Context. Observations of center-to-limb variations (CLVs) of spectral lines and continua provide a good test for the accuracy of models with a solar and stellar atmospheric structure and spectral line formation. They are also widely used to constrain elemental abundances, and are becoming increasingly more important in atmospheric studies of exoplanets. However, only a few such data sets exist for chromospheric lines.

Aims. We aim to create a set of standard profiles by means of mosaics made with the CRISP and CHROMIS instruments of the Swedish 1-m Solar Telescope (SST), as well as to explore the robustness of said profiles obtained using this method.

Methods. For each spectral line, we used a mosaic that ranges from the center to the limb. Each of these mosaics were averaged down to 50 individual spectral profiles and spaced by 0.02 in the μ scale. These profiles were corrected for p-mode oscillations, and their line parameters (equivalent width, line shift, full-width at half-maximum, and line depth) were then compared against literature values whenever possible.

Results. We present a set of 50 average profiles that are spaced equidistantly along the cosine of the heliocentric angle (μ) by steps of 0.02 for five continuum points between 4001 and 7772 Å, as well as ten of the most commonly observed spectral lines at the SST (Ca II H & K, Ηβ, Mg I 5173 Å, C I 5380 Å, Fe I 6173 Å, Fe I 6301 Å, Ha, O I 7772 Å, and Ca II 8542 Å).

Conclusions. The CLV of line profiles and continua are shared in the CDS as machine readable tables, providing a quantitative constraint on theoretical models that aim to model stellar atmospheres.

Context. Fabry–Pérot interferometers (FPIs) have become very popular in solar observations because they offer a balance between cadence, spatial resolution, and spectral resolution through a careful design of the spectral sampling scheme according to the observational requirements of a given target. However, an efficient balance requires knowledge of the expected target conditions, the properties of the chosen spectral line, and the instrumental characteristics.

Aims. Our aim is to find a method that allows the optimal spectral sampling of FPI observations in a given spectral region to be found. The selected line positions must maximize the information content in the observation with a minimal number of points.

Methods. In this study, we propose a technique based on a sequential selection approach in which a neural network is used to predict the spectrum (or physical quantities, if the model is known) from the information at a few points. Only those points that contain relevant information and improve the model prediction are included in the sampling scheme.

Results. We have quantified the performance of the new sampling schemes by showing the lower errors in the model parameter reconstructions. The method adapts the separation of the points according to the spectral resolution of the instrument, the typical broadening of the spectral shape, and the typical Doppler velocities. The experiments that use the Ca II 8542 Å line show that the resulting wavelength scheme naturally places more points in the core than in the wings (by almost a factor of 4), consistent with the sensitivity of the spectral line at each wavelength interval. As a result, observations focused on magnetic field analysis should prioritize a denser grid near the core, while those focused on thermodynamic properties would benefit from a larger coverage. The method can also be used as an accurate interpolator to improve the inference of the magnetic field when using the weak-field approximation.

Conclusions. Overall, this method offers an objective approach for designing new instrumentation or observing proposals with customized configurations for specific targets. This is particularly relevant when studying highly dynamic events in the solar atmosphere with a cadence that preserves spectral coherence without sacrificing much information.

Context. A proper estimate of the chromospheric magnetic fields is thought to improve modelling of both active region and coronal mass ejection evolution. However, because the chromospheric field is not regularly obtained for sufficiently large fields of view, estimates thereof are commonly obtained through data-driven models or field extrapolations, based on photospheric boundary conditions alone and involving pre-processing that may reduce details and dynamic range in the magnetograms.

Aims. We investigate the similarity between the chromospheric magnetic field that is directly inferred from observations and the field obtained from a magnetohydrostatic (MHS) extrapolation based on a high-resolution photospheric magnetogram.

Methods. Based on Swedish 1-m Solar Telescope Fe I 6173 Å and Ca II 8542 Å observations of NOAA active region 12723, we employed the spatially regularised weak-field approximation (WFA) to derive the vector magnetic field in the chromosphere from Ca II, as well as non-local thermodynamic equilibrium (non-LTE) inversions of Fe I and Ca II to infer a model atmosphere for selected regions. Milne-Eddington inversions of Fe I serve as photospheric boundary conditions for the MHS model that delivers the three-dimensional field, gas pressure, and density self-consistently.

