*4.2. Jet Structure and the Late Afterglow*

As soon as the reverse shock has disappeared, the shocked jet shell transfers most of its energy to the shocked ISM [83,91] and the system turns into a *blastwave*, that is, a forward shock whose memory of the details of the original explosion is lost. In this phase, its dynamics depend solely on the (angle-dependent) Sedov length *l*S(*θ*) ∝ (*E*(*θ*)/*n*) 1/3 and can be described by the evolution of the Lorentz factor of the shocked ISM immediately behind the forward shock, as a function of the forward shock radius (and the angle, given the anisotropy) Γ(*R*, *θ*). As long as the blastwave is relativistic, Γ(*R*, *θ*) 1, its radial structure and evolution are well-described by the self-similar Blandford–McKee solution [92], stably against perturbations [83], and hence Γ(*R*, *θ*) = (*R*/*l*S(*θ*))−3/2 [24,92,240]. Lateral energy transfer (which typically proceeds from the jet axis outwards) starts being relevant as soon as transverse causal contact is established, which happens at each angle *θ* when [Γ(*R*, *θ*)]−<sup>1</sup> & *θ*. From that radius on, lateral pressure waves will start transferring shock energy laterally, gradually smearing out the angular energy profile *E*(*θ*) [31,32,43,241]. As soon as the entire shocked region reaches transverse causal contact, it expands laterally [11,93,95,242], gradually reaching isotropy. At late times, when the whole shock becomes non-relativistic and the expansion quasi-spherical, the shock structure and evolution is well-described [93,94,243] by the Sedov-Taylor self-similar solution [244–246]. This qualitative description should make clear that most of the memory of the angular structure is wiped out during the lateral spreading phase.

The shock dynamics and emission in the structured jet scenario in this phase have been addressed by several studies, e.g., [24,31,32,37,43,57,64,199,241,247–251]. At any time in the evolution, the emission is typically modeled within a parametrized relativistic leptonic diffusive shock acceleration and synchrotron/inverse-Compton emission model [227,228].

Shock-accelerated electrons are assumed to be injected in the shock downstream with a power-law distribution of Lorentz factors, d*n*e/d*γ* ∝ *γ* −*p* (where *n*<sup>e</sup> is the comoving electron number density in the immediate downstream, and we focus on the case where *p* > 2), above a minimum Lorentz factor *γ*m. Their number is assumed to be a fraction. Most often, the fraction is set to *χ*<sup>e</sup> = 1, despite the theoretical expectation being *<sup>χ</sup>*<sup>e</sup> <sup>∼</sup> few <sup>×</sup> <sup>10</sup>−<sup>2</sup> [233,252]. Yet, in some GRB afterglows with broad-band, highcadence datasets, *χ*<sup>e</sup> < 1 has been shown to provide a substantially better fit to the data, e.g., [236,253]. *χ*<sup>e</sup> of the total shocked ISM electrons, and their energy density is assumed to be a fraction *e*<sup>e</sup> of the internal energy in the shock downstream (which in a strong shock depends only on the shock velocity/Lorentz factor and on the adiabatic index, being set by shock-jump conditions [254]): the minimum Lorentz factor *γ*<sup>m</sup> is entirely determined once the shock Lorentz factor Γ and the *χ*e, *e*<sup>e</sup> and *p* parameters are given. A random magnetic field generated by small-scale turbulence behind the shock [255] is assumed to hold a fraction *e*<sup>B</sup> of the energy density. The electron population in the shock downstream evolves as fresh electrons are injected, and as older electrons cool down due to synchrotron, inverse-Compton and adiabatic losses: as a result, the electron distribution in phase space takes approximately the form [228]

$$\frac{\mathbf{d}n\_{\mathbf{e}}}{\mathbf{d}\gamma} \propto \begin{cases} \gamma^{-q} & \gamma\_{\mathbf{p}} \le \gamma < \gamma\_{0} \\ \gamma^{-p-1} & \gamma \ge \gamma\_{0} \end{cases},\tag{10}$$

