*4.1. Jet Structure and the Early Afterglow*

Initially, as the expanding jet material is highly relativistic, there is no causal contact between regions at angular distances & 1/Γ and hence the shock dynamics only depends on local quantities. Calling *E*(*θ*) = 4*π*d*E*/dΩ(*θ*), as soon as the isotropic-equivalent ISM

mass is swept by the jet, *M*(*θ*) = *nm*p4*πR*(*θ*) <sup>3</sup>/3 (where *m*<sup>p</sup> is the proton mass and *R* is the outer radius of the jet or, more precisely, that of the forward shock) becomes comparable to *E*(*θ*)/Γ 2 (*θ*)*c* 2 , a reverse shock starts to propagate backwards (as seen in the rest-frame of the contact discontinuity that separates the shocked ISM and jet materials) through the jet, initiating the deceleration of the latter. Assuming a jet radial width ∆<sup>0</sup> ∼ *cT*CE after breakout, the initial deceleration phase proceeds differently depending on the local Sedov length *l*S(*θ*) = (3*E*(*θ*)/4*πnm*p*c* 2 ) 1/3 and bulk Lorentz factor Γ(*θ*). Assuming free expansion, the "sound-crossing" radius at which radial sound waves (assuming a relativistic sound speed *c*<sup>s</sup> = *c*/ √ 3) cross the shell is *R*s(*θ*) ∼ √ 3Γ(*θ*) <sup>2</sup>∆0. The "deceleration" radius at which the interaction with the ISM becomes relevant is *R*d(*θ*) ∼ *l*S(*θ*)Γ <sup>−</sup>2/3(*θ*). If *R*s(*θ*) > *R*d(*θ*), the portion of the jet is said to be in the "thick shell" regime, where the deceleration starts before radial pressure waves can smooth out radial inhomogeneities and cause any significant radial spreading [80,90]: in this case, the reverse shock is relativistic (i.e., the velocity of the unshocked jet is relativistic as seen from the contact discontinuity that separates the shocked ISM and shocked jet material) and crosses the whole jet shell at a radius *R*cross(*θ*) ∼ *l*S(*θ*) 3/4∆ 1/4 0 . Conversely, if *R*s(*θ*) < *R*d(*θ*) the jet portion is in the "thin shell" regime, where it reaches the deceleration radius after undergoing a significant radial spread, which washes out radial inhomogeneities and leads to an effective jet radial with ∆(*θ*) ∼ *R*/Γ(*θ*) 2 . In this case, the reverse shock remains Newtonian and crosses the shell at *R*cross(*θ*) ∼ *R*d(*θ*). Interestingly, as shown in Figure 8, assuming *E*(0) = 10<sup>54</sup> erg and Γ(0) = 1000 (the dependence on these values is weak) and adopting a Gaussian profile ∝ exp(−(*θ*/*θ*c) <sup>2</sup>/2) for both quantities, for most short GRBs (with *T*CE . 2 s and *n* . 1 cm−<sup>3</sup> ) the deceleration is entirely in the thin shell regime (see also [229]), while for long GRB jets (typically with *T*CE ∼ 30 s and *<sup>n</sup>* <sup>∼</sup> <sup>1</sup> cm−<sup>3</sup> ) it proceeds in the thick shell regime within an inner region *θ* < *θ*thick which corresponds typically to the jet core, *θ*thick/*θ*<sup>c</sup> ∼ 1. More generally, within the above Gaussian structured jet assumption, the existence of a transition angle *θ*thick corresponds to the condition Γ(0) > (*l*S(0)/ √ 3*cT*CE) 3/8 <sup>∼</sup> <sup>430</sup> *<sup>E</sup>* 1/8 <sup>54</sup> *n* −1/8 0 *T* −3/8 CE,1 , in which case *θ*thick = (2*θ*c/ √ 7)[8 ln Γ(0) − 3 ln(*l*S(0)/ √ 3*cT*CE)]1/2 .

**Figure 8.** Angle *θ*thick within which the deceleration takes place in the thick shell regime, in units of the core angle *θ*c, in a structured jet with Gaussian energy d*E*/dΩ(*θ*) and bulk Lorentz factor Γ(*θ*) profiles ∝ exp(−(*θ*/*θ*c) <sup>2</sup>/2), assuming 4*π*d*E*/dΩ(0) = 10<sup>54</sup> erg and Γ(0) = 1000, as a function of the ISM number density *n* and central engine duration *T*CE. Boxes show the regions of the plane where most long (red box) and short (grey box) GRBs are expected to lie.

During this phase, diffusive shock acceleration of electrons [230–233] can take place at both the forward and reverse shocks, leading to synchrotron [234] and possibly inverse Compton radiation. For viewing angles close to the jet axis, where the emission is dominated by material moving close to the line of sight, the reverse shock emission is expected to peak at the observer time *t*pk,RS ∼ (1 + *z*)*R*cross(*θ*v)/Γ(*θ*v) 2 *c* that corresponds to the shock crossing time. In the thin shell regime, this matches the peak time of the forward shock emission. The light curve of the reverse shock emission in the Optical (where the peak of the synchrotron spectrum at *t*pk,RS is expected to lie for "standard" parameters, [234]) and X-rays are expected to display a rapid rise and decay before and after the peak, therefore appearing as a flare. In the radio, the expected decay is slower (as the synchrotron peak moves rapidly to lower frequencies after the peak), with possible late-time bumps [235], even though this critically depends on how rapidly the shock-generated magnetic field decays after the reverse shock has disappeared [236]. The emission as seen by a far off-axis observer may be instead dominated by material moving at a different angle, or more generally consist of a comparable amount of radiation from a broader portion of the jet, resulting in a delayed and smoother light curve (see [229] for examples in short GRBs).

In parts of the jet where the deceleration proceeds in the thick shell regime, the radial structure plays a role in the reverse-forward shock dynamics, and thus in shaping the early afterglow emission. This is particularly relevant in far off-axis parts of the jet that contain blown-out cocoon material (which is expected to feature a broad velocity profile, e.g., [79,116,138]) and/or if the central engine does not turn off abruptly at *T*CE, but rather decays slowly, resulting in a jet with a low-velocity tail that contains a non-negligible amount of energy. In that case, the reverse shock can be long-lived, with slower material gradually catching up, with modified dynamics of both the reverse and forward shocks: this is often called a *refreshed* shock [237] scenario. This remains currently one of the leading explanations for the X-ray *plateaux* (see Section 3.7). The latter phenomenon is therefore mostly explained as resulting from either a radial or an angular jet structure [238]: the degeneracy between these two options when trying to explain "non-standard" decays in afterglow light curves has been addressed in [239].
