Context. Sunspots represent a key feature in the solar atmosphere to explore how magnetic fields interact with plasma flows, exhibiting large variations in physical parameters over very small spatial scales (< 100 km), and sometimes featuring dynamic phenomena such as oscillatory umbral flashes. To fully understand the thermodynamic, magnetic and kinematic structure of these regions, from the stable photosphere to the shock-dominated chromosphere, Stokes inversion techniques are employed to jointly model these layers.

Aims. We aim to determine the average thermal, magnetic, and kinematic properties of a sunspot from the photosphere to the chromosphere and to deepen our understanding of the properties of umbral flashes.

Methods. We analysed high-resolution spectropolarimetric data acquired with the CRISP instrument at the Swedish Solar Telescope (SST). The dataset includes full Stokes measurements of the Mg I 517.2 nm, Na I 589.5 nm, Fe I 630.2 nm, and Ca II 854.2 nm spectral lines. We performed inversions using the FIRTEZ code, which includes non-local thermodynamic equilibrium (NLTE) and 3D magnetohydrostatic (MHS) equilibrium to constrain the gas pressure and density.

Results. We successfully inferred the physical parameters in a three-dimensional (x, y, z) domain and provide their average values as a function of the radial distance from the sunspot’s center at different heights. Among other findings, we determine that the photospheric Evershed flow is found to reverse into the inverse Evershed inflow in the upper photosphere. In contrast, the moat flow outside the sunspot persists as an outflow at similar heights, suggesting that it is not a direct continuation of the Evershed flow. Furthermore, analysis of an umbral flash event reveals supersonic upflows (Mach numbers ∥M∥≥1.5) and thermodynamic conditions consistent with shock fronts.

Conclusions. The application of 3D MHS equilibrium and NLTE effects combined with multiple lines sensing different layers of the atmosphere allows for the reliable retrieval of atmospheric parameters, which are typically difficult to simultaneously constrain in the photosphere and chromosphere. The inferred properties of umbral flash show clear evidence of shock dynamics, coinciding with previous theoretical and observational studies that point to converging supersonic flows that move the optical depth iso-surfaces as the driving mechanism behind umbral flashes.

The magnetic fields in the solar polar region are important to our understanding of the internal dynamo process, the global coronal structure, and the origin of the solar wind. The inference of polar fields based on spectropolarimetric observation is highly model-dependent and can suffer from various systematic effects. Here, we analyze a raster map of the southern polar region taken by the Hinode Spectro-Polarimeter, utilizing the Stokes Inversion based on Response functions code. The inversions provide height-dependent vector magnetic field maps between optical depths log₁₀τ = −2 and 0. We examine the impacts on the total magnetic flux estimates from adopting: (1) one-versus-two-component atmospheric models via a “filling factor” parameter; and (2) different analysis schemes. At log₁₀τ = −1.5, the polar magnetic flux is estimated to be (1.84 ± 0.03) × 10²¹ Mx and (1.38 ± 0.02) × 10²¹ Mx under the one- and two-component atmosphere assumptions, respectively. The magnetic flux is approximately constant or increases slightly with height, respectively. We find that the two-component (one-component) configuration is preferred for 58.3% (32.3%) of the pixels. Different initial guesses—including the input atmosphere model and the filling factor—as well as different inversion settings can significantly affect the results, especially for locations with weaker polarization signals. Our work highlights the importance of considering unresolved magnetic structures or stray light. Model degeneracy and the convergence to local minima limit the precision of the polar magnetic flux inference (no better than several tens of percent, in this case). Higher-resolution observations and advanced inversion and disambiguation algorithms may alleviate these limitations.

Context. The study of magnetic activity in the Sun's polar regions is essential for understanding the solar cycle. However, measuring polar magnetic fields presents challenges due to projection effects and their intrinsically weak magnetic field strength. Faculae, bright regions on the visible solar surface associated with increased magnetic activity, offer a valuable proxy for measuring polar fields.

Aims. This research aims to analyze the magnetic activity of the Sun's polar regions through the use of polar faculae.

Methods. A neural network model (U-Net) was employed to detect polar faculae in images from the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). The model was trained on synthetic data, eliminating the need for manual labeling, and was used to analyze 14 years of data from May 2010 to May 2024.

