2.3.4. Synchrotron and Inverse Compton Emission

Following [82], the synchrotron spectrum emitted by an electron with Lorentz factor *γ*, averaged over an isotropic pitch angle distribution, is:

$$P\_{\nu'}^{\text{syn}}(\nu',\gamma) = \frac{2\sqrt{3}\,\text{e}^3 \,\text{B}'}{m\_\varepsilon \,\text{c}^2} \,\text{x}^2 \left[\text{K}\_{4/3}(\text{x})\,\text{K}\_{1/3}(\text{x}) - 0.6\,\text{x}(\text{K}\_{4/3}^2(\text{x}) - \text{K}\_{1/3}^2(\text{x}))\right],\tag{41}$$

where *x* ≡ *ν* 0 4*π mec*/(6 *e B*0*γ* 2 ) and *K<sup>n</sup>* are the modified Bessel functions of order *n*. The total power emitted at the frequency *ν* 0 is obtained integrating over the electron distribution:

$$P\_{\nu'}^{\prime \text{sym}}(\nu') = \int P\_{\nu'}^{\prime \text{sym}}(\nu', \gamma) \frac{d\mathcal{N}}{d\gamma} d\gamma \,. \tag{42}$$

The SSC radiation emitted by an electron with Lorentz factor *γ* can be calculated as:

$$P\_{\nu'}^{\prime \text{SSC}}(\nu',\gamma) = \frac{3}{4} \hbar \sigma\_T c \frac{\nu'}{\gamma^2} \int \frac{d\tilde{\nu}'}{\tilde{\nu'}} n\_{\tilde{\nu'}} \mathcal{K}(\gamma,\nu',\tilde{\nu'})\,,\tag{43}$$

where *n* ˜*<sup>ν</sup>* <sup>0</sup> is the photon density of synchrotron photons and the integration is performed over the entire synchrotron spectrum. Integration over the electron distribution provides the total SSC emitted power at frequency *ν* 0 .

## 2.3.5. Pair Production

Pair production by photon-photon annihilation is particularly important for a correct estimate of the radiation spectrum in the GeV-TeV band. Indeed, some of the emitted VHE photons are lost due to their interaction with photons at lower energies (typically X-ray photons). As a result, the observed flux is attenuated and the resulting spectrum at VHE is modified. Here we follow the treatment presented in [83]. The cross section of the process *σγγ* as a function of *β* 0 , the centre-of-mass speed of the electron and positron is given by:

$$
\sigma\_{\mathcal{V}\mathcal{I}}(\boldsymbol{\beta}') = \frac{3}{16} \sigma\_{\mathcal{V}}(1-\boldsymbol{\beta}'^2) \left[ (3-\boldsymbol{\beta}'^4) \ln \left( \frac{1+\boldsymbol{\beta}'}{1-\boldsymbol{\beta}'} \right) - 2\boldsymbol{\beta}'(2-\boldsymbol{\beta}'^2) \right] \tag{44}
$$

where:

$$\beta'(\omega\_{l\prime}\omega\_{s\prime}\mu) = \left[1 - \frac{2}{\omega\_l\omega\_s(1-\mu)}\right]^{\frac{1}{2}}\tag{45}$$

and *ω<sup>t</sup>* = *hν* 0 *<sup>t</sup>*/*mec* <sup>2</sup> with *ν* 0 *t* being the target photon frequency, *ω<sup>s</sup>* = *hν* 0/*mec* <sup>2</sup> with *ν* 0 being the source photon frequency and *µ* = cos *φ*, where *φ* is the scattering angle. Then, it is possible to derive the annihilation rate of photons into electron-positron pairs as:

$$R(\omega\_{\rm t}, \omega\_{\rm s}) = c \int\_{-1}^{\mu\_{\rm max}} \frac{d\mu}{2} (1 - \mu) \sigma\_{\gamma\gamma}(\omega\_{\rm t}, \omega\_{\rm s}, \mu) \, , \tag{46}$$

where *µmax* = max(−1, 1 − 2/*ωsωt*) coming from the requirement *β* <sup>0</sup><sup>2</sup> > 0. Considering *x* = *ωtω<sup>s</sup>* , it is possible to derive asymptotic limits for *R*(*ω<sup>t</sup>* , *ωs*) ≡ *R*(*x*) in two regimes. For *x* → 1 (i.e., near the threshold condition) *R*(*x*) → *cσT*/2(*x* − 1) 3/2, while for *<sup>x</sup>* <sup>1</sup> (i.e., ultra-relativistic limit) *<sup>R</sup>* <sup>→</sup> <sup>3</sup> 4 *cσ<sup>T</sup>* ln *x*/*x*. An accurate and simple approximation, which takes into account both regimes, is given by:

