2.3.1. Synchrotron and SSC Cooling

The synchrotron power emitted by an electron with Lorentz factor *γ* depends on the pitch angle, i.e., the angle between the electron velocity and the magnetic field line. In the following, we assume that the electrons have an isotropic pitch angle distribution and we use equations that are averaged over the pitch angle (e.g., [80]). The synchrotron cooling rate of an electron with Lorentz factor *γ* is given by:

$$\dot{\gamma}\_{syn} \equiv \frac{d\gamma}{dt'}\Big|\_{syn} = -\frac{\sigma\_T \gamma^2 B'^2}{6\pi \, m\_c c} \tag{32}$$

The cross section for the inverse Compton mechanism is constant and equal to the Thomson cross section (*σT*) as long as the photon energy in the frame of the electron is smaller than the rest mass electron energy *m<sup>e</sup> c* 2 . For higher photon energies, the cross section decreases as a function of the energy and is described by the Klein-Nishina (KN) cross section. To estimate SSC losses, we adopt the formulation proposed in [81], which is valid for both regimes. Defining the SSC kernel as:

$$\mathcal{K}(\gamma, \nu', \tilde{\nu}') = \begin{cases} \frac{\varepsilon}{\tilde{\varepsilon}} - \frac{1}{4\gamma^2} & \frac{\varepsilon}{4\gamma^2} < \varepsilon < \tilde{\varepsilon} \\\ 2q \ln q + (1 + 2q)(1 - q) + \frac{1}{2}(1 - q)\frac{(4\gamma \sharp q)^2}{(1 + 4\gamma \sharp q)} & \tilde{\varepsilon} < \varepsilon < \frac{4\gamma^2 \varepsilon}{1 + 4\gamma \sharp} \end{cases} \tag{33}$$

where:

$$\mathfrak{E} = \frac{h\tilde{\nu}'}{m\_{\varepsilon}c^2} \quad \varepsilon = \frac{h\nu'}{m\_{\varepsilon}c^2} \quad q = \frac{\varepsilon}{4\gamma\varepsilon(\gamma-\varepsilon)}\,. \tag{34}$$

*ε*˜ and *ε* are the energies of the photons (normalized to the rest mass electron energy) before and after the scattering process, respectively. The two terms of Equation (33) account, respectively, for the down-scattering (i.e., *ε* < *ε*˜) and the up-scattering (i.e., *ε* > *ε*˜) process. The energy loss term for the SSC can now be calculated with the equation:

$$\dot{\gamma}\_{\rm SSC} = \frac{d\gamma}{dt'}\bigg|\_{\rm SSC} = -\frac{\Im h \sigma\_{\rm t}}{4m\_{\rm c}c\gamma^2} \int d\nu' \nu' \int \frac{d\tilde{\nu}'}{\tilde{\nu}'} n\_{\tilde{\nu}'}(t') \mathcal{K}(\gamma\_\nu \nu', \tilde{\nu}') \,. \tag{35}$$
