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
$E_0\leq10^{54}$ erg and baryonic loading parameter $B\leq10^{-2}$. It means
that only time integrated spectra may be observed from such GRBs.