Results. For the line-of-sight component, the MHS chromospheric field generally agrees with the non-LTE inversions and WFA, but tends to be weaker by 16% on average than these when larger in magnitude than 300 G. The observationally inferred transverse component is systematically stronger, up to an order of magnitude in magnetically weaker regions, but the qualitative distribution with height is similar to the MHS results. For either field component, the MHS chromospheric field lacks the fine structure derived from the inversions. Furthermore, the MHS model does not recover the magnetic imprint from a set of high fibrils connecting the main polarities.

Conclusions. The MHS extrapolation and WFA provide a qualitatively similar chromospheric field, where the azimuth of the former is better aligned with Ca II 8542 Å fibrils than that of the WFA, especially outside strong-field concentrations. The amount of structure as well as the transverse field strengths are, however, underestimated by the MHS extrapolation. This underscores the importance of considering a chromospheric magnetic field constraint in data-driven modelling of active regions, particularly in the context of space weather predictions.

Aims. The non-uniform surface temperature distribution of rotating active stars is routinely mapped with the Doppler imaging technique. Inhomogeneities in the surface produce features in high-resolution spectroscopic observations that shift in wavelength because of the Doppler effect, depending on their position on the visible hemisphere. The inversion problem has been systematically solved using maximum a posteriori regularized methods assuming smoothness or maximum entropy. Our aim in this work is to solve the full Bayesian inference problem by providing access to the posterior distribution of the surface temperature in the star compatible with the observations.

Methods. We use amortized neural posterior estimation to produce a model that approximates the high-dimensional posterior distribution for spectroscopic observations of selected spectral ranges sampled at arbitrary rotation phases. The posterior distribution is approximated with conditional normalizing flows, which are flexible, tractable, and easy-to-sample approximations to arbitrary distributions. When conditioned on the spectroscopic observations, these normalizing flows provide a very efficient way of obtaining samples from the posterior distribution. The conditioning on observations is achieved through the use of Transformer encoders, which can deal with arbitrary wavelength sampling and rotation phases.

Results. Our model can produce thousands of posterior samples per second, each one accompanied by an estimation of the log-probability. Our exhaustive validation of the model for very high-signal-to-noise observations shows that it correctly approximates the posterior, albeit with some overestimation of the broadening. We apply the model to the moderately fast rotator II Peg, producing the first Bayesian map of its temperature inhomogenities. We conclude that conditional normalizing flows are a very promising tool for carrying out approximate Bayesian inference in more complex problems in stellar physics, such as constraining the magnetic properties using polarimetry.

Stokes inversion techniques are very powerful methods for obtaining information on the thermodynamic and magnetic properties of solar and stellar atmospheres. In recent years, highly sophisticated inversion codes have been developed that are now routinely applied to spectro-polarimetric observations. Most of these inversion codes are designed to find an optimum solution to the nonlinear inverse problem. However, to obtain the location of potentially multimodal cases (ambiguities), the degeneracies and the uncertainties of each parameter inferred from the inversions algorithms – such as Markov chain Monte Carlo (MCMC) – require evaluation of the likelihood of the model thousand of times and are computationally costly. Variational methods are a quick alternative to Monte Carlo methods, and approximate the posterior distribution by a parametrized distribution. In this study, we introduce a highly flexible variational inference method to perform fast Bayesian inference, known as normalizing flows. Normalizing flows are a set of invertible, differentiable, and parametric transformations that convert a simple distribution into an approximation of any other complex distribution. If the transformations are conditioned on observations, the normalizing flows can be trained to return Bayesian posterior probability estimates for any observation. We illustrate the ability of the method using a simple Milne-Eddington model and a complex non-local thermodynamic equilibrium (NLTE) inversion. The method is extremely general and other more complex forward models can be applied. The training procedure need only be performed once for a given prior parameter space and the resulting network can then generate samples describing the posterior distribution several orders of magnitude faster than existing techniques.

Context. Solar flares release an enormous amount of energy (∼10³² erg) into the corona. A substantial fraction of this energy is transported to the lower atmosphere, which results in chromospheric heating. The mechanisms that transport energy to the lower solar atmosphere during a flare are still not fully understood.

Aims. We aim to estimate the temporal evolution of the radiative losses in the chromosphere at the footpoints of a C-class flare, in order to set observational constraints on the electron beam parameters of a RADYN flare simulation.

Methods. We estimated the radiative losses from hydrogen, and singly ionized Ca and Mg using semiempirical model atmospheres, which were inferred from a multiline inversion of observed Stokes profiles obtained with the CRISP and CHROMIS instruments on the Swedish 1-m Solar Telescope. The radiative losses were computed taking into account the effect of partial redistribution and non-local thermodynamic equilibrium. To estimate the integrated radiative losses in the chromosphere, the net cooling rates were integrated between the temperature minimum and the height where the temperature reaches 10 kK. We also compared our time series of radiative losses with those from the RADYN flare simulations.

