*3.1. Hot Fireball Model*

The hot fireball model assumes the expansion of a fireball composed mostly of photons, electron-positron pairs, and neutrinos [85–87], where magnetic field is energetically subdominant. As the fireball expands adiabatically from a very small radius, the energy of photons and pairs is transferred to protons, which are accelerated to large Lorentz factors [88]. At large distances from the central engine, the kinetic energy of the jet is transformed back to thermal energy, and gamma-rays are produced [89].

Using conservation of energy and entropy, it can be shown (e.g., [15]) that the acceleration of the jet is linear with distance from the base of the fireball *R*0, namely Γ(*r*) ∝ *r*. The acceleration proceeds until the outflow reaches the saturation radius *R<sup>s</sup>* = *R*0Γ*<sup>s</sup>* , where Γ*<sup>s</sup>* is the terminal Lorentz factor. This is true as long as the photons are coupled to electrons in the outflow, therefore, the photospheric radius *R*ph plays an important role. (i) If *R<sup>s</sup>* < *R*ph, then the jet reaches Γ*<sup>s</sup>* at *R<sup>s</sup>* . (ii) On the other hand, if *R*ph < *R<sup>s</sup>* , then the acceleration mostly stops at *R*ph [90,91].

The luminosity of the photospheric component in the hot fireball model depends on cases (i) and (ii) mentioned above. For case (i), since the photospheric radius is larger than the saturation radius, the photon temperature decreases due to adiabatic cooling beyond the saturation radius and the thermal luminosity is expected to be lower than in case (ii). The observed photospheric emission is expected to be at a few MeV [85–88]. The emerging spectrum would not be as simple as *f<sup>ν</sup>* ∝ *ν* <sup>2</sup> below the peak, where *ν* is the observed

frequency and *f<sup>ν</sup>* is the flux density, however integration from different radii would flatten it slightly to approximately *f<sup>ν</sup>* ∝ *ν* 1.4−1.5 [83,92–94] and an angular dependence of the jet Lorentz factor can flatten it to ∝ *ν* 0 [95], see below. The presence of a strong thermal component in the gamma-ray spectrum would point to the hot fireball model. There is some evidence for a photospheric component in a number of GRB spectra, e.g., [96]. On the other hand, the lack of this clear component has been used to support the magnetically dominated jet model [97], which is described below.
