4.2.1. Synchrotron Emission from Structured Jets

The polarization curves for a smooth top-hat jet are presented in Figure 9 for different B-field configurations as well as for different levels of smoothness of the edges. The behavior is similar for *θ*obs < *θ<sup>j</sup>* , but significant differences between the top-hat jet case appear for *θ*obs > *θ<sup>j</sup>* . Now that the spectral luminosity does not fall off so sharply for off-axis observers in the latter case, there is always some emission beamed along the LOS. For B-field configurations that show a larger degree of symmetry of the direction of polarization vectors around the LOS (e.g., *<sup>B</sup>*<sup>⊥</sup> and *<sup>B</sup>*<sup>k</sup> ), the net polarization starts to decline as the edges of the jet are made smoother. This occurs due to the increase in symmetry that was broken sharply in the top-hat jet. A completely opposite behavior is seen in ordered B-field configurations, where the polarization increases with increasing smoothness. This arises since for a very sharp edge the observed flux is dominated by the core and once most of it has a similar weight (i.e., beaming and Doppler factor) then a significant amount of canceling occurs, while for a very smooth or gradual edge the flux is dominated by the region near the line of sight where the B-field is ordered, resulting in very little averaging out of the polarization.

The right column of Figure 9 shows the polarization curves for structured jets. When compared with polarization curves from top-hat jets or even smooth top-hat jets, these are broadly similar. Note that the *δ* = 2 smooth top-hat jet (left panel of Figure 9) is broadly similar in structure to (*a* = 2, *b* = 0) structured jet (right panel), where both show similar polarization behavior, and therefore a *δ* = 2 smooth top-hat jet can also be considered a structured jet. In all cases, the curves are now stretched towards larger viewing angles. This means that appreciable polarization can now be measured when the LOS falls outside of the brighter core. In addition to that, the drop in fluence for viewing angles outside of the core is not so severe, as was found for the top-hat jet. Therefore, depending on the exact angular profile, off-axis observers with *q* = *θ*obs/*θ<sup>c</sup>* . few to several can still detect the GRB and measure high levels of polarization. This is demonstrated in Figure 9 using a dotted line where the solid to dotted line transition occurs when the off-axis (*θ*obs > 0) to on-axis (*θ*obs = 0) fluence ratio has dropped to 1%. Nevertheless, there are additional constraints on the detectability of such off-axis bursts. For example, when the bulk-Γ is non-uniform and declines with *θ*, the viewing angle out to which the prompt emission can be observed may be limited by compactness e.g., [24,145,226]. This is shown using a thick dot in the figure beyond which the Thomson optical depth of the *e* <sup>±</sup>-pairs (*τT*) produced due to *γγ*-annihilation becomes greater than 10. As a result, the polarization is rather limited to <sup>Π</sup> . 20% for *<sup>B</sup>*<sup>⊥</sup> and *<sup>B</sup>*<sup>k</sup> , but it can be much higher for the ordered field in *B*tor.