Results. We obtained a high spatial-resolution map of integrated radiative losses around the flare peak time. The stratification of the net cooling rate suggests that the Ca IR triplet lines are responsible for most of the radiative losses in the flaring atmosphere. During the flare peak time, the contribution from Ca II H and K and Mg II h and k lines are strong and comparable to the Ca IR triplet (∼32 kW m⁻²). Since our flare is a relatively weak event, the chromosphere is not heated above 11 kK, which in turn yields a subdued Lyα contribution (∼7 kW m⁻²) in the selected limits of the chromosphere. The temporal evolution of total integrated radiative losses exhibits sharply rising losses (0.4 kW m⁻² s⁻¹) and a relatively slow decay (0.23 kW m⁻² s⁻¹). The maximum value of total radiative losses is reached around the flare peak time, and can go up to 175 kW m⁻² for a single pixel located at footpoint. After a small parameter study, we find the best model-data consistency in terms of the amplitude of radiative losses and the overall atmospheric structure with a RADYN flare simulation in the injected energy flux of 5 × 10¹⁰ erg s⁻¹ cm⁻².

Context. Our knowledge of the heating mechanisms that are at work in the chromosphere of plage regions remains highly unconstrained from observational studies. While many heating candidates have been proposed in theoretical studies, the exact contribution from each of them is still unknown. The problem is rather difficult because there is no direct way of estimating the heating terms from chromospheric observations. 

Aims: The purpose of our study is to estimate the chromospheric heating terms from a multi-line high-spatial-resolution plage dataset, characterize their spatio-temporal distribution and set constraints on the heating processes that are at work in the chromosphere. 

Methods: We used nonlocal thermodynamical equilibrium inversions in order to infer a model of the photosphere and chromosphere of a plage dataset acquired with the Swedish 1-m Solar Telescope (SST). We used this model atmosphere to calculate the chromospheric radiative losses from the main chromospheric cooler from H I, Ca II, and Mg II atoms. In this study, we approximate the chromospheric heating terms by the net radiative losses predicted by the inverted model. In order to make the analysis of time-series over a large field of view computationally tractable, we made use of a neural network which is trained from the inverted models of two non-consecutive time-steps. We have divided the chromosphere in three regions (lower, middle, and upper) and analyzed how the distribution of the radiative losses is correlated with the physical parameters of the model. 

Results: In the lower chromosphere, the contribution from the Ca II lines is dominant and predominantly located in the surroundings of the photospheric footpoints. In the upper chromosphere, the H I contribution is dominant. Radiative losses in the upper chromosphere form a relatively homogeneous patch that covers the entire plage region. The Mg II also peaks in the upper chromosphere. Our time analysis shows that in all pixels, the net radiative losses can be split in a periodic component with an average amplitude of amp̅Q = 7.6 kW m−2 and a static (or very slowly evolving) component with a mean value of −26.1 kW m−2. The period of the modulation present in the net radiative losses matches that of the line-of-sight velocity of the model. 

Conclusions: Our interpretation is that in the lower chromosphere, the radiative losses are tracing the sharp lower edge of the hot magnetic canopy that is formed above the photosphere, where the electric current is expected to be large. Therefore, Ohmic current dissipation could explain the observed distribution. In the upper chromosphere, both the magnetic field and the distribution of net radiative losses are room-filling and relatively smooth, whereas the amplitude of the periodic component is largest. Our results suggest that acoustic wave heating may be responsible for one-third of the energy deposition in the upper chromosphere, whereas other heating mechanisms must be responsible for the rest: turbulent Alfvén wave dissipation or ambipolar diffusion could be among them. Given the smooth nature of the magnetic field in the upper chromosphere, we are inclined to rule out Ohmic dissipation of current sheets in the upper chromosphere.

