2.2.2. Description of Shocks in GRB Afterglow Modeling

The theory of relativistic shock acceleration is applied to the GRB afterglow by introducing several unknown parameters in the model. These are the fractions *e<sup>e</sup>* and *e<sup>B</sup>* of dissipated energy gained by the accelerated particles and amplified magnetic field, the spectral index *p* of the accelerated particle spectrum and the fraction *ξ<sup>e</sup>* of particles, which efficiently enter the Fermi mechanism and populate the non-thermal distribution.

Recalling that the shock-dissipated energy (in the comoving frame) is given by *dE*0 *sh* = (<sup>Γ</sup> <sup>−</sup> <sup>1</sup>)*dmc*<sup>2</sup> (see Section 2.1), the corresponding energy density is *u* 0 sh = (Γ − 1)*ρ* 0 *c* 2 . From shock jump conditions, the density in the comoving frame *ρ* 0 is related to the density of the unshocked medium (measured in the rest frame) by the equation:

*ρ*

$$\mathbf{'} = \mathbf{4}\Gamma\rho \tag{20}$$

which is valid in both the ultra-relativistic and non-relativistic limits (see e.g., [35]). In the GRB afterglow scenario, it is usually assumed that pairs are unimportant and then the density of protons and electrons is the same: *n<sup>p</sup>* = *n<sup>e</sup>* = *n*. This implies that the mass is dominated by protons: *ρ* = *nmp*. In this case, the available energy density that will be distributed to the accelerated particles (electrons and protons) and to the magnetic field can be expressed as:

$$
\mu\_{\rm sh}' = 4\Gamma(\Gamma - 1)nm\_p c^2 \tag{21}
$$

A fraction *e<sup>B</sup>* of this energy will be conveyed to the magnetic field:

$$
\mu\_B' = \varepsilon\_B \mu\_{\rm sh}' = \varepsilon\_B 4 \Gamma(\Gamma - 1) m m\_p c^2 \tag{22}
$$

from which it follows that the magnetic field strength *B* 0 is:

$$B' = \sqrt{32\pi\epsilon\_B m\_p c^2 n(\Gamma - 1)\Gamma} \tag{23}$$

Similarly, for the accelerated electrons:

$$
\mu\_{\varepsilon}' = \varepsilon\_{\varepsilon} \mu\_{\text{sh}}' = \varepsilon\_{\varepsilon} 4\Gamma(\Gamma - 1) nm\_p c^2 = \langle \gamma \rangle m\_{\varepsilon} c^2 4\Gamma \xi\_{\varepsilon} n \tag{24}
$$

where h*γ*i is the average random Lorentz of the accelerated electrons:

$$
\langle \gamma \rangle = \frac{\epsilon\_\varepsilon}{\tilde{\xi}\_\varepsilon} \frac{m\_p}{m\_\varepsilon} (\Gamma - 1) \tag{25}
$$

The accelerated non-thermal electrons are assumed to have a power-law spectrum as a result of shock acceleration. Their energy distribution can be described by a powerlaw *N*(*γ*)*dγ* ∝ *γ* <sup>−</sup>*pd<sup>γ</sup>* for *<sup>γ</sup>min* <sup>≤</sup> *<sup>γ</sup>* <sup>≤</sup> *<sup>γ</sup>max* where *<sup>γ</sup>min* is the *minimum Lorentz factor* of the injected electrons and *γmax* is the *maximum Lorentz factor* at which electrons can be accelerated. To derive the relation between *γ*min, *γ*max and the model parameters, we consider the definition of the average Lorentz factor h*γ*i:

$$\langle \gamma \rangle = \frac{\int\_{\gamma\_{\min}}^{\gamma\_{\max}} N(\gamma)\gamma d\gamma}{\int\_{\gamma\_{\min}}^{\gamma\_{\max}} N(\gamma)d\gamma} \tag{26}$$

and solve the integrals. Equations (25) and (26) leads to (for *p* 6= 1):

