Polarized Line Formation in Multi-dimensional Media. IV. A Fourier Decomposition Technique to Formulate the Transfer Equation with Angle-dependent Partial Frequency Redistribution

Details Polarized Line Formation in Multi-dimensional Media. IV. A Fourier Decomposition Technique to Formulate the Transfer Equation with Angle-dependent Partial Frequency Redistribution Polarized Line Formation in Multi-dimensional Media. IV. A Fourier Decomposition Technique to Formulate the Transfer Equation with Angle-dependent Partial Frequency Redistribution

Sep 2011

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2011apj...739...40a&link_type=abstract

The Astrophysical Journal, Volume 739, Issue 1, article id. 40 (2011).

Astronomy and Astrophysics

Astronomy

2

Line: Formation, Magnetic Fields, Polarization, Radiative Transfer, Scattering, Sun: Atmosphere

Scientific paper

To explain the linear polarization observed in spatially resolved structures in the solar atmosphere, the solution of polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries is essential. For strong resonance lines, partial frequency redistribution (PRD) effects also become important. In a series of papers, we have been investigating the nature of Stokes profiles formed in multi-D media including PRD in line scattering. For numerical simplicity, so far we have restricted our attention to the particular case of PRD functions which are averaged over all the incident and scattered directions. In this paper, we formulate the polarized RT equation in multi-D media that takes into account the Hanle effect with angle-dependent PRD functions. We generalize here to the multi-D case the method for Fourier series expansion of angle-dependent PRD functions originally developed for RT in one-dimensional geometry. We show that the Stokes source vector S = (SI , SQ , SU ) T and the Stokes vector I = (I, Q, U) T can be expanded in terms of infinite sets of components \ssty{\tilde{\bm { {S}}}}^{(k)}, \ssty{\tilde{\bm { {I}}}}^{(k)}, respectively, k in [0, +∞). We show that the components \ssty{\tilde{\bm { {S}}}}^{(k)} become independent of the azimuthal angle (phiv) of the scattered ray, whereas the components \ssty{\tilde{\bm { {I}}}}^{(k)} remain dependent on phiv due to the nature of RT in multi-D geometry. We also establish that \ssty{\tilde{\bm { {S}}}}^{(k)} and \ssty{\tilde{\bm { {I}}}}^{(k)} satisfy a simple transfer equation, which can be solved by any iterative method such as an approximate Lambda iteration or a Bi-Conjugate Gradient-type projection method provided we truncate the Fourier series to have a finite number of terms.

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