Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit

Abstract

Context. The spectrum of the linear polarization, which is formed by scattering and observed on the solar disk close to the limb, is very different from the intensity spectrum and thus able to provide new information, in particular about anisotropies in the solar surface plasma and magnetic fields. In addition, a large number of lines show far wing polarization structures assigned to partial redistribution (PRD), which we prefer to denote as Rayleigh/Raman scattering. The two-level or two-term atom approximation without any lower level polarization is insufficient for many lines. Aims. In the previous paper of this series, we presented our theory generalized to the multilevel and multiline atom and comprised of statistical equilibrium equations for the atomic density matrix elements and radiative transfer equation for the polarized radiation.The present paper is devoted to applying this theory to model the second solar spectrum of the Na i D1 and D2 lines.Methods. The solution method is iterative, of the lambda-iteration type. The usual acceleration techniques were considered or even applied, but we found these to be unsuccessful, in particular because of nonlinearity or large number of quantities determining the radiation at each depth. Results. The observed spectrum is qualitatively reproduced in line center, but the convergence is yet to be reached in the far wings and the observed spectrum is not totally reproduced there. Conclusions. We need to investigate noniterative resolution methods. The other limitation lies in the one-dimensional (1D) atmosphere model, which is unable to reproduce the intermittent matter structure formed of small loops or spicules in the chromosphere. This modeling is rough, but the computing time in the presence of hyperfine structure and PRD prevents us from envisaging a threedimensional (3D) model at this instant.

Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit

“…In the present paper, the redistribution function derivation has been made under the unpolarized and infinitely sharp lower level approximation. An innovative numerical approach alternatively based on the numerical solution of the statistical equilibrium is currently in progress (Bommier 2016b), which enables a complete account of the collisional effects. This approach does not make use of a redistribution function.…”

“…A numerical solution of the statistical equilibrium system of equations is required for the solution of the full problem where all required rates are considered. A numerical solution of the statistical equilibrium equations coupled with the radiative transfer equation is under way (Bommier 2016b). This new numerical approach to the solution of the coupled problem does not make use of a redistribution function.…”

Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit

“…These equations are derived from the formalism described above and are at the basis of the new code XTAT that we are developing for non-LTE polarized line formation with redistribution, and will be described in the next paper of this series (Bommier 2016). As in Bommier (1980), the statistical equilibrium equations for the multilevel atomic density matrix ρ are given by…”

“…They thermalize the level and the velocity distribution in this level remains close to the Maxwellian. As an example, we performed a numerical solution of the velocity-dependent statistical equilibrium, which we present in the following paper of this series (Bommier 2016). We chose the pair of Na i D lines for our example.…”

“…4, we show that for PRD studies, the statistical equilibrium equations have to be resolved for each atomic velocity class. The following paper of this series (Bommier 2016) is devoted to a numerical application of this formalism to the second solar spectrum of the Na i D 1 and D 2 lines.…”