3.1.2. Internal Shock Scenario and Photospheric Emission

In the case of LGRBs, the progenitor continuously emits energy at a rate *L*, over a longer duration *<sup>t</sup> <sup>r</sup>*0/*c*, and ejects mass at a rate *<sup>M</sup>*˙ <sup>=</sup> *<sup>L</sup>*/(*η<sup>c</sup>* 2 ). In this case, the scaling laws for the instantaneous release are still valid, provided that *E* is replaced by *L* and *M* by *M*˙ , and a further equation for the mass conservation of the baryons (within the spherical symmetry assumption) is required [72]

$$m\_p'(r) = \frac{\dot{M}}{4\pi r^2 m\_p c \Gamma(r)} = \frac{L}{4\pi r^2 m\_p c^3 \eta \Gamma(r)}\tag{16}$$

where *n* 0 *p* (*r*) is the comoving number density of baryons and *m<sup>p</sup>* is the proton mass.

For longer activity of the inner engine, fluctuations in the energy emission rate would result in the propagation of independent shells, each of them with analogous thickness *r*<sup>0</sup> and dynamics. For two consecutive shells with a difference in their Lorentz factors *δ*Γ ∼ *η* or velocities *δv* ≈ *c*/(2*η* 2 ), collisions become possible after a typical time *tcol* = *r*0/*δv* and an observer frame radius [73]

$$
\sigma\_{\rm col} = \upsilon t\_{\rm col} \simeq \mathfrak{ct}\_{\rm col} \simeq 2\eta^2 r\_0 \,. \tag{17}
$$

Above *rcol*, which is a factor *η* larger than *r<sup>s</sup>* , collisions occur, dissipate the kinetic energy, and convert it into the observed radiation [74,75]. The advantages of the internal shock scenario are listed below:


and different adiabatic energy losses of photons [62,78]. This emission explains the thermal-like emission embedded in the non-thermal spectra of some GRBs [62,79,80].

However, the internal shock scenario manifests some drawbacks.