$$R(\mathbf{x}) \approx 0.652 \alpha \sigma\_T \frac{\mathbf{x}^2 - 1}{\mathbf{x}^3} \ln \left( \mathbf{x} \right) \text{H}(\mathbf{x} - 1) \text{ \AA} \tag{47}$$

where *H*(*x* − 1) is the Heaviside function [83]. The approximation accurately reproduces the behavior near the peak at *<sup>x</sup>peak* <sup>∼</sup> 3.7 and over the range 1.3 <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; <sup>10</sup><sup>4</sup> , which usually dominates during the calculations. A comparison between Equations (46) and (47) is given in Figure 3, where the goodness of the approximation adopted in the mentioned *x* range can be observed.

**Figure 3.** Comparison between the exact annihilation rate (Equation (46)) and the approximated formula (Equation (47)). The ratio between the two curves in the range (1.3 < *x* < 10<sup>4</sup> ) is shown in the bottom panel. In this range, the ratio is always . 7%.

The impact of the flux attenuation due to pair production mechanism on the GRB spectra is estimated in terms of the optical depth value *τγγ*. From its definition:

$$
\pi\_{\gamma\gamma}(\upsilon') = \sigma\_{\upsilon'\upsilon'\_t} \mathfrak{n}'(\upsilon'\_t) \Delta R' \tag{48}
$$

where *n* 0 (*ν* 0 *t* ) is the number density of the target photons per unit of volume, *σ<sup>ν</sup>* 0*ν* 0 *t* is the cross section and ∆*R* 0 is the width of the emission region. Introducing the cross section in terms of the annihilation rate *R*(*x*) in its approximated formula and integrating over all the possible target photon frequencies:

$$
\pi\_{\gamma\gamma}(\nu') = \frac{\Delta\mathcal{R}'}{c} \int \mathcal{R}(\nu', \nu\_t') n\_{\nu'}'(\nu\_t') d\nu\_t' \tag{49}
$$

where *ν* 0 and *ν* 0 *t* are the frequencies of the source and of the target interacting photons. The pair production attenuation factor can be then introduced simply multiplying the

flux by a factor (1 − *e* <sup>−</sup>*τγγ* )/*τγγ*. This attenuation factor will modify the GRB spectrum, giving a non-negligible contribution in the VHE domain, in particular. An example of the modification of a GRB spectrum due to pair production can be observed in Figure 4. Here, the flux emitted in the afterglow external forward shock scenario by synchrotron and SSC radiation and the flux attenuation due to pair production have been calculated with a numerical code. For a set of quite standard afterglow parameters and assuming ∆*R* 0 = *R*/Γ, the attenuation of the observed flux due to pair production become relevant above 0.2 TeV, and it reduces the flux by ∼ 30% at 1 TeV and by ∼ 70% at 10 TeV.

**Figure 4.** Spectral energy distribution in the GRB afterglow external forward shock scenario estimated with the numerical code presented in this section for *tobs* = 170 s. The effect of the pair production attenuation is clearly evident in the VHE tail. The set of afterglow parameters used are the following: *<sup>E</sup><sup>k</sup>* <sup>=</sup> <sup>1</sup> <sup>×</sup> <sup>10</sup><sup>53</sup> erg, *<sup>s</sup>* <sup>=</sup> 0, *<sup>A</sup>*<sup>0</sup> <sup>=</sup> 1 cm−<sup>3</sup> , *e<sup>e</sup>* = 0.2, *e<sup>B</sup>* = 0.02, Γ<sup>0</sup> = 300, *p* = 2.5 and *z* = 0.5.

