A Markovian description of the multi-level source function and its application to the Lyman series in the Sun

TL;DR

This paper introduces a Markov chain-based method to analyze multi-level atomic source functions, providing new insights into the formation of Lyman series lines in the Sun's atmosphere, especially regarding interlocking effects.

Contribution

The paper presents a novel Markov chain approach to compute and interpret multi-level source functions, enhancing understanding of atomic level interlocking in stellar atmospheres.

Findings

Interlocking effects increase with Lyman series order.

Ly-α has minimal interlocking contribution, while Ly-β and Ly-γ have significant contributions.

The method offers a more physical interpretation of spectral line formation.

Abstract

Aims. We introduce a new method to calculate and interpret indirect transition rates populating atomic levels using Markov chain theory. Indirect transition rates are essential to evaluate interlocking in a multi-level source function, which quantifies all the processes that add and remove photons from a spectral line. A better understanding of the multi-level source function is central to interpret optically thick spectral line formation in stellar atmospheres, especially outside local thermodynamical equilibrium (LTE). Methods. We compute the level populations from a hydrogen model atom in statistical equilibrium, using the solar FALC model, a 1D static atmosphere. From the transition rates, we reconstruct the multi-level source function using our new method and compare it with existing methods to build the source function. We focus on the Lyman series lines and analyze the different contributions to the source functions and synthetic spectra. Results. Absorbing Markov chains can represent the level-ratio solution of the statistical equilibrium equation and can therefore be used to calculate the indirect transition rates between the upper and lower levels of an atomic transition. Our description of the multi-level source function allows a more physical interpretation of its individual terms, particularly a quantitative view of interlocking. For the Lyman lines in the FALC atmosphere, we find that interlocking becomes increasingly important with order in the series, with Ly-{\alpha} showing very little, but Ly-\b{eta} nearly 50% and Ly-{\gamma} about 60% contribution coming from interlocking. In some cases, this view seems opposed to the conventional wisdom that these lines are mostly scattering, and we discuss the reasons why.