The Hanle effect

Free access article

Issue		A&A Volume 476, Number 2, December III 2007


Page(s)		665 - 674
Section		Astrophysical processes
DOI		http://dx.doi.org/10.1051/0004-6361:20077980

A&A 476, 665-674 (2007)
DOI: 10.1051/0004-6361:20077980

Decomposition of the Stokes parameters into irreducible components

H. Frisch

Laboratoire Cassiopée (CNRS), Université de Nice, Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France
e-mail: frisch@obs-nice.fr

(Received 31 May 2007 / Accepted 14 September 2007)

Abstract
Context.It has been shown for the weak-field Hanle effect that the Stokes parameters I, Q, and U can be represented by a set of six cylindrically symmetrical functions. The proof relies on azimuthal Fourier expansions of the radiation field and of the Hanle phase matrix. It holds for a plane-parallel atmosphere and scattering processes that can be described by a redistribution matrix where redistribution in frequency is decoupled from angle redistribution and polarization.
Aims.We give a simpler and more general proof of the Stokes parameter decomposition using powerful new tools introduced for polarimetry, in particular the Landi Degl'Innocenti spherical tensors ${\mathcal T}^K_Q(i,\mbox{\boldmath$\displaystyle\Omega$ })$ .
Methods.The elements of the Hanle phase matrix are written as a sum of terms that depend separately on the magnetic field vector and the directions $\mbox{\boldmath$\displaystyle\Omega$ }$ and $\mbox{\boldmath$\displaystyle\Omega$ }'$ of the incoming and scattered beams. The dependence on $\mbox{\boldmath$\displaystyle\Omega$ }$ and $\mbox{\boldmath$\displaystyle\Omega$ }'$ is expressed in terms of the spherical tensors ${\mathcal T}^K_Q(i,\mbox{\boldmath$\displaystyle\Omega$ })$ where i refers to the Stokes parameters ( $i=0,\ldots,3$ ). A multipolar expansion in terms of the ${\mathcal T}^K_Q(i,\mbox{\boldmath$\displaystyle\Omega$ })$ is then established for the source term in the transfer equation for the Stokes parameters.
Results.We show that the Stokes parameters have a multipolar expansion that can be written as $I_i(\nu,\mbox{\boldmath$\displaystyle\Omega$ })= \sum_{KQ}{\mathcal T}^K_Q(i,\mbox{\boldmath$\displaystyle\Omega$ })I_Q^K(\nu,\theta)$ (K=0,1,2, $-K\le Q\le +K$ ) where the I_Q^K are nine cylindrically symmetrical, irreducible tensors, $\theta$ being the inclination of $\mbox{\boldmath$\displaystyle\Omega$ }$ with respect to the vertical in the atmosphere. The proof is generalized to frequency-dependent phase matrices. It is applied both to partial frequency redistribution with angle-averaged scalar frequency redistribution functions and to complete frequency redistribution with the Hanle effect in the line core and Rayleigh scattering in the wings. Non-LTE transfer equations for the I_Q^K and integral equations for the associated source functions S_Q^K are established. Formal vectors and matrices constructed with I_Q^K, S_Q^K, and ${\mathcal T}_Q^K$ are introduced in order to present the results in a compact matrix notation. In particular, a simple factorized form is proposed for the Hanle phase matrix.

Key words: line: formation -- polarization -- magnetic fields -- radiative transfer

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