Results. The U-Net model demonstrates superior performance and efficiency over existing methods, enabling automated large-scale studies. We find that polar faculae numbers exhibit cyclical behavior with distinct minima and maxima, showing similar patterns between poles but with notable temporal delays (south pole: minimum early 2014, maximum late 2016; north pole: minimum late 2014, maximum mid-2019). Polar faculae magnetic fields remain consistent in magnitude (∼±75 G) across both poles and throughout the solar cycle. A strong linear correlation was found between the polar faculae count and the overall polar magnetic field strength. The spatio-temporal evolution reveals systematic migration of field polarity reversals from mid-latitudes toward the poles at rates of 3−8 m/s. During solar minimum, we observe a small relative increase in stronger-field faculae compared to solar maximum, suggesting either the coexistence of two magnetic distributions or subtle solar cycle dependence in faculae properties.

Context. Spatially resolved observations of the Sun and the astronomical sample size of stellar bodies are the respective key strengths of solar and stellar observations. However, the large difference in object brightness between the Sun and other stars has led to distinctly different instrumentation and methodologies between the two fields.Aims. We produced and analyzed synthetic full-disk spectra derived from 19 small area field-of-view optical observations of solar flares acquired by the Swedish 1-m Solar Telescope (SST) between 2011 and 2024. These were used to investigate what can and cannot be inferred about physical processes on the Sun from Sun-as-a-star observations.Methods. The recently released Numerical Empirical Sun-as-a-Star Integrator (NESSI) code provides synthetic full-disk integrated spectral line emission based on smaller field-of-view input while accounting for center-to-limb variations and differential rotation. We used this code to generate pseudo-Sun-as-a-star spectra from the SST observations.Results. We show that limited-area solar observations can be extrapolated to represent the full disk accurately in a manner close to what is achievable with Sun-as-a-star telescopes. Additionally, we identify nine spectral features, four of which are caused by instrumental effects. Most notably, we find a relation between the heliocentric angle of flares and the width of the excess emission left by them as well as a source of false positive coronal mass ejections-like signatures, and we defined an energy scaling law based on chromospheric line intensities that shows that the peak flare contrast roughly scales with the square root of the bolometric energy.Conclusions. The presented method of creating pseudo-Sun-as-a-star observations from limited field-of-view solar observations allows for the accurate comparison of solar flare spectra with their stellar counterparts while allowing for the detection of signals at otherwise unachievable noise levels.

Context. Sunspots survive on the solar surface for timescales ranging from days to months. This requires them to be in an equilibrium involving magnetic fields and hydrodynamic forces. Unfortunately, theoretical models of sunspot equilibrium are very simplified as they assume that spots are static and possess a self-similar and axially symmetric magnetic field. These assumptions neglect the role of small-scale variations of the magnetic field along the azimuthal direction produced by umbral dots, light bridges, penumbral filaments, and so forth.

Aims. We aim to study whether sunspot equilibrium is maintained once azimuthal fluctuations in the magnetic field, produced by the sunspot fine structure, are taken into account.

Methods. We apply the FIRTEZ Stokes inversion code to spectropolarimetric observations to infer the magnetic and thermodynamic parameters in two sunspots located at the disk center and observed with two different instruments: one observed from the ground with the 1.5-meter German GREGOR Telescope and another with the Japanese spacecraft Hinode. We compare our results with three-dimensional radiative magnetohydrodynamic simulations of a sunspot carried out with the MuRAM code.

Results. We infer clear variations in the gas pressure and density of the plasma directly related to fluctuations in the Lorentz force and associated with the filamentary structure in the penumbra. Similar results are obtained in the umbra despite its lack of an observed filamentary structure. Results from the two observed sunspots are in excellent qualitative and quantitative agreement with the numerical simulations.

Conclusions. Our results indicate that the magnetic topology of sunspots along the azimuthal direction is very close to magnetohydrostatic equilibrium, thereby helping to explain why sunspots are such long-lived structures capable of surviving on the solar surface for days or even full solar rotations.

