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### Scattering Matrix

The calculation of Thomson scattering including polarization was first performed by Chandrasekhar [16]; here we show a much simpler derivation employing the spin harmonics. The Thomson differential scattering cross section depends on angle as where and are the incoming and outgoing polarization vectors respectively in the electron rest frame. Radiation polarized perpendicular to the scattering plane scatters isotropically, while that in the scattering plane picks up a factor of where is the scattering angle. If the radiation has different intensities or temperatures at right angles, the radiation scattered into a given angle will be linearly polarized.

Now let us evaluate the scattering term explicitly. The angular dependence of the scattering gives
 (38)
where the U transformation follows from its definition in terms of the difference in intensities polarized from the scattering plane. With the relations and , the angular dependence in the representation of Eqn. (44) becomes,
 (39)
where the overall normalization is fixed by photon conservation in the scattering. To relate these scattering frame quantities to those in the frame defined by , we must first perform a rotation from the frame to the scattering frame. The geometry is displayed in Fig. 1, where the angle separates the scattering plane from the meridian plane at spanned by and . After scattering, we rotate by the angle between the scattering plane and the meridian plane at to return to the frame. Eqn. (43) tells us these rotations transform as . The net result is thus expressed as
 (40)
where we have employed the explict spin-2, forms in Tab. 1. Integrating over incoming angles, we obtain the collision term in the electron rest frame
 (41)
where the two terms on the rhs account for scattering out of and into a given angle respectively. Here the differential optical depth sets the collision rate in conformal time with ne as the free electron density and as the Thomson cross section.

The transformation from the electron rest frame into the background frame yields a Doppler shift in the temperature of the scattered radiation. With the help of the generalized addition relation for the harmonics Eqn. (7), the full collision term can be written as
 (42)
The vector describes the isotropization of distribution in the electron rest frame and is given by
 (43)
The matrix encapsulates the anisotropic nature of Thomson scattering and shows that as expected polarization is generated through quadrupole anisotropies in the temperature and vice versa
 (44)
where and and the unprimed harmonics are with respect to .These components correspond to the scalar, vector and tensor scattering terms as discussed in §IIC and IIIC.

Next: Gravitational Redshift Up: Radiation Transport Previous: Stokes Parameters
Wayne Hu
9/9/1997