Full non-LTE multi-level radiative transfer I. An atom with three bound infinitely sharp levels

The standard nonlocal thermodynamic equilibrium (non-LTE) multi-level radiative transfer problem only takes into account the deviation of the radiation field and atomic populations from their equilibrium distribution. We aim to show how to solve for the full non-LTE (FNLTE) multi-level radiative transfer problem, also accounting for deviation of the velocity distribution of the massive particles from Maxwellian. We considered, as a first step, a three-level atom with zero natural broadening. In this work, we present a new numerical scheme. Its initialisation relies on the classic, multi-level approximate Lambda-iteration (MALI) method for the standard non-LTE problem. The radiative transfer equations, the kinetic equilibrium equations for atomic populations, and the Boltzmann equations for the velocity distribution functions were simultaneously iterated in order to obtain self-consistent particle distributions. During the process, the observer’s frame absorption and emission profiles were re-computed at every iterative step by convolving the atomic frame quantities with the relevant velocity distribution function. We validate our numerical strategy by comparing our results with the standard non-LTE solutions in the limit of a two-level atom with Hummer’s partial redistribution in frequency, and with a three-level atom with complete redistribution. In this work, we considered the so-called cross-redistribution problem. We then show new FNLTE results for a simple three-level atom while evaluating the assumptions made for the emission and absorption profiles of the standard non-LTE problem with partial and cross-redistribution.

💡 Research Summary

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The paper presents a comprehensive extension of the traditional non‑local thermodynamic equilibrium (non‑LTE) radiative transfer framework by incorporating the full kinetic description of massive particles, i.e., the velocity distribution functions (VDFs) of atoms, into the self‑consistent solution of the radiative transfer problem. While standard non‑LTE calculations treat only the radiation field and the level populations as deviating from equilibrium, they assume that the atomic velocities follow a Maxwell‑Boltzmann distribution at all depths. The authors argue that this assumption can be violated in stellar atmospheres, circumstellar disks, and other astrophysical plasmas where velocity‑changing collisions (VCC) and strong line broadening mechanisms may distort the VDF.

To address this, the authors adopt the kinetic theory formalism developed by Oxenius (1986) and later extended by Hubeny, Hummer, and collaborators (the so‑called HOS I and HOS II papers). In this formalism, each bound level i of an atom is described by a distribution function F_i = n_i f_i(𝑣), where n_i is the level population and f_i(𝑣) is the normalized VDF. The radiative transfer equation (RTE) for a transition i → j retains its familiar form μ∂I_{ij}/∂τ = I_{ij} − η_{ji} χ_{ij}, but the absorption coefficient χ_{ij} and the emission coefficient η_{ji} now depend explicitly on the VDFs through convolution integrals of the atomic frame absorption and emission profiles (α_{ij} and β_{ji}) with f_i and f_j. Consequently, the line profiles in the observer’s frame are no longer simple Voigt or Doppler shapes; they are dynamically reshaped at each iteration by the current VDFs.

The kinetic description of the massive particles is provided by a Boltzmann equation that includes three distinct collisional operators: (1) a radiative term accounting for photon absorption and spontaneous emission, (2) an inelastic term describing electron‑impact excitation and de‑excitation, and (3) a velocity‑changing collision term that drives the VDF toward a Maxwellian distribution with a rate Q_{V,i}. The VCC term has the form n_i Q_{V,i}