*4.2. Polarization from Structured Jets*

The angular structure of the relativistic jet in GRBs becomes particularly important for relatively nearby events, e.g., GRB170817A (*D* ' 40 Mpc), which can be detected with the current cadre of instruments when the observer is relatively far off-axis and the emission is dim. For distant GRBs, as mentioned earlier, it is challenging to detect emission from significantly off-axis jets. Still, there will be some events in which the LOS is just outside the quasi-uniform core that may not be sharp, as found otherwise in a top-hat jet, but instead be smoother. Then, it becomes important to model the angular structure and compare polarization measurements with accurate theoretical models.

The first level of correction for an idealized top-hat jet model is the consideration of smooth wings of comoving spectral luminosity while the bulk-Γ remains uniform [225]. Like the top-hat jet, *L* 0 *ν* <sup>0</sup> = *L* 0 *ν* 0 ,0 for *ξ* ≤ *ξ<sup>j</sup>* (*θ* ≤ *θ<sup>j</sup>* ), but outside of this uniform core the spectral luminosity can have either exponential or power-law wings:

$$\frac{L\_{\nu'}^{\prime}}{L\_{\nu',0}^{\prime}} = \begin{cases} \exp[(\sqrt{\xi}\_{\not\!j} - \sqrt{\xi})/\Delta], & \xi > \xi\_{\not\!j} \quad \text{(exponential rings)}\\ \left(\frac{\xi}{\xi\_{\not\!j}}\right)^{-\delta/2}, & \xi > \xi\_{\not\!j} \quad \text{(power-law weights)}. \end{cases} \tag{13}$$

Here again it is assumed that Γ, *θ<sup>j</sup>* , *θ*obs, and the spectrum do not have any radial dependence.

In a more realistic structured jet the core is no longer uniform. Instead, the spectral luminosity as well as the bulk-Γ depend on polar angle *θ*. In general, the properties of the flow can also depend on the azimuthal angle *φ*, but here the discussion makes the simplifying and physically reasonable assumption of axisymmetric jets. Two different types of structured jets are considered here:

$$\frac{L\_{\nu'}'(\theta)}{L\_{\nu',0}'} = \frac{\Gamma(\theta) - 1}{\Gamma\_c - 1} = \exp\left(-\frac{\theta^2}{2\theta\_c^2}\right) \tag{Gaussian \text{Jet}} \tag{14}$$

$$\frac{L\_{\nu'}^{\prime}(\theta)}{L\_{\nu',0}^{\prime}} = \Theta^{-a}, \quad \frac{\Gamma(\theta) - 1}{\Gamma\_c - 1} = \Theta^{-b}, \quad \Theta = \sqrt{1 + \left(\frac{\theta}{\theta\_c}\right)^2} \qquad (\text{Power-Law Jet}) \tag{15}$$

Here, *L* 0 *ν* 0 ,0 and Γ*<sup>c</sup>* are the core spectral luminosity and bulk-Γ at *θ* = 0.
