Context. The Milne-Eddington approximation provides an analytic and simple solution to the radiative transfer equation. It can be easily implemented in inversion codes used to fit spectro-polarimetric observations and infer average values of the magnetic field vector and the line-of-sight velocity of the solar plasma. However, in principle, it is restricted to spectral lines that are formed under local thermodynamic conditions, namely, photospheric and optically thin lines.
Aims: We show that a simple modification to the Milne-Eddington approximation is sufficient to infer relevant physical parameters from spectral lines that deviate from local thermodynamic equilibrium, such as those typically observed in the solar chromosphere.
Methods: We modified the Milne-Eddington approximation by including several exponential terms in the source function to reproduce the prototypical shape of chromospheric spectral lines. To check the validity of such an approximation, we first studied the influence of these new terms on the profile shape by means of the response functions. Then we tested the performance of an inversion code including the modification against the presence of noise. The approximation was also tested with realistic spectral lines generated with the RH numerical radiative transfer code. Finally, we confronted the code with synthetic profiles generated from magneto-hydrodynamic simulations carried out with the Bifrost code. For the various tests, we focused on the vector magnetic field and the line-of-sight velocity. We compared our results with the weak-field approximation and center of gravity technique as well.
Results: The response function corresponding to the new terms in the source function have no trade-offs with the response to the different components of the magnetic field vector and line-of-sight velocity. This allows us to perform a robust inference of the physical parameters from the interpretation of spectral line shapes. The strategy has been successfully applied to synthetic chromospheric Stokes profiles generated with both standard models and realistic magnetohydrodynamic (MHD) simulations. The magnetic field vector and velocity can be successfully recovered with the modified Milne-Eddington approximation.
Conclusions: Milne-Eddington model atmospheres that include exponential terms are not new to the solar community but have been overlooked for quite some time. We show that our modification to the Milne-Eddington approximation succeeds in reproducing the profile shape of two chromospheric spectral lines, namely, the Mg I b2 line and the Ca II at 854.2 nm. The results obtained with this approach are in good agreement with the results obtained from the weak field approximation (for magnetic field) and the center of gravity (for velocity). However, the Milne-Eddington approximation possesses a great advantage over classical methods since it is not limited to weak magnetic fields or to a restricted range of velocities.