The Hanle effect The density matrix and scattering approaches to the √ - law

A √-law was demonstrated by Landi Degl'Inno-centi & Bommier (1994) for resonance polarization in a magnetic atmosphere where the primary source of photons is of thermal origin (isotropic and unpolarized). In this paper we propose a generalized form of this law by dropping the hypothesis on the primary source of photons. We restrict ourselves to the case of weak magnetic fields (Hanle effect). For spectral lines formed with complete redistribution, it has been shown by Landi Degl'Innocenti et al (1990), using the density matrix theory in its irreducible tensorial operator version , that the Hanle effect can be reduced to an integral equation of the convolution type for a six-component source vector. As shown by Faurobert-Scholl (1991), a similar equation can be obtained by performing an azimuthal Fourier decomposition of the transfer equation for the Stokes parameters. In the first part of the paper we recall the main steps of the two methods and establish the correspondence between the con-volution equations that they provide. In the second part we use these equations to obtain a generalized √-law. For the equation coming from the density matrix formalism, we essentially follow the original proof of Landi Degl'Innocenti & Bommier (1994). For the equation coming from the Fourier decomposition , because of a lack of symmetry in operator describing the action of the magnetic field, we use as intermediate step the Hopf-Bronstein-Rybicki relation established by Ivanov (1995) for transport operators which are not self-adjoint.