2.3.2. Adiabatic Cooling

As discussed in Section 2.1, particles lose their energy adiabatically due to the spreading of the emission region. This energy loss term should be inserted in the kinetic equation governing the evolution of the particle distribution. To derive the adiabatic losses, we rewrite Equation (10) as a function of energy losses *dγ* in a comoving time *dt*0 :

$$
\dot{\gamma}\_{ad} = \frac{d\gamma}{dt'}\Big|\_{ad} = -\frac{\gamma\beta^2}{3} \frac{d\ln V'}{dt'}\Big,\tag{36}
$$

with *β* being the random velocity of particles in a unit of *c*. The comoving volume *V* 0 of the emission region can be estimated considering that the contact discontinuity is moving away from the shock at a velocity *c*/3. After a time *t* 0 = R *dR*/Γ(*R*) *c*, the comoving volume is:

$$V' = 4\pi R^2 \frac{ct'}{3} \,' \,. \tag{37}$$

and:

$$\gamma\_{ad} = \frac{d\gamma}{dt'}\Big|\_{ad} = -\frac{\gamma\beta^2}{3} \left(\frac{2\Gamma c}{R} + \frac{1}{t'}\right). \tag{38}$$

#### 2.3.3. Synchrotron Self-Absorption (SSA)

Electrons can re-absorb low energy photons before they escape from the source region. The absorption coefficient *α<sup>ν</sup>* can be expressed as [80]:

$$\mathfrak{a}\_{\upsilon} = -\frac{1}{8\pi\upsilon'^2 m\_{\varepsilon}} \int d\gamma P'(\gamma\_{\prime}\nu')\gamma^2 \frac{\partial}{\partial \gamma} \left[\frac{N(\gamma)}{\gamma^2}\right] \tag{39}$$

valid for any radiation mechanism at the emission frequency *ν* 0 , with *P* 0 (*γ*, *ν* 0 ) being the specific power of electrons with Lorentz factor *γ* at frequency *ν* 0 and assuming *hν* <sup>0</sup> *γmec* 2 . Thus, the SSA mechanism will mostly affect the low frequency range. This results in a modification of the lower frequency tail of the synchrotron spectrum as:

$$P'\_{\nu} \propto \begin{cases} \nu^{\prime 5/2} & \nu'\_{\mathrm{i}} < \nu' < \nu'\_{\mathrm{SSA}} \\ \nu^{\prime 2} & \nu' < \nu'\_{\mathrm{SSA}} < \nu'\_{\mathrm{i}} \end{cases} \tag{40}$$

assuming a power-law distribution of electrons, with *ν* 0 *<sup>i</sup>* = min(*ν* 0 *<sup>m</sup>*, *ν* 0 *cool*) and *ν* 0 *SSA* the frequency below which the synchrotron flux is self-absorbed and the source becomes optically thick.
