General theory of radiative transfer in a magnetized atmosphere with scattering by electrons

Here we consider the general radiative transfer theory in a magnetized atmosphere for any value of parameter $x=\omega _B/\omega \simeq 0.933\times 10^{-8}\lambda (\mu \mathrm{m}) B(\text{G})$, where ωB is the cyclotron frequency of electron rotation and ω is the angular frequency of considered monochromatic radiation. The main term of the radiative transfer equations $\textbf {J}_{\alpha \beta }$ for the Stokes parameters I, V, U and Q describes the scattering of radiation coming from all directions and distances. All Stokes parameters of the incident radiation mutually transform into each other along their path due to interference and different cross-sections for them. To find this transformation of the Stokes parameters one has to solve the complex system of transfer equations without the sources and term $\textbf {J}_{\alpha \beta }$. This is done in our paper. Firstly, we present the general solution and then give the solution for the case of a homogeneous magnetic field, where the formulas have clear algebraic form. We note that for small parameter x our formulas describe the known Faraday rotation. Our formulas allow us to derive an integral equation for the density of polarized radiation, multiple scattered in a magnetized atmosphere for any values of the parameter x. The obtained correct radiation transfer equation allows us to calculate the Stokes parameters of radiation emerging from an atmosphere, in particular, for the Milne problem.