Hanle effect for stellar dipoles and quadrupoles

We derive exact expressions for the degree of lineal polarization over a resolved or integrated stellar disc due to resonance scattering and the Hanle effect from a dipolar or quadrupolar distribution of magnetic fields. We apply the theory of scattering polarization within the formalism of the spherical tensors representation for the density matrix and radiation field. The distribution of linear polarization over the stellar disk for different configurations of the magnetic field is studied and its topology discussed. For an unresolved dipole, the resulting polarization can be expressed in terms of just three functions (of the inclination angle and effective dipole strength), that are calculated numerically and their behaviour discussed. Dipolar and (aligned) quadrupoles are considered in some detail, but the techniques here ---in particular, the extensive use of the spherical tensor formalism for polarization---, can easily be applied to more general field configurations.

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