4.2.2. Photospheric Emission from Structured Jets

Photospheric emission yields negligible polarization in a uniform jet unless the viewing angle is less than one beaming cone away from the edge of the jet, i.e., |*q* − 1| . *ξ* −1/2 *<sup>j</sup>* ⇔ Γ|*θ*obs − *θ<sup>j</sup>* | . 1. One way to obtain finite net polarization is by having a structured jet (see Figure 10). This was initially demonstrated in Monte Carlo (MC) simulations of photospheric emission emerging from axisymmetric relativistic outflows [56,57] that featured sheared layers outside of the uniform core with gradients in bulk-Γ as a function of the polar angle *θ*. It was shown that narrow jets with Γ*θ<sup>c</sup>* ≈ 1 and steep gradients in bulk-Γ with Γ(*θ*) ∝ *θ* −*p* for *θ* > *θ<sup>c</sup>* (some works use the symbol *θ<sup>j</sup>* instead of *θ<sup>c</sup>* to refer to the half-opening angle of the uniform core) and *p* ∼ 4 can yield polarization Π . 40% for *q* = *θ*obs/*θ<sup>c</sup>* & 1. A more realistic scenario would have Γ*θ<sup>c</sup>* ≈ 10 in which case Π . 10% is expected. A similar conclusion is reached by carrying out a radial integration of the radiation transfer equations for the Stokes parameters in a steady flow having angular structure in the comoving emissivity and bulk-Γ [24]. The results of this work are shown in the bottom-left panel of Figure 10, and even here it was realized that steep gradients in the bulk-Γ profile are required to achieve significant polarization with Π . 15%.

**Figure 9.** (**Left**) Pulse-integrated polarization for smooth jets with uniform core and exponential or power-law wings in spectral luminosity while the bulk-Γ remains uniform. The edges of the uniform jet become smoother with increasing (decreasing) ∆ (*δ*) for exponential (power law) wings. (**Right**) Polarization curves for structured jets. Two cases for the Gaussian jet (GJ) are shown, where in one both *L* 0 *ν* <sup>0</sup> and Γ vary with *θ* and in the other Γ is kept uniform. For the power-law jet (PLJ), the power-law index for *L* 0 *ν* <sup>0</sup> is fixed (*a* = 2), but that for the bulk-Γ (*b*) is varied. The curve for *b* = 0 is mostly overlapped by that of *b* = 1. The dotted lines show the polarization curves for viewing angles at which the fluence has declined to values smaller than 1% of that expected at *θ*obs = 0. The thick dots mark critical viewing angles beyond which the emission region becomes too compact to *γγ*-annihilation, causing the emission to be optically thick to Thomson scattering of the produced *e* ±-pairs. Figure adapted from [24] and some results for the smoothed top-hat jets were first presented in [225].

**Figure 10.** Polarization from non-dissipative photospheric emission model in a structured jet. (**Top-left**) Polarization from the Monte Carlo (MC) simulation of Ito et al. [56] shown for different viewing angles *θ*obs and different gradients in bulk-Γ (here *η*). (**Top-right**) MC simulation results from Lundman et al. [57] featuring a uniform core with half-opening angle *θ<sup>j</sup>* and power-law shear (Γ(*θ*) ∝ *θ* −4 ) layer in bulk-Γ. The off-axis spectral luminosity normalized by the on-axis value (viewing angle *θ<sup>v</sup>* = 0) is shown with dashed red line. (**Bottom-left**) Polarization of photospheric emission from a structured jet obtained from semi-analytic radiation transfer calculation of Gill et al. [177] that features angular structure in both the comoving emissivity (*L* 0 *ν* <sup>0</sup>(*θ*) ∝ Θ−*<sup>a</sup>* , see Equation (10)) and bulk-Γ (Γ(*θ*) ∝ Θ−*<sup>b</sup>* ) with √ *ξ<sup>c</sup>* = Γ*cθ<sup>c</sup>* = 3 where *θ<sup>c</sup>* is the core angle. The solid lines fix *a* = 2 and dotted lines set *b* = 2 to disentangle the effect of the two profiles. (**Bottom-right**) Polarization derived from a MC simulation with outflow properties obtained from a 2D special relativistic hydrodynamic simulation of a jet launched inside a Wolf-Rayet star (from Parsotan et al. [58]). The top-panel shows the lightcurve and the bottom panel shows the temporal evolution of Π and position angle *χ*.

A more realistic scenario was explored in Parsotan et al. [58] who carried out twodimensional (2D) special relativistic hydrodynamic simulations of a jet launched inside a Wolf Rayet star. The flow dynamics and angular structure thus obtained from the simulation were then used with a MC code to obtain the polarization of photospheric emission at the last scattering surface. The results are shown in the bottom-right panel of Figure 10 that shows the lightcurve and temporal evolution of the polarization and PA, with the conclusion that Π . 2.5% and PA remained steady within the uncertainties. In other cases, where the outflow showed more structure, a slightly larger time-resolved polarization of Π . 5% and time-variable PA was obtained.
