The Hanle effect

Decomposition of the Stokes parameters into irreducible components

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 .

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  and of the incoming and scattered beams. The dependence on and is expressed in terms of the spherical tensors  where i refers to the Stokes parameters (). A multipolar expansion in terms of the 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 (, ) where the are nine cylindrically symmetrical, irreducible tensors, θ being the inclination of 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 and integral equations for the associated source functions  are established. Formal vectors and matrices constructed with , , and 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.