The Hanle effect

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 T Κ Q(i, Ω). 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 Ω and Ω' of the incoming and scattered beams. The dependence on Ω and Ω' is expressed in terms of the spherical tensors T Κ Q (i, Ω) where i refers to the Stokes parameters (i = 0,..., 3). A multipolar expansion in terms of the T K Q,(Ω) 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 (ν,Ω) =1Σ KQ τΚ Q (i,Ω)l K Q )(ν,θ) (K = 0,1, 2, -K ≤ Q < +K) where the I Q K are nine cylindrically symmetrical, irreducible tensors, θ being the inclination of Ω 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 Κ Q are established. Formal vectors and matrices constructed with I Q K , S K Q , and τ K Q 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.