*3.3. Internal Shock Model*

In the context of GRBs, the first ideas for interpreting the observed highly irregular temporal pattern of radiation came soon after establishing the extragalactic origin of GRBs [10]. It was suggested that energy and matter injection by the compact central object (<10<sup>7</sup> cm) does not occur at a steady rate. The resulting outflow would in that case consist of a sequence of "shells" with fluctuating Lorentz factors ([126,127]). In the interaction of a faster shell and a slower one emitted earlier, a shock would develop, which would accelerate electrons to relativistic velocities.

Emission from internal shocks in a relativistic wind with varying Lorentz factors has been studied extensively, e.g., [128–132]. The initial kinetic energy is dissipated in collisions of a series of successive shells emitted from the central engine, having a nonuniform distribution of Lorentz factors Γ(*t*). In the model described by [129], shells interact only by direct collisions, and one shock wave is discretized by the series of shocks (for a comparison with a detailed hydrodynamical calculation, see e.g., [133,134]). The dynamic phase is described by the following parameters:The total duration of the energy ejection by the central engine, the distribution Γ(*t*), and the injected kinetic power during the ejection phase.

For each collision, one can calculate the radius, collision time, Lorentz factor of the shocked material, and the energy dissipated in the collision. The advantage of this model is that the variability time of the energy injection roughly translates into the observed variability time in the GRB lightcurve [135]. The fraction of the thermal energy dissipated in collisions is deposited in electrons in the two colliding shells, while the remaining energy goes into proton acceleration and magnetic field amplification. The efficiency of the energy dissipation process is typically low, .15% [129,136,137], which is the main drawback of the internal shock model.

The microphysics related to a shocked medium is usually parametrized by assuming that a fraction *e<sup>e</sup>* of the dissipated energy is given to the ambient electrons [126,129,131,135]. The electrons are assumed to be accelerated to a power-law, *n*(*γe*) ∝ *γ* −*p e* , above their initial thermal distribution (typical Lorentz factor denoted by *γm*). The slope of the electron distribution *p* depends on the details of the acceleration process [129]. Under the assumption that the leading radiative process is synchrotron emission from these power-law distributed electrons, the observed high energy photon spectral index *β* of the "Band"-fitting function, provides the indication for the steepness of the particle distribution, *p* ≈ 2.5 [138]. For typical parameters, the synchrotron emission produced by the accelerated electrons in a magnetic field *B* would occur at observed energy Esyn = 50(Γ∗/300)(*B*/1000 *G*)(*γe*/100) 2 eV [129]. To obtain higher electron Lorentz factors (103–10<sup>4</sup> ) in order to reach an observed peak energy at a <sup>∼</sup>few <sup>×</sup> 100 keV, several authors have suggested that only a fraction *<sup>ζ</sup>* <sup>∼</sup>10−<sup>3</sup> of electrons are accelerated [129,131,139].

Note that there is a large uncertainty in the value of the magnetic field. During the prompt phase, there can be two sources of magnetic field: (i) a strong magnetic field may be associated with the central engine (e.g., [140]). Its strength will decay with distance, however it may still be considerable if the source is highly magnetized, and the dissipation does not occur at too large a distance. (ii) In addition, the magnetic field may be generated at the shock front, obtaining an uncertain fraction (referred to as *eB*) of the dissipated energy at the shock.

The accelerated relativistic electrons cool mainly by the synchrotron process, and the associated inverse Compton radiation. The high energy portion of the spectrum is attenuated by photon-photon annihilation, and by the EBL (extragalactic background light) absorption. The low energy portion of the spectrum has a steep cutoff due to self-absorption. The temporal profiles of the prompt emission can be obtained when the contributions from all collisions are taken into account. One example of such study is shown in Figure 1. Here the calculation was performed neglecting the interaction between photons emitted in a shocked region and electrons/photons present in another region (see, e.g., [141]); in addition, the possible contribution of the shock accelerated protons was not considered. 698 Ž. Bošnjak et al.: Prompt HE emission from GRBs in the internal shock model

**Fig. 15.** A single pulse burst in the "synchrotron case" with a low magnetic field. Same as in Fig. 14 except for the microphysics parameters:

**Fig. 16.** A single pulse burst in the "inverse Compton case". Same as in Fig. 14 except for the initial distribution of the Lorentz factor that varies from 100 to 600 and for the microphysics parameters: !<sup>B</sup> = 10−2, !<sup>e</sup> = 1/3, ζ = 1 and *p* = 3.5. In *the left panel*, the lightcurves observed both in

discontinuity).

synchrotron component.

sults in *t* # syn % *t* #

to an immediate violent shock (for instance with an initial

scatterings. The emission at high energy is dominated by the

ex and a low efficiency for inverse Compton

**–** In a second phase (around the peak of the pulse in the GBM range), the shock becomes stronger, Γ<sup>m</sup> increases and the synchrotron timescale decreases. On the other hand, as the radius increases, the dynamical timescale increases. This re-

!<sup>B</sup> = 5 × 10−3, !<sup>e</sup> = 1/3, ζ = 2 × 10−<sup>3</sup> and *p* = 2.5. **Figure 1.** A single pulse burst: the main emission peak is due to the synchrotron radiation. The microphysics parameters used in the simulations are *<sup>e</sup><sup>B</sup>* = 5<sup>×</sup> <sup>10</sup>−<sup>3</sup> , *<sup>e</sup><sup>e</sup>* = 1/3, *<sup>ζ</sup>* = 2 <sup>×</sup> <sup>10</sup>−<sup>3</sup> , *p* = 2.5, and dE/dt = 5 <sup>×</sup> <sup>10</sup><sup>53</sup> erg s−<sup>1</sup> . As the assumed magnetic field is low, the non-negligible signatures of inverse Compton scatterings are favored in the *Fermi* LAT (the Large Area Telescope) energy range LAT energy range. The process included in calculation are the following: adiabatic cooling, synchrotron emission and synchrotron-self absorption, inverse Compton scatterings, and *γγ*-annihilation. The effects of secondary pairs were not taken into account. *Left*: observed light curves in *Fermi*-GBM (Gamma-ray Burst Monitor and the LAT range. The synchrotron (thin solid line) and inverse Compton (thin dashed line) components are shown. *Right*: observed time-integrated spectrum during the rise, early decay, and whole duration of the pulse. From Credit: [131], reproduced with permission ©ESO [131].

the *Fermi*-GBM+LAT energy range are entirely dominated by inverse Compton emission.

ex and the efficiency of inverse Compton

electron Lorentz factors Γm, and therefore large synchrotron timescales. On the other hand, these early times correspond to small radii so the dynamical timescale is still small. In this

scatterings is large, as a large fraction of the shocked region is populated by relativistic electrons (see Sect. 3). It results in a weak precursor in the GeV lightcurve. This precursor can disappear if a different initial distribution of the Lorentz factor in the outflow is adopted, especially if it leads

first phase, *t* # syn <∼ *t* #

Some authors have pointed out difficulties within the internal shock model when applied to the 'naked-eye' burst GRB080319B for which variable prompt optical emission is present [142,143]. The main issue seems to be that the observations point to a very large radius of emission: at these large distances, the gamma-ray flux would be much smaller than observed. These difficulties have served as motivation for alternative models [142].