where *γ*<sup>p</sup> = min(*γ*m, *γ*c), *γ*<sup>0</sup> = max(*γ*m, *γ*c),

$$\mathfrak{q} = \left\{ \begin{array}{cc} p & \gamma\_{\mathfrak{m}} \le \gamma\_{\mathfrak{c}} \\ 2 & \gamma\_{\mathfrak{m}} > \gamma\_{\mathfrak{c}} \end{array} \right. \tag{11}$$

and *γ*<sup>c</sup> is the Lorentz factor above which electrons lose most of their energy through synchrotron, inverse-Compton and adiabatic losses in a dynamical time *t* 0 dyn ∼ *R*/Γ*c*. This is typically given by [227,228]

$$\gamma\_{\rm c} = \frac{6\pi m\_{\rm e}c^2 \Gamma}{\sigma\_{\rm T}B^2 R(1+Y)},\tag{12}$$

where *Y* = *u*rad/*u*<sup>B</sup> is the ratio of radiation energy density. This includes the synchrotron radiation produced by the electrons–which emit and cool both by synchrotron and by inverse-Compton scattering of their own synchrotron photons, i.e., synchrotronself-Compton [256,257]–and possibly an external radiation field, which can be upscattered by relativistic electrons in the shock downstream giving rise to an additional emission component [258–260]. to magnetic energy density as measured in the shock downstream comoving frame. More complicated electron energy distributions arise when Klein-Nishina effects are important and most electrons cool rapidly, e.g., [178,180].

The above simple model of electron acceleration and cooling thus depends on the "microphysical" parameters *p*, *χ*e, *e*<sup>e</sup> and *e*B, and on the shock Lorentz factor Γ. Once the shock dynamics Γ(*R*, *θ*) is determined, the luminosity emitted towards an observer at a viewing angle *θ*<sup>v</sup> from electrons in the entire shock can thus be computed for a fixed set of microphysical parameters by integrating the radiative transfer equation after computing the synchrotron (and possibly inverse-Compton) emissivity (and absorption coefficient) over the shock downstream, the shock radial structure is given by the appropriate selfsimilar solution. In order for the resulting luminosity to be appropriately compared to the observed flux, the integration must be performed over the appropriate equal-arrival-time volume or, in other words, at the appropriate 'retarded' time to account for the different photon paths that lead to the same arrival time to the observer [7,67,261–263]. If the shock is approximated as infinitely thin, this reduces to equal-arrival-time surfaces [43,199,228,264].

Even when limiting the discussion to an isotropic explosion and to synchrotron emission only, the above model leads to rather complex light curves [263]. At any fixed time, the SED is composed of various smoothly-connected power law segments, corresponding

to different spectral regimes: the main critical frequencies are the synchrotron frequencies *ν*syn(*γ*) = *δγ*<sup>2</sup> *eB*/2*πm*e*c* (where *δ* is the Doppler factor related to bulk motion) corresponding to the electron distribution break Lorentz factors *γ*<sup>m</sup> and *γ*<sup>c</sup> (typically referred to as *ν*<sup>m</sup> = *ν*syn(*γ*m) and *ν*<sup>c</sup> = *ν*syn(*γ*c)), and frequency *ν*<sup>a</sup> below which synchrotron self-absorption becomes important (i.e., the synchrotron self-absorption optical depth *τ*ssa(*ν*a) = 1). The comoving specific synchrotron emissivity at a comoving frequency *ν* 0 can be approximated by a series of power-law segments, namely

$$j'\_{\boldsymbol{\nu}'} = j'\_{\boldsymbol{\nu}', \max} \begin{cases} (\boldsymbol{\nu}'/\boldsymbol{\nu}'\_{\mathbf{p}})^{1/3} & \boldsymbol{\nu}' \le \boldsymbol{\nu}'\_{\mathbf{p}} \\ (\boldsymbol{\nu}'/\boldsymbol{\nu}'\_{\mathbf{p}})^{-(q-1)/2} & \boldsymbol{\nu}'\_{\mathbf{p}} < \boldsymbol{\nu}' \le \boldsymbol{\nu}'\_{0} \\ (\boldsymbol{\nu}'\_{0}/\boldsymbol{\nu}'\_{\mathbf{p}})^{-(q-1)/2}(\boldsymbol{\nu}'/\boldsymbol{\nu}'\_{0})^{-p/2} & \boldsymbol{\nu}' > \boldsymbol{\nu}'\_{0} \end{cases} \tag{13}$$