$$\begin{cases} \frac{\ln\left(\frac{2\max}{\gamma\_{\min}}\right)}{\left(\frac{1}{\gamma\_{\min}} - \frac{1}{\gamma\_{\max}}\right)} = \frac{\varepsilon\_{\varepsilon}}{\xi\_{\varepsilon}} \frac{m\_p}{m\_\ell} (\Gamma - 1) & \text{if } p = 2\\ \begin{bmatrix} \frac{p-1}{p-2} \frac{\gamma\_{\min}^{-p+2} - \gamma\_{\max}^{-p+2}}{\gamma\_{\min}^{-p+1} - \gamma\_{\max}^{-p+1}} \end{bmatrix} = \frac{\varepsilon\_{\varepsilon}}{\xi\_{\varepsilon}} \frac{m\_p}{m\_\ell} (\Gamma - 1) & \text{if } p \neq 2 \end{cases} \tag{27}$$

A simplified equation for *γmin* can be obtained assuming that *γ* −*p*+2 *max γ* −*p*+2 *min* :

$$\gamma\_{\rm min} = \frac{\epsilon\_{\varepsilon}}{\tilde{\xi}\_{\varepsilon}} \frac{m\_p}{m\_\varepsilon} \frac{p-2}{p-1} (\Gamma - 1) \tag{28}$$

Since *p* is expected to be 2 < *p* < 3, this condition is verified for *γmax γmin*. The minimum Lorentz factor is then not treated as a free parameter of the model, as it is calculated from Equation (28) as a function of the free parameters *e<sup>e</sup>* , *ξ<sup>e</sup>* and *p*. Concerning the prescription for the value of *γmax* (for details see Section 3.5), it usually relies on the condition that radiative losses between acceleration episodes are equal to the energy gains, where energy gains proceed at the Bhom rate. As we mentioned in the previous section, PIC simulations, however, have shown that this might not be the case.

A similar treatment can be adopted also for protons simply substituting *e<sup>e</sup>* with *ep*, *m<sup>e</sup>* with *m<sup>p</sup>* and assuming a power-law energy distribution with spectral index *q*. As a result, the minimum Lorentz factor for protons can be derived as:

$$\begin{aligned} \left[\frac{\ln\left(\frac{\gamma mx\_p}{\gamma\_{\min,p}}\right)}{\frac{1}{\gamma\_{\min,p}} - \frac{1}{\gamma\_{\max,p}}}\right] &= \frac{\epsilon\_p}{\xi\_p} (\Gamma - 1) & \text{if } q = 2\\ \begin{bmatrix} \frac{q-1}{q-2} \frac{\gamma\_{\min,p}^{-q+2} - \gamma\_{\max}^{-q+2}}{\gamma\_{\min,p} - \gamma\_{\max,p}} \\ \gamma\_{\min,p} - \gamma\_{\max,p} \end{bmatrix} &= \frac{\epsilon\_p}{\xi\_p} (\Gamma - 1) & \text{if } q \neq 2 \end{aligned} \tag{29}$$

Solving the equations assuming that *γmax*,*<sup>p</sup> γmin*,*<sup>p</sup>* leads to:

$$
\gamma\_{\min,p} = \frac{\epsilon\_p}{\mathfrak{T}\_p} \frac{q-2}{q-1} (\Gamma - 1) \tag{30}
$$

The equations for *γmin* and Equation (23) for *B*, coupled with the description of the blast-wave dynamics described in Section 2.1, provides all the necessary equations to derive the radiative output for a jet with energy *E* and initial bulk Lorentz factor Γ<sup>0</sup> expanding in a medium with density *n*(*R*). The derivation of the radiative output is detailed in Section 2.3. To conclude the discussion about particle acceleration, in the next section we anticipate which constraints can be inferred on the physics of particle acceleration from multi-wavelength observations, once the afterglow model is adopted.

#### 2.2.3. Constraints to the Acceleration Mechanism Provided by Observations

Assuming that accelerated particles have a power-law spectrum (*dN*acc/*dγ* ∝ *γ* −*p* ) and the cooling is dominated by synchrotron radiation, the spectral slope *p* can be inferred from observations of the synchrotron spectrum and/or from the temporal decay of the lightcurves if observations are performed at frequencies higher than the typical frequency *ν*<sup>m</sup> of photons emitted by electrons with the Lorentz factor *γmin* (this is correct in both cases of fast and slow cooling regime). The estimated value of *p* from the afterglow modeling are spread on a wide range, from *p* ∼ 2 to *p* ∼ 3, suggesting that the spectrum of injected particles does not seem to have a typical slope, at odds with theoretical predictions. The determination of *p*, however, suffers from the uncertainties on the spectral index inferred from optical and X-ray observations, where the observed spectra are subject to unknown dust and metal absorption. A derivation of *p* from the decay rate of the lightcurves is also subject to the correct identification of the spectra regime, and partially also to the assumption on the density profile of the external medium, which is often unconstrained (see Section 3.2).