Context. Inferences of the magnetic field in the solar atmosphere by means of spectropolarimetric inversions (i.e., Stokes inversion codes) yield magnetic fields that are non-solenoidal (∇ B ≠ 0). Because of this, results obtained by such methods are sometimes put into question. Aims. We aim to develop and implement a new technique that, in conjunction with Stokes inversion codes, can retrieve magnetic fields that are simultaneously consistent with observed polarization signals and with the null divergence condition. Methods. The method used in this work strictly imposes ∇ B=0 by determining the vertical component of the magnetic field (Bz) from the horizontal ones (BxBy). We implement this technique, which we refer to as solenoidal inversion, into the FIRTEZ Stokes inversion code and apply it to spectropolarimetric observations of a sunspot observed with the Hinode/SP instrument. Results. We show that the solenoidal inversion retrieves a vertical component of the magnetic field that is consistent in 80% of the analyzed three-dimensional (x, y, z) domain, with the vertical component of the magnetic field inferred from the non-solenoidal inversion. We demonstrate that the solenoidal inversion is capable of a better overall fitting to the observed Stokes vector than the non-solenoidal inversion. In fact, the solenoidal magnetic field fits Stokes V worse, but this is compensated by a better fit to Stokes I. We find a direct correlation between the worsening in the fit to the circular polarization profiles by the solenoidal inversion and the deviations in the inferred Bz with respect to the non-solenoidal inversion. Finally, we also show that the spatial distribution of the electric currents given by ∇x B does not change significantly after imposing the null divergence condition. Conclusions. In spite of being physically preferable, solenoidal magnetic fields are topologically very similar in 80% of the analyzed three-dimensional domain to the non-solenoidal fields obtained from spectropolarimetric inversions. These results support the idea that common Stokes inversion techniques fail to reproduce ∇ B=0 mainly as a consequence of the uncertainties in the determination of the individual components of the magnetic field. In the remaining 20% of the analyzed domain, where the Bz inferred by the solenoidal and non-solenoidal inversions disagree, it remains to be proven that the solenoidal inversion is to be preferred because even though the overall fit to the Stokes parameters improves, the fit to Stokes V worsens. It is in these regions where the application of the Stokes inversion constrained by the null divergence condition can yield new insights about the topology of the magnetic field in the solar photosphere.

Context. One-dimensional, semi-empirical models of the solar atmosphere are widely employed in numerous contexts within solar physics, ranging from the determination of element abundances and atomic parameters to studies of the solar irradiance and from Stokes inversions to coronal extrapolations. These models provide the physical parameters (i.e. temperature, gas pressure, etc.) in the solar atmosphere as a function of the continuum optical depth τc. The transformation to the geometrical z scale (i.e. vertical coordinate) is provided via vertical hydrostatic equilibrium. Aims. Our aim is to provide updated, one-dimensional, semi-empirical models of the solar atmosphere as a function of z, but employing the more general case of three-dimensional magneto-hydrostatic equilibrium (MHS) instead of vertical hydrostatic equilibrium (HE). Methods. We employed a recently developed Stokes inversion code that, along with non-local thermodynamic equilibrium effects, considers MHS instead of HE. This code is applied to spatially and temporally resolved spectropolarimetric observations of the quiet Sun obtained with the CRISP instrument attached to the Swedish Solar Telescope. Results. We provide average models for granules, intergranules, dark magnetic elements, and overall quiet-Sun as a function of both τc and z from the photosphere to the lower chromosphere. Conclusions. We demonstrate that, in these quiet-Sun models, the effect of considering MHS instead of HE is negligible. However, employing MHS increases the consistency of the inversion results before averaging. We surmise that in regions with stronger magnetic fields (i.e. pores, sunspots, network) the benefits of employing the magneto-hydrostatic approximation will be much more palpable.

The focus of this investigation is to quantify the conversion of magnetic to thermal energy initiated by a quiet Sun cancellation event and to explore the resulting dynamics from the interaction of the opposite-polarity magnetic features. We used imaging spectroscopy in the Hα line, along with spectropolarimetry in the Fe I 6173 Å and Ca II 8542 Å lines from the Swedish Solar Telescope (SST) to study a reconnection-related cancellation and the appearance of a quiet Sun Ellerman bomb (QSEB). We observed, for the first time, QSEB signature in both the wings and core of the Fe I 6173 Å line. We also found that, at times, the Fe I line-core intensity reaches higher values than the quiet Sun continuum intensity. From FIRTEZ-dz inversions of the Stokes profiles in Fe I and Ca II lines, we found enhanced temperature, with respect to the quiet Sun values, at the photospheric (log τ_c  = −1.5; ∼1000 K) and lower chromospheric heights (log τ_c  = −4.5; ∼360 K). From the calculation of total magnetic energy and thermal energy within these two layers, it was confirmed that the magnetic energy released during the flux cancellation can support heating in the aforesaid height range. Further, the temperature stratification maps enabled us to identify cumulative effects of successive reconnection on temperature pattern, including recurring temperature enhancements. Similarly, Doppler velocity stratification maps revealed impacts on plasma flow pattern, such as a sudden change in the flow direction. 

In this work, we study the accuracy that can be achieved when inferring the atmospheric information from realistic numerical magnetohydrodynamic simulations