3.1.1. Photon-Dominated Scenario

The simplest scenario considers a photon-dominated expanding shell of width *δr* 0 "instantaneously" releasing its energy. From here on, prime symbols indicate quantities measured in the comoving frame of the shell, namely from an observer within it. On the other hand, *r* is the radial coordinate of the laboratory frame, a frame outside the shell where the observer is sitting on the central engine.

Enforcing energy and entropy conservation laws, the shell keeps accelerating up to Γmax ' *η*, which is attained at at the so-called dissipation radius *r<sup>s</sup>* ∼ *ηr*0; beyond it, most of the internal energy of the shell has been converted into the kinetic one, so the flow no longer accelerates and it coasts. Thus, the fireball obeys the following scaling laws of the shell comoving temperature, Lorentz factor, and comoving volume, respectively

$$\begin{cases} \;T'(r) \propto r^{-1} \\\;T'(r) \propto r^{-2/3} \end{cases}, \quad \Gamma(r) \propto r \quad , \quad V'(r) \propto r^3 \quad , \quad r < r\_s \tag{14}$$

from which it follows that, as the shell accelerates (as Γ increases with *r*), its internal energy drops (as *T* 0 decreases with *r*). Finally, the evolution of the observed temperature is given by

$$T^{ob}(r) = \Gamma(r)T'(r) = \begin{cases} \begin{array}{c} T\_0 \\\ T\_0(r/r\_s)^{-2/3} \end{array} \end{cases}, \begin{array}{c} r < r\_s \\\ r \ge r\_s \end{array} . \tag{15}$$