The typical value of *e*<sup>e</sup> inferred from the afterglow modeling is around 0.1, meaning that 10% of the shock-dissipated energy is gained by the electrons, spanning from 0.01 to large values, such as 0.8. Although this seems a large uncertainty, *e<sup>e</sup>* is perhaps the most well-constrained parameter of the model, and is in good agreement with the values predicted by numerical investigations [42]. For the fraction *e*B, on the contrary, the inferred values varies in a very wide range, typically from 10−<sup>5</sup> to 10−<sup>1</sup> [67–69]. Recent studies that incorporate Fermi-LAT GeV observations [24,70] have demonstrated that the typical values estimated for *<sup>e</sup><sup>B</sup>* can be even smaller, in the range <sup>∼</sup>10−7–10−<sup>2</sup> . These values are needed in order to model GeV radiation self-consistently with radiation detected at lower frequencies, with repercussions on the estimates of the other parameters, such as *n* and *E*. These small values of the *e<sup>B</sup>* needed to model the radiation have been tentatively interpreted as the sign of turbulence decay in the downstream [65,66]. As a consequence, even though the turbulence is strong (*e<sup>B</sup>* ' 0.1) in the vicinity of the shock where the particle is accelerated, it becomes weaker at larger distances, in the region where particles cool (see Section 3.3). Small values of *e<sup>B</sup>* are confirmed by the modeling of recent TeV detections of afterglow radiation from GRBs ([71,72], see Section 4).

Another parameter that one would like to constrain from observations is the fraction of particles *ξ<sup>e</sup>* that are injected into the Fermi process. In the vast majority of the studies, this parameter is not included (i.e., it is implicitly assumed that all the electrons are accelerated, *ξ<sup>e</sup>* = 1). This parameter is indeed difficult to constrain, as it is degenerate with all the other parameters [73].

Observations so far have not been able to identify the location of a high-energy cutoff in the synchrotron spectrum that would reveal the maximal energy of the synchrotron photons and then the maximum energy *γmax* of the accelerated electrons. Observations by Fermi-LAT are in general consistent with a single power-law extending up to at least 1 GeV. Photons with energies in excess of 1 GeV have been detected from several GRBs, the record holder for Fermi-LAT being a 95 GeV photon [74]. These photons cannot be

safely associated to synchrotron radiation on the basis of spectral analysis, as their paucity makes it difficult to assess from spectral analysis whether they are consistent with the power-law extrapolation of the synchrotron spectrum or if they are indicative of the rising of a distinct spectral component. In any case, the Fermi-LAT detections are suggesting that synchrotron photons should be produced at least up to a few GeV. This is consistent with the limit commonly invoked for particle acceleration: if the acceleration proceeds at the Bhom rate (*tacc* ' *rL*/*c*) with *r<sup>L</sup>* = *E*/*eB* being the Larmor radius, and is limited by synchrotron cooling (*tsyn* ' 6 *π mec*/*σTB* <sup>2</sup>*γ*) then *<sup>γ</sup>max* <sup>∼</sup> <sup>10</sup><sup>7</sup> <sup>−</sup> <sup>10</sup><sup>8</sup> can be reached. Even though this does not necessarily imply that acceleration must proceed at the Bhom limit, the value of *γmax* inferred from the detection of GeV photons is quite large and barely consistent with what is found by PIC simulations. Whether or not the observations are in tension with the present derivation of *γmax* from PIC simulations and theoretical arguments, strongly depends on a clear identification of the origin of photons in the GeV-TeV energy range. Present and future observations with Imaging Atmospheric Cherenkov Telescopes (IACTs) are the main candidates to shed light on this issue.
