Context. Inversion codes are computer programs that fit a model atmosphere to the observed Stokes spectra, thus retrieving the relevant atmospheric parameters. The rising interest in the solar chromosphere, where spectral lines are formed by scattering, requires developing, testing, and comparing new non-local thermal equilibrium (NLTE) inversion codes. Aims. We present a new NLTE inversion code that is based on the analytical computation of the response functions. We named the code SNAPI, which is short for spectropolarimetic NLTE analytically powered inversion. Methods. SNAPI inverts full Stokes spectrum in order to obtain a depth-dependent stratification of the temperature, velocity, and the magnetic field vector. It is based on the so-called node approach, where atmospheric parameters are free to vary in several fixed points in the atmosphere, and are assumed to behave as splines in between. We describe the inversion approach in general and the specific choices we have made in the implementation. Results. We test the performance on one academic problem and on two interesting NLTE examples, the Ca II 8542 and Na I D spectral lines. The code is found to have excellent convergence properties and outperforms a finite-difference based code in this specific implementation by at least a factor of three. We invert synthetic observations of Na lines from a small part of a simulated solar atmosphere and conclude that the Na lines reliably retrieve the magnetic field and velocity in the range −3 < log τ < −0.5 Conclusions. .

Aims. One of the necessary parameters needed for the interpretation of the light curves of transiting exoplanets or eclipsing binary stars (as well as interferometric measurements of a star or microlensing events) is how the intensity and polarization of light changes from the center to the limb of a star. Scattering and absorption processes in the stellar atmosphere affect both the center-to-limb variation of intensity (CLVI) and polarization (CLVP). In this paper, we present a study of the CLVI and CLVP in continuum spectra, taking into consideration the different contributions of scattering and absorption opacity for a variety of spectral type stars with spherical atmospheres. Methods. We solve the radiative transfer equation for polarized light in the presence of a continuum scattering, taking into consideration the spherical model of a stellar atmosphere. To cross-check our results, we developed two independent codes that are based on Feautrier and short characteristics methods, respectively, Results. We calculate the center-to-limb variation of intensity (CLVI) and polarization (CLVP) in continuum for the Phoenix grid of spherical stellar model atmospheres for a range of effective temperatures (4000−7000 K), gravities (log g = 1.0−5.5), and wavelengths (4000−7000 Å), which are tabulated and available at the CDS. In addition, we present several tests of our codes and compare our calculations for the solar atmosphere with published photometric and polarimetric measurements. We also show that our two codes provide similar results in all considered cases. Conclusions. For sub-giant and dwarf stars (log g = 3.0−4.5), the lower gravity and lower effective temperature of a star lead to higher limb polarization of the star. For giant and supergiant stars (log g = 1.0−2.5), the highest effective temperature yields the largest polarization. By decreasing the effective temperature of a star down to 4500−5500 K (depending on log g), the limb polarization decreases and reaches a local minimum. It increases again with a corresponding decrease in temperature down to 4000 K. For the most compact dwarf stars (log g = 5.0−5.5), the limb polarization degree shows a maximum for models with effective temperatures in the range 4200−4600 K (depending on log g) and decreases toward higher and lower temperatures.

Context. Response functions provide us with a quantitative measure of sensitivity of the emergent spectrum to perturbations in the solar atmosphere and are thus the method of choice for interpreting spectropolarimetric observations. For the lines formed in the solar chromosphere, it is necessary to compute these responses taking into account nonlocal thermodynamic equilibrium (NLTE) effects. Aims. We show how to analytically compute the response of the level populations in NLTE to a change of a given physical quantity at a given depth in the atmosphere. These responses are then used to compute opacity and emissivity responses, which are then propagated to obtain the response of the emergent intensity. Methods. Our method is based on the derivative of the rate equations, where we explicitly incorporate spatial coupling in the radiative rate terms. After considering and collecting all interdependencies, the problem reduces to a linear system of equations with a dimension equal to the product of the number of spatial points and the number of energy levels. Results. We compare analytically computed response functions with those obtained using a finite difference approach and find very good agreement. In addition, a more accurate way of propagating opacity and emissivity perturbations through the numerical solution of the radiative transfer equation was developed. Conclusions. This method allows for the fast evaluation of the response of the emergent spectrum to perturbations of a given quantity at a given depth, and thus is a significant step towards more efficient NLTE inversions.

Tools for computing detailed optically thick spectral line profiles out of local thermodynamic equilibrium have always been focused on speed, due to the large computational effort involved. With the Lightweaver framework, we have produced a more flexible, modular toolkit for building custom tools in a high-level language, Python, without sacrificing speed against the current state of the art. The goal of providing a more flexible method for constructing these complex simulations is to decrease the barrier to entry and allow more rapid exploration of the field. In this paper we present an overview of the theory of optically thick nonlocal thermodynamic equilibrium radiative transfer, the numerical methods implemented in Lightweaver including the problems of time-dependent populations and charge-conservation, as well as an overview of the components most users will interact with, to demonstrate their flexibility.