Similar considerations can also be conducted for the electron/positron production. Assuming that the electron and positron arises with equal Lorentz factor *γ* and that *xpeak* ∼ 3.7, a photon with energy *ω<sup>s</sup>* 1 will mostly interact with a target photon of energy *ω<sup>t</sup>* ≈ 3/*ω<sup>s</sup>* . Then, from the energy conservation condition:

$$2\gamma = \omega\_s + \frac{3}{\omega\_s} \approx \omega\_s = \frac{h\nu'}{m\_c c^2} \tag{50}$$

The *e* ± production can be observed as an additional source term for the distribution of accelerated particles. As a result, an additional injection term *Q pp e* to be inserted in the kinetic equation (Equation (31)) is calculated as:

$$Q\_{\varepsilon}^{pp}(\gamma\_{\prime}t^{\prime}) = 4 \frac{m\_{\varepsilon}c^2}{h} n\_{\nu^{\prime}}(\frac{2\gamma m\_{\varepsilon}c^2}{h}, t^{\prime}) \int dv\_{t}^{\prime} n\_{\nu^{\prime}}(v\_{t\prime}^{\prime}t^{\prime}) \mathcal{R}(\frac{2\gamma m\_{\varepsilon}c^2}{h}, v\_{t}^{\prime}) \tag{51}$$

#### 2.3.6. Comparison with Analytical Approximations

In order to compare results from the numerical method described in the previous section and analytical prescriptions available in the literature, we give an example in Figure 5. The analytical prescriptions for the synchrotron and the SSC component are calculated following [38,48]. In [38], the synchrotron spectra and light-curves are derived assuming a power-law distribution of electrons in an expanding relativistic shock, cooling only by synchrotron emission. The dynamical evolution is described following BM76 equations for an adiabatic blast-wave expanding in a constant density medium. The resulting emission spectrum (green dashed lines in Figure 5) is described with a series of sharp broken power-laws. The SSC component associated to the synchrotron emission was computed, as a function of the afterglow parameters, in [48]. In this work, calculations are performed assuming that the scatterings occur in Thomson regime. Modifications to the synchrotron spectrum caused by strong SSC electron cooling are also detailed.

**Figure 5.** Comparison between spectra estimated with analytical approximations and numerical calculations. For the analytical method, the synchrotron emission (green dashed line) is estimated starting from [38] (SPN98, see legend) while the SSC (black dashed line) is taken from [48] (SE01). Vertical lines mark the break frequencies. The results from the numerical code described in Section 2.3 are shown with solid blue and red lines. This example shows the spectrum calculated at *t* = 10<sup>4</sup> s for *<sup>s</sup>* <sup>=</sup> 0, *<sup>p</sup>* <sup>=</sup> 2.3, *<sup>e</sup><sup>e</sup>* <sup>=</sup> 0.05, *<sup>e</sup><sup>B</sup>* <sup>=</sup> <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> , *E<sup>k</sup>* = 10<sup>52</sup> erg, *n*<sup>0</sup> = 1 cm−<sup>3</sup> , Γ<sup>0</sup> = 400 and *z* = 1.

From the comparison proposed in Figure 5, it can be clearly observed that analytical and numerical results are in general in good agreement. Both curves follow the same behavior except for the high-energy part of the SSC component. Here, the KN scattering regime, which is not taken into account in the analytical approximation, becomes relevant. As a result, the numerical calculations differ from the analytical ones showing a peak and a cutoff in the SSC spectrum due to the KN effects.

Nevertheless, there are a few minor discrepancies between the two methods. The numerically-derived spectrum is very smooth around the break frequencies, with the result that the theoretically expected slope (e.g., the one predicted by the analytical approximations) is reached only in regions of the spectrum that lie far from the breaks, i.e., is reached only asymptotically. This puts into questions simple methods for discriminating among different regimes and different density profiles based on closure relations, which are relations between the spectral and the temporal decay indices [10,38]. Regarding the flux normalization, there are minor discrepancies between the numerical and analytical results. This is due to the fact that in analytical prescriptions it is assumed that the radiation is entirely emitted at the characteristic synchrotron frequency. On the contrary, in the numerical derivation, the full synchrotron spectrum of a single electron is summed up over the whole electron distribution. Similar considerations apply to the SSC component when comparing with the analytical spectra. Moreover, the discrepancies observed between analytical and numerical SSC spectra are amplified by the differences observed in the target synchrotron spectra.

In general, this comparison shows that the numerical treatment is a powerful tool able to predict the multi-wavelength GRB emission in a more accurate way than the analytical prescriptions. The latter ones, however, are still giving valid approximations of the overall spectral shape. The main limitation of analytical estimates arises when TeV observations are involved. The importance of KN corrections is evident in this band and should be properly treated for a correct interpretation of the TeV spectra, as will be shown in Section 4.
