*3.7. Fraction of Accelerated Particles*

As described in Section 2, the representation of the relativistic shock acceleration in GRB afterglow relies on some free parameters. These values describe the energy equipartition between particles and magnetic field and the non-thermal accelerated particle distribution.

In particular, the parameter *ξ<sup>e</sup>* is responsible to account for the fraction of particles (here we consider electrons, but the same considerations are also valid for protons) accelerated into a non-thermal distribution. This means that from relativistic shock theory it is expected that a fraction 1 − *ξ<sup>e</sup>* of electrons is heated into a thermal distribution rather than a non-thermal one (see Figure 9). For simplicity, it is usually assumed in afterglow studies that *ξ<sup>e</sup>* = 1, i.e., all the particles are accelerated into a non-thermal distribution. Such an assumption is used in order to avoid the large degeneracy which affects the GRB parameters when including this additional free value. In particular, afterglow modeling predictions obtained assuming *ξ<sup>e</sup>* = 1 for parameters *E<sup>k</sup>* , *n*, *e<sup>e</sup>* and *e<sup>B</sup>* cannot be distinguished from those estimated for any *ξ<sup>e</sup>* in the range *me*/*m<sup>p</sup>* < *ξ<sup>e</sup>* < 1 and afterglow parameters *E* 0 *<sup>k</sup>* = *Ek*/*ξ<sup>e</sup>* , *n* 0 = *n*/*ξ<sup>e</sup>* , *e* 0 *<sup>e</sup>* = *ξee<sup>e</sup>* and *e* 0 *<sup>B</sup>* = *ξee<sup>B</sup>* [73]. This can be proven considering the dependencies of the jet dynamics and shock energy equipartition on the model parameters. As shown in BM76 and by previous calculations on the evolution of a relativistic blastwave, in the self-similar regime the bulk Lorentz factor evolve as Γ ∝ (*Ek*/*n*) 1/2. As a result, the same flow evolution can be obtained with different values of *E<sup>k</sup>* and *n* as long as their ratio is preserved. The fraction of energy given to the magnetic field is reduced by a factor *ξ<sup>e</sup>* but at the same time the energy density given to the shock is increased by a factor 1/*ξ<sup>e</sup>* so that the magnetic field energy density is the same in the scenario when including *ξ<sup>e</sup>* or not. Analogous considerations can be also conducted for the number and the energy density of the electrons so that their values are preserved. As a result, in principle it is not possible to distinguish between the two parameter sets obtained for *ξ<sup>e</sup>* or for any value *me*/*m<sup>p</sup>* < *ξ<sup>e</sup>* < 1. This implies that afterglow model parameters, when considering *ξ<sup>e</sup>* 6= 1, are estimated with an uncertainty of factor of *me*/*mp*, and the afterglow observations do not directly constrain their values (e.g., *E<sup>k</sup>* or *eB*) but rather are a fraction of their value multiplied or divided by *ξ<sup>e</sup>* (e.g., *Ek*/*ξ<sup>e</sup>* or *eBξe*).

It is possible to obtain information on the value of *ξ<sup>e</sup>* through PIC simulations or indirect features of the thermal component on the synchrotron afterglow spectra. As mentioned in the previous section, PIC simulations of weakly magnetized shocks have found that the downstream population include around ∼3% of the electrons, which are accelerated into a non-thermal distribution. In case the efficiency is low (around ∼10% or less) the presence of a large population of thermal electrons may affect the afterglow radiation spectra. The thermal electrons are distributed at lower energies than the non-thermal ones since *ηγmec* <sup>2</sup> *<sup>γ</sup>mp<sup>c</sup>* 2 , where *η mp*/*m<sup>e</sup>* is a factor describing the ignorance on the plasma physics governing electron heating beyond *γmec* 2 . As a result, the synchrotron radio component emission may be affected through the production of a new emission component from thermal electrons (for *η* 1 and moderate 1/*ξe*) or a large self-absorption optical depth (for *ξ<sup>e</sup>* 1), which may be visible in a time scale of few hours. Possible effects of the thermal component are discussed in [136–138].

**Figure 9.** Distribution of accelerated electrons from a relativistic shock. The two bumps correspond, respectively, to the distribution of the (1 − *ξ*) fraction of particles heated into a thermal component with post shock energy ∼ *γmec* <sup>2</sup> and the fraction *ξ* of particles accelerated into a non-thermal component with post shock energy ∼ *γmpc* 2 . Adapted from [73]. ©AAS. Reproduced with permission.

Information from the TeV component cannot completely solve the degeneracy between afterglow parameters and cannot provide additional clues on the non-thermal electron distribution. However, it can provide unique fundamental data useful to constrain the other afterglow parameters less constrained such as *e<sup>B</sup>* and the density. This will also impact the *ξ<sup>e</sup>* calculation since it can help to reduce the degeneracy between the sets of solutions available in the parameter space. Indeed, the multi-wavelength modeling of GRB 190829A (detected at TeV energies by H.E.S.S.) showed that the only way to explain all the detected radiation, from radio to TeV, is to introduce the parameter *ξ<sup>e</sup>* in the modeling, which is constrained by data to be *ξ<sup>e</sup>* < 0.13 [72].
