A modified Milne-Eddington approximation for a qualitative interpretation of chromospheric spectral lines

TL;DR

This paper introduces a simple modification to the Milne-Eddington approximation, enabling it to interpret chromospheric spectral lines formed under non-local thermodynamic equilibrium conditions, thus broadening its applicability.

Contribution

It presents a modified Milne-Eddington approximation that can be used for spectral lines outside LTE, validated through response functions, noise tests, and realistic simulations.

Findings

The modification accurately reproduces spectral line profiles.

The approach performs well with noisy and realistic data.

It extends the utility of the Milne-Eddington approximation to chromospheric lines.

Abstract

The Milne-Eddington approximation provides an analytic and simple solution to the radiative transfer equation. It can be easily implemented in inversion codes that are used to fit spectro-polarimetric observations to infer average values of the magnetic field vector and the line-of-sight velocity of the solar plasma. However, it is in principle restricted to spectral lines formed under local thermodynamic conditions. We show that a simple modification in the linear source function of the Milne-Eddington approximation is sufficient to infer relevant physical parameters from spectral lines that deviate from local thermodynamic equilibrium. This is not a new modification for the solar community but it has been forgotten for quite some time To check the validity of such approximation we make use of the Mg I b2 and the Ca II lines. We first study the influence of these new terms on the profile shape by means of the response functions. Then, we test the performance of an inversion code including such modification against the presence of noise. The approximation is also tested with realistic spectral lines generated with the RH numerical radiative transfer code. Finally, we confront the code with synthetic profiles generated from magneto-hydrodynamic simulations.