where *ν*<sup>0</sup> = *ν*syn(*γ*0), *ν*<sup>p</sup> = *ν*syn(*γ*p), and *q*, *γ*<sup>0</sup> and *γ*<sup>p</sup> have the same meaning as before. The maximum specific emissivity *j* 0 *ν* 0 ,max depends on the microphysical parameters, ISM density and shock Lorentz factor [199,227]. When considering optically thin portions of the shock, the intensity received by the observer is just *I<sup>ν</sup>* = *δ* 3 *j* 0 *ν* <sup>0</sup>(*ν*/*δ*)∆*R* 0 , where ∆*R* 0 is the shock effective thickness, typically of order ∆*R* <sup>0</sup> ∼ *R*/Γ 2 . Hence, if the emission is dominated by a small, optically thin portion of the shock, the observed flux density has the same shape as the emissivity. Synchrotron self-absorption [265] suppresses the emission when *τ*ssa > 1, introducing low-frequency power law segments with *I<sup>ν</sup>* ∝ *ν <sup>α</sup>* with *α* = 2 if *ν* < *ν*<sup>p</sup> and *α* = 5/2 otherwise. If *γ*<sup>c</sup> < *γ*m, an additional power law segment with *α* = 11/8 emerges below *ν*<sup>a</sup> due to the inhomogeneous cooling stage of electrons behind the shock [266]. Useful figures summarizing all the possible synchrotron spectral regimes in GRB afterglows can be found in [263]. Reference [178] treats additional cases where the electron distribution (and hence the synchrotron spectrum) is modified by a non-monotonic dependence of the cooling rate on the electron Lorentz factor *γ* due to Klein–Nishina effects.

For a given jet structure, observed off the jet core (*θobs* > *θc*), the emission is dominated, at different times, by different portions of the emitting surface [249,251]. Figure 9 shows that as time increases, the flux seen by the observer is dominated by emissions produced progressively close to the jet axis as a consequence of the competition between the decrease in the shock velocity (implying the increase in the beaming angle and the decrease in the emissivity) and the increasing shock initial energy towards the jet axis with respect to the observer line of sight (center of the coordinate system in Figure 9). This effect has important consequences on the observed afterglow light curves for different jet structures and viewing angles. Figure 10 shows some examples of mono-chromatic afterglow light curves at three characteristic frequencies corresponding to the radio, optical and X-ray bands (from top to bottom). Three different jet structures are considered here: uniform (solid lines), Gaussian (dashed line) power-law with both the energy and the Lorentz factor decreasing as *θ* −3 (dotted line). In all three cases, a jet core opening angle of 3 degrees is considered. As long as the observer's line of sight is within the beaming cone of the jet core, *θ*<sup>v</sup> − *θ*<sup>c</sup> < 1/Γ(*R*, 0), the light curves corresponding to different structures are almost indistinguishable. This applies to the entire light curve as long as *θ*<sup>v</sup> < *θ*c. If the viewing angle is larger, differences among the three structures are apparent mainly in the rising phases of the light curves, with the two non-uniform structures presenting similar shallow rising phases preceding the peak, after which all structures join into the same, core-dominated decay. The enormous difference in the light curves at intermediate viewing angles *θ*<sup>c</sup> < *θ*<sup>v</sup> . few × *θ*<sup>c</sup> stems from the fact that, in the Gaussian and power-law structure cases, the emission is initially dominated by material moving along the line of sight, which is absent in the uniform jet case.

