*2.3. Derivation of the Radiative Output*

The expected radiative output can be estimated by means of analytical approximations, which provide prescriptions for the location of the synchrotron self-absorption frequency *νsa*, the characteristic frequency *ν<sup>m</sup>* emitted by electrons with Lorentz factor *γmin*, the cooling frequency *ν<sup>c</sup>* emitted by electrons with Lorentz factor *γ<sup>c</sup>* and the overall synchrotron flux [38,47]. In these approaches, the synchrotron spectrum is in general approximated with power-laws connected by sharp breaks, but more sophisticated analytical approximations of numerically derived synchrotron spectra have also been proposed [54]. The associated SSC component in the Thomson regime [48] and corrections to be applied to the synchrotron and SSC spectra to account for the effects of the Klein–Nishina [75] cross section (see Section 2.3.1) are also available in the literature. These prescriptions are usually developed for the deceleration phase, when the Blandford–McKee solution [35] for the blast-wave dynamics applies, i.e., as long as the blast-wave is still relativistic. These models take as input parameters the kinetic energy content of the blast-wave *E<sup>k</sup>* , the external density *n*(*R*) = *n*<sup>0</sup> *R* −*s* (with *s* = 0 or *s* = 2), the fraction of shock-dissipated energy gained by electrons (*ee*) and by the amplified magnetic field (*eB*), and the spectral slope of the accelerated electrons *p*. During the deceleration phase, the initial bulk Lorentz factor Γ<sup>0</sup> does not play any role, but its value determines the radius (or time) at which the deceleration begins.

An alternative approach to estimate the expected spectra and their evolution in time consists in numerically solving the differential equation describing the evolution of the particle spectra and estimating the associated emission [76–78]. In this section, we describe a radiative code that simultaneously solves the time evolution of the electron and photon distribution. The code has been adopted, e.g., for the modeling of GRB 190114C presented in [71].

The temporal evolution of the particle distribution is described by the differential equation:

$$\frac{\partial \mathcal{N}(\gamma\_{\prime}t^{\prime})}{\partial t^{\prime}} = \frac{\partial}{\partial \gamma} \left[ \dot{\gamma} \mathcal{N}(\gamma\_{\prime}t^{\prime}) \right] + \mathcal{Q}(\gamma) \,. \tag{31}$$

where *γ*˙ = *∂γ ∂t* 0 is the rate of change of the Lorentz factor *γ* of an electron caused by adiabatic, synchrotron and SSC losses and energy gains by synchrotron self-absorption. In the SSC mechanism, the synchrotron photons produced by electrons in the emission region act as seed photons that are up-scattered at higher energies by the same population of electrons. Such a scenario will generate a very high energy spectral component, which is the target of searches by IACTs such as MAGIC<sup>1</sup> and H.E.S.S.<sup>2</sup> . In principle, up-scattering of an external population of seed photons can also be considered and included in the cooling term, but here we will ignore this mechanism (external Compton) and focus only on SSC. The source term *Q*(*γ*, *t* 0 ) = *Qacc*(*γ*, *t* 0 ) + *Qpp*(*γ*, *t* 0 ) describes the injection of freshly accelerated

particles (*Qacc*(*γ*, *t* 0 ) = *dNacc*/*dγ dt*<sup>0</sup> ) and the injection of pairs *Qpp*(*γ*, *t* 0 ) produced by photon–photon annihilation.

In the next sections, we explicitly define each one of the terms included in Equation (31) and how to estimate the synchrotron and SSC emission. To solve the equation, an implicit finite difference scheme based on the discretization method proposed by [79] can be adopted.
