A Theory of Mulicolor Black Body Emission from Relativistically Expanding Plasmas

We consider the emission of photons from the inner parts of a relativistically expanding plasma outflow, characterized by a constant Lorentz factor, Gamma. Photons that are injected in regions of high optical depth are advected with the flow until they escape at the photosphere. Due to multiple scattering below the photosphere, the locally emerging comoving photon distribution is thermal. However, as an observer sees simultaneously photons emitted from different angles, hence with different Doppler boosting, the observed spectrum is a multi-color black-body. We calculate here the properties of the observed spectrum at different observed times. Due to the strong dependence of the photospheric radius on the angle to the line of sight, for parameters characterizing gamma-ray bursts (GRBs) thermal photons are seen up to tens of seconds following the termination of the inner engine. At late times, following the inner engine termination, both the number flux and energy flux of the thermal spectrum decay as F ~ t^{-2}. At these times, the multicolor black body emission results in a power law at low energies (below the thermal peak), with power law index F_\nu ~ \nu^{0}. This result is remarkably similar to the average value of the low energy spectral slope index (``\alpha'') seen in fitting the spectra of large GRB sample.