin, i.e. display no spectral attenuation, is only moderately dependent on the shell thickness and its opening solid angle; these new limits on required velocity for given geometries will aid in placing realistic constraints on GRB source models.