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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 *n*_{e} 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*