The fact that the emission is dominated by material progressively closer to the jet axis impacts also the apparent displacement of the source centroid as seen in Very Long Baseline Interferometry (VLBI) imaging [67,248,267–270]. The surface brightness of the shock, as seen by a distant observer, corresponds to the intensity *I<sup>ν</sup>* described above. Its distribution *Iν*(*θx*, *θy*) on the plane of the sky (where *θ<sup>x</sup>* and *θ<sup>y</sup>* represent two suitably

chosen angular coordinates on the relevant sky patch) is most commonly referred to as the 'image' of the source. See the next section for some example surface brightness distributions, which can be measured through VLBI imaging within the limited resolution that can be reached with current facilities. The image centroid (i.e., the mean of the distribution) lies on the projection of the jet axis on the (*θx*, *θy*) plane because the jet axisymmetry induces a reflection symmetry in the image. Such symmetry can also be exploited to speed up the computation of *I<sup>ν</sup>* for the calculation of light curves [199]. The displacement of the centroid before the light curve peak (after which the emission becomes core-dominated) is directly related to the shape of the jet structure.

The fact that the evolution of the pre-peak light curves and in principle also that of the image centroid position contain some information on the jet structure suggests that it is possible to reconstruct (at least partially) the latter information from the observations. This is typically achieved by fitting an analytical structured jet afterglow model to the observed light curves (and centroid displacement), in order to recover the parameters that describe the structure, e.g., [63,67,248,269,271,272]. As demonstrated by [273], the lightcurve-based reconstruction can be performed more explicitly by integrating a differential equation derived from standard afterglow theory. Unfortunately [273,274], the accuracy and cadence required for a detailed reconstruction are highly demanding, and global degeneracies remain unbroken unless the evolution of the emission in multiple spectral regimes is observed see also [251].

**Figure 9.** Angular map showing the intensity distribution per unit solid angle (color coded) of emission from the afterglow forward shock, centered on the line of sight. The position of the jet axis is marked by the white cross symbol. The green cross and red (blue) contours show the peak of the intensity and the region containing 50 (80%) of the total flux. A power-law structured jet with core values Γ<sup>c</sup> = 1000 and *θ*<sup>c</sup> = 0.03 rad is considered. The power-law slopes are *a* = 4 and *b* = 2 for the energy and Γ structure, the observer viewing angle is 10 times the core opening angle and the external medium density is constant. The maps correspond to different observing times (as measured in the rest frame of the central engine), from 500 s (**top left**) to 580 days (**bottom right**). Reproduced from [251].

**Figure 10.** Example synthetic afterglow light curves for jets at *z* = 1 with differing structures, seen under various viewing angles (color-coded as reported in the legend), computed within the "standard afterglow" model described in Section 4.2 assuming *e*<sup>e</sup> = 0.1, *e*<sup>B</sup> = 10−<sup>3</sup> , *p* = 2.2, and a homogeneous external medium with number density *n* = 0.1 cm−<sup>3</sup> . Each structure can be defined as d*E*/dΩ = (*E*c/4*π*)*f*(*θ*/*θ*c) and Γ = 1 + (Γ<sup>c</sup> − 1)*f*(*θ*/*θ*c): (i) the Uniform structure (solid lines) has *f*(*x*) = Θ(1 − *x*), where Θ(*x*) is the Heaviside step function; (ii) Gaussian jet (dashed lines), with *f*(*x*) = exp(−*x* <sup>2</sup>/2); (iii) Power-law jet, with *f*(*x*) = (1 + *x* 3 ) −1 . For all jets, *E*<sup>c</sup> = 10<sup>53</sup> erg, Γ<sup>c</sup> = 300 and *θ*<sup>c</sup> = 3 ◦ . Each panel shows light curves computed at a different observing frequency: 1.4 GHz (top panel), r-band (i.e., 4.6 <sup>×</sup> <sup>10</sup><sup>14</sup> Hz, central panel) and 1 keV (i.e., 2.4 <sup>×</sup> <sup>10</sup><sup>17</sup> Hz, bottom panel). In the bottom panel, we show for comparison the slopes expected for the X-ray (mostly also valid for the Optical) post-peak (for viewing angles close to the jet axis) and post-jet-break light curves. The late-time peak produced by the emission of the counter-jet is also annotated.
