*3.4. Impact of Jet Structure on the Prompt Emission Observables*

One big difficulty in interpreting the prompt emission properties of GRBs in terms of the jet structure comes from the fact that the prompt emission mechanism is not well understood: for this reason, a typical approach is to assume that the prompt emission simply transforms some fraction of the kinetic (or magnetic) energy into radiation. Given a prompt emission efficiency *ηγ*(*θ*) (which represents the fraction of the available kinetic/magnetic energy at *θ* that is radiated in gamma-rays), the dependence of the prompt emission properties (*E*peak, *E*iso, *L*iso) on the viewing angle *θ*<sup>v</sup> (the *apparent structure* in the language of [125,196]) is set by the energy and Lorentz factor angular profiles, Equations (1) and (2). In particular, the bolometric isotropic-equivalent energy can be computed as [125,197,198]

$$E\_{\rm iso}(\theta\_{\rm V}) = \int\_0^{2\pi} \mathrm{d}\phi \int\_0^{\pi/2} \sin\theta \,\mathrm{d}\theta \,\frac{\delta^3(\theta,\phi,\theta\_{\rm V})}{\Gamma(\theta)} \eta\_{\uparrow}(\theta) \frac{\mathrm{d}E}{\mathrm{d}\Omega}(\theta),\tag{6}$$

where *δ* = Γ −1 (1 − *β* cos *α*) −1 is the Doppler factor, with *α* being the angle between the line of sight and the radial expansion direction, which can be expressed as [125]

$$
\cos \alpha = \cos \theta \cos \theta\_\mathbf{v} + \sin \theta \sin \phi \sin \theta\_\mathbf{v}. \tag{7}
$$

Figure 5 shows *E*iso(*θ*v) corresponding to a few different d*E*/dΩ and Γ profiles, assuming a constant *η<sup>γ</sup>* at all angles. A general feature that is demonstrated in the figure is that at some viewing angles (typically close to the jet axis) the emission is dominated by material moving along the line of sight, resulting in *E*iso(*θ*v) ∼ 4*π*d*E*/dΩ(*θ* = *θ*v). Far off-axis, on the other hand, the flux received by the observer is spread over a larger portion of the jet, corresponding to regions with the most favorable combination of a large intrinsic luminosity and a sufficiently low Lorentz factor as to avoid a too severe de-beaming of radiation away from the line of sight. The steeper the Lorentz factor decay as a function of *θ*, the shallower the *E*iso decay at large *θ*v, as the de-beaming is less severe in broader and more energetic portions of the jet.

**Figure 5.** Apparent and intrinsic structure for a uniform and a Gaussian structured jet. Black dashed lines represent 4*π*d*E*/dΩ at an angle *θ* = *θ*<sup>v</sup> from the jet axis. Colored solid lines show *E*iso(*θ*v) for four different models: a uniform jet model with d*E*/dΩ = 1053/4*π* erg and Γ = 300 within an angle *θ*<sup>j</sup> = 3 ◦ (blue) and three Gaussian models with <sup>d</sup>*E*/d<sup>Ω</sup> = (1053/4*π*) exp[−(*θ*/*θ*c) <sup>2</sup>/2] and Γ*β* = 300 exp[−(*θ*/*θ*Γ) <sup>2</sup>/2], with *θ*<sup>c</sup> = 3 ◦ and three different values of *θ*<sup>Γ</sup> (orange, red and purple, with the corresponding *θ*Γ/*θ*<sup>c</sup> ratios given in the legend). Adapted from [125].

In order to compute *E*peak(*θ*v), one needs to make a further assumption about the spectral shape *S* 0 *ν* (*ν* 0 , *θ*) of the radiation as measured in the jet comoving frame (e.g., [197,199]): the simplest assumption, often adopted in the literature, is that of a fixed spectrum at all angles, which yields *E*peak(*θ*v) ∝ Γ(*θ* = *θ*v) for viewing angles at which the emission is dominated by material on the line of sight, and a shallower decrease at larger viewing angles (see, e.g., Figure 2 in [199]). In long GRBs, this simple assumption (along with a Gaussian ansatz for the jet structure) is sufficient [125] to reproduce the observed *E*iso − *E*peak correlation [191], while in short GRBs it leads to a clear tension with the observed *E*peak of GRB170817A [197]. A more 'physical' approach to the modeling of the expected peak spectral energy would require one to assume a particular prompt emission scenario, such as, e.g., that of internal shocks, and explicitly compute the typical comoving photon energy at each angle for a given jet structure [200].

The computation of *L*iso(*θ*v) also requires additional assumptions [144,145]. The duration of the prompt emission is usually assumed to be linked to that of the central engine activity. This is based on the idea that the prompt emission is composed of short pulses, each corresponding to a dissipation event in the jet. The events happen around a typical radius *R*, where the jet material travels with a typical bulk Lorentz factor Γ, and their intrinsic duration is very short, so that the observed duration is set by the (Dopplercontracted) angular time scale *t*ang ∼ *R*/Γ 2 *c* (i.e., the maximal arrival time difference between photons emitted within an angle 1/Γ with respect to the line of sight), which is

assumed to be much shorter than the central engine activity duration *T*CE. As a result, the observed duration is *T*GRB ∼ *T*CE and the average isotropic-equivalent luminosity is simply *L*iso ∼ *E*iso/*T*CE. If the spread in *R* and Γ is not too large, then this holds true also for off-axis observers, as long as the single-pulse duration remains much shorter than *T*CE [201], and in such situation Equation (6) divided by *T*CE can be used to compute the average *L*iso(*θ*v) [114,197,199,202,203]. Pulse overlap can result in a shallower dependence on the viewing angle [201], but the dependence will eventually steepen at sufficiently large angles such that the duration of single pulses will become comparable or longer to *T*CE.

Useful discussions on the transformation between on- and off-axis isotropic-equivalent energies, durations, luminosities and peak photon energies, along with useful analytical approximations, can be found in [204,205].
