4.1.3. Compton Drag

Inverse-Compton scattering of anisotropic radiation yields high levels of polarization for the scattered radiation field with Π ≤ 100%. This is very different from Comptonization since the polarization vector of the scattered photon can now be aligned with a particular direction, which is transverse to the plane containing the wave vectors,~*k* 00 1 and~*k* 00 2 , of the incoming and scattered photons, respectively, in the rest frame of the electron (hence the double primes). If the scattering angle is *θ* 00 sc = arccos( ~*k* 00 1 · ~*k* 00 2 ), then Thomson scattering of radiation imparts local polarization

$$\text{II} = \frac{1-\cos^2\theta\_{\text{sc}}^{\prime\prime}}{1+\cos^2\theta\_{\text{sc}}^{\prime\prime}} \xrightarrow[\text{electrons}]{\text{cold}} \frac{1-\cos^2\theta\_{\text{sc}}^{\prime}}{1+\cos^2\theta\_{\text{sc}}^{\prime}} \xrightarrow[\text{flow}]{\text{radial}} \frac{1-\cos^2\theta^{\prime}}{1+\cos^2\theta^{\prime}} \tag{12}$$

to the outgoing photon. Indeed, if *θ* 00 sc = *π*/2, then Π¯ = 100%. Here it was assumed that the electrons are cold and therefore their rest frame is the fluid frame (*θ* 00 sc = *θ* 0 sc) that is moving with velocity ~*v*, and if it is moving everywhere in the radial direction (*v*ˆ = *r*ˆ), then *θ* 0 sc = ˜*<sup>θ</sup>* 0 . In general, the local polarization depends on the angle *θ* 0 0 between the wave vector of the incoming photon and the velocity vector of the electron. If the electrons have a finite internal energy density, which means that they have a velocity distribution, then the local polarization is obtained by performing a weighted integral over all—see *θ* 0 0 [224] for details.

The expected polarization when assuming cold electrons in the comoving frame of an ultrarelativistic top-hat jet is shown in Figure 8. The polarization curves are very similar to that obtained for synchrotron emission for the *B*<sup>⊥</sup> field configuration, but for Compton drag the normalization is (nearly exactly) higher by Π−<sup>1</sup> max(*α*) as given by Equation (10). Similar results were first obtained by Lazzati et al. [134] for narrower jets with *ξ<sup>j</sup>* ≤ 25

where they showed that when *ξ* = 0.04 very high polarization with Π . 95% can be obtained with Compton drag.

**Figure 8.** Pulse-integrated polarization of prompt GRB radiation generated by the Compton drag mechanism. The electrons were assumed to be cold in the comoving frame. Figure adapted from [24], but also see [134] for results for a narrower top-hat jet.
