*3.1. The Fireball Model*

The GRB standard model considers a homogeneous *fireball* [69]. For a pure radiation fireball, a large fraction of the initial energy released by the newly-formed BH is converted directly into photons. Close to the BH, at a radius *r*<sup>0</sup> larger than the Schwarzschild radius, *R<sup>S</sup>* = 2*GM*/*c* 2 , the photon temperature is

$$T\_0 = \left(\frac{L}{4\pi a \, c \, r\_0^2}\right)^{1/4} = 1.2 \, L\_{52}^{1/4} r\_{0, \mathcal{I}}^{-1/2} \, \text{MeV} \tag{13}$$

where *a* is the radiation constant, and the luminosity *L* and the radius *r*<sup>0</sup> are expressed, respectively, as *L*<sup>52</sup> = *L*/10<sup>52</sup> erg/s and *r*0,7 = *r*0/10<sup>7</sup> cm. In the following, to understand the order of magnitude of the key physical parameters characterizing GRBs, we use the notation *Q<sup>x</sup>* = *Q*/10*<sup>x</sup>* , where the quantity *Q* is given in cgs units. The temperature *T*<sup>0</sup> is above the threshold for pair production, hence a large number of *e* ± pairs are created via photon–photon interactions, leading to a fully thermalized pairs-photons plasma with the opacity in Equation (12) 15 .

GRB luminosities are many orders of magnitude above the Eddington luminosity, *<sup>L</sup><sup>E</sup>* <sup>=</sup> 1.25 <sup>×</sup> <sup>10</sup>38(*M*/*<sup>M</sup>*) erg s−<sup>1</sup> ; therefore, the radiation pressure is much larger than self gravity and the fireball expands under its own pressure up to <sup>Γ</sup> <sup>≈</sup> <sup>10</sup>2–10<sup>3</sup> [48,49]. Since the final kinetic energy cannot exceed the initial explosion energy *E*tot, the maximum attainable Lorentz factor is defined as Γmax = *E*tot/*Mc*<sup>2</sup> and depends upon the amount of baryons (baryon load) of rest mass *M* within the fireball [69].
