Theory of photospheric emission from relativistic outflows

In this paper we reexamine the optical depth of ultrarelativistic spherically symmetric outflows and reevaluate the photospheric radius for each model during both the acceleration and coasting phases. It is shown that for both the wind and the shell models there are two asymptotic solutions for the optical depth during the coasting phase of the outflow. In particular we show that quite counterintuitively a geometrically thin shell may appear as a thick wind for photons propagating inside it. For this reason we introduce notions of photon thick and photon thin outflows, which appear more general and better physically motivated with respect to winds and shells. Photosphere of relativistic outflow is a dynamic surface. We study its geometry and find that the photosphere of photon thin outflow has always a convex shape, while in the photon thick one it is initially convex (there is always a photon thin layer in any outflow) and then it becomes concave asymptotically approaching the photosphere of an infinitely long wind. We find that both instantaneous and time integrated observed spectra are very close to the thermal one for photon thick outflows, in line with existing studies. It is our main finding that the photospheric emission from the photon thin outflow produces non thermal time integrated spectra, which may be described by the Band function well known in the GRB literature. We find that energetic GRBs should produce photon thin outflows with photospheric emission lasting less than one second for the total energy $E_0leq10^{54}$ erg and baryonic loading parameter $Bleq10^{-2}$. It means that only time integrated spectra may be observed from such GRBs.