The European Solar Telescope (EST) is a project aimed at studying the magnetic connectivity of the solar atmosphere, from the deep photosphere to the upper chromosphere. Its design combines the knowledge and expertise gathered by the European solar physics community during the construction and operation of state-of-the-art solar telescopes operating in visible and near-infrared wavelengths: the Swedish 1m Solar Telescope, the German Vacuum Tower Telescope and GREGOR, the French Télescope Héliographique pour l’Étude du Magnétisme et des Instabilités Solaires, and the Dutch Open Telescope. With its 4.2 m primary mirror and an open configuration, EST will become the most powerful European ground-based facility to study the Sun in the coming decades in the visible and near-infrared bands. EST uses the most innovative technological advances: the first adaptive secondary mirror ever used in a solar telescope, a complex multi-conjugate adaptive optics with deformable mirrors that form part of the optical design in a natural way, a polarimetrically compensated telescope design that eliminates the complex temporal variation and wavelength dependence of the telescope Mueller matrix, and an instrument suite containing several (etalon-based) tunable imaging spectropolarimeters and several integral field unit spectropolarimeters. This publication summarises some fundamental science questions that can be addressed with the telescope, together with a complete description of its major subsystems.

Context. The evolution of the photospheric magnetic field plays a key role in the energy transport into the chromosphere and the corona. In active regions, newly emerging magnetic flux interacts with the pre-existent magnetic field, which can lead to reconnection events that convert magnetic energy into thermal energy.

Aims. We aim to study the heating caused by a strong reconnection event that was triggered by magnetic flux cancelation.

Methods. We use imaging and spectropolarimetric data in the Fei6301& 6302 Å, Caii8542 Å, and CaiiK spectral lines obtained with the CRISP and CHROMIS instruments at the Swedish 1-m Solar Telescope. These data were inverted with the STiC code by performing multi-atom, multi-line, non-local thermodynamic equilibrium inversions. These inversions yielded a three-dimensional model of the reconnection event and surrounding atmosphere, including temperature, velocity, microturbulence, magnetic field, and radiative loss rate.

Results. The model atmosphere shows the emergence of magnetic loops with a size of several arcseconds into a pre-existing pre-dominantly unipolar field. Where the reconnection region is expected to be, we see an increase in the chromospheric temperature of roughly 2000 K as well as bidirectional flows of the order of 10 km s−1 emanating from there. We see bright blobs of roughly 0.2 arcsec in diameter in the CaiiK, moving at a plane-of-the-sky velocity of the order of 100 km s−1 and a blueshift of 100 km s−1, which we interpret as ejected plasmoids from the same region. This scenario is consistent with theoretical reconnection models, and therefore provides evidence of a reconnection event taking place. The chromospheric radiative losses at the reconnection site are as high as160 kW m−2, providing a quantitative constraint on theoretical models that aim to simulate reconnection caused by flux emergence in the chromosphere.

Context. Obtaining an accurate measurement of magnetic field vector in the solar atmosphere is essential for studying changes in field topology during flares and reliably modelling space weather.

Aims. We tackle this problem by applying various inversion methods to a confined X2.2 flare that occurred in NOAA AR 12673 on 6 September 2017 and comparing the photospheric and chromospheric magnetic field vector with the results of two numerical models of this event.

Methods. We obtained the photospheric magnetic field from Milne-Eddington and (non-)local thermal equilibrium (non-LTE) inversions of Hinode SOT/SP FeI 6301.5 angstrom and 6302.5 angstrom. The chromospheric field was obtained from a spatially regularised weak-field approximation (WFA) and non-LTE inversions of CaII 8542 angstrom observed with CRISP at the Swedish 1 m Solar Telescope. We investigated the field strengths and photosphere-to-chromosphere shear in the field vector.

Results. The LTE- and non-LTE-inferred photospheric magnetic field components are strongly correlated across several optical depths in the atmosphere, with a tendency towards a stronger field and higher temperatures in the non-LTE inversions. For the chromospheric field, the non-LTE inversions correlate well with the spatially regularised WFA, especially in terms of the line-of-sight field strength and field vector orientation. The photosphere exhibits coherent strong-field patches of over 4.5 kG, co-located with similar concentrations exceeding 3 kG in the chromosphere. The obtained field strengths are up to two to three times higher than in the numerical models, while the photosphere-to-chromosphere shear close to the polarity inversion line is more concentrated and structured.

Conclusions. In the photosphere, the assumption of LTE for FeI line formation does not yield significantly different magnetic field results in comparison to the non-LTE case, while Milne-Eddington inversions fail to reproduce the magnetic field vector orientation where FeI is in emission. In the chromosphere, the non-LTE-inferred field is excellently approximated by the spatially regularised WFA. Our inversions confirm the locations of flux rope footpoints that have been predicted by numerical models. However, pre-processing and lower spatial resolution lead to weaker and smoother field in the models than what our data indicate. This highlights the need for higher spatial resolution in the models to better constrain pre-eruptive flux ropes.