2000ApJ...537..255D

Sari, Piran, & Narayan have derived analytic formulae to model the spectra from gamma-ray burst blast waves that are energized by sweeping up material from the surrounding medium. We extend these expressions to apply to general radiative regimes and to include the effects of synchrotron self-absorption. Electron energy losses due to the synchrotron self-Compton process are also treated in a very approximate way. The calculated spectra are compared with detailed numerical simulation results. We find that the spectral and temporal breaks from the detailed numerical simulation are much smoother than the analytic formulae imply and that the discrepancies between the analytic and numerical results are greatest near the breaks and endpoints of the synchrotron spectra. The expressions are most accurate (within a factor of ∼3) in the optical/X-ray regime during the afterglow phase and are more accurate when ε_e, the fraction of swept-up particle energy that is transferred to the electrons, is ≲0.1. The analytic results provide at best order-of-magnitude accuracy in the self-absorbed radio/infrared regime and give poor fits to the self-Compton spectra because of complications from Klein-Nishina effects and photon-photon opacity.