Temporal evolution of thermal emission from relativistically expanding plasma

Propagation of photons in relativistically expanding plasma outflows characterized by steady Lorentz factor Γ is considered. Photons that are injected in regions of high optical depth are advected with the flow until they escape at the photosphere. Below the photosphere, the photons are coupled to the plasma via Compton scattering. I show here that as a result of the slight misalignment of the scattering electrons' velocity vectors, the ( local) comoving photon energy decreases with radius as ∈′(r) ∝ r ^-2/3. This mechanism dominates the photon cooling in scenarios of faster adiabatic cooling of the electrons. I then show that the photospheric radius of a relativistically expanding plasma wind strongly depends on the angle to the line of sight, θ. For θ ≲ Γ^-1, the photospheric radius is θ-independent, while for θ ≳ Γ^-1, r_ph(θ) ∝ θ². I show that the θ-dependence of the photosphere implies that for flow parameters characterizing gamma-ray bursts (GRBs), thermal photons originating from below the photosphere can be observed up to tens of seconds following the inner engine activity decay. I calculate the probability density function P(r, θ) of a thermal photon to escape the plasma at radius r and angle θ. Using this function, I show that following the termination of the internal photon injection mechanism, the thermal flux decreases as F_BB^ob(t) ∝ t^-2, and that the decay of the photon energy with radius results in a power-law decay of the observed temperature, T^ob(t) ∝ t ^-2/3 at early times, which changes to t^-1/2 later. Detailed numerical results are in very good agreement with the analytical predictions. I discuss the consequences of this temporal behavior in view of the recent evidence for a thermal emission component observed during the prompt emission of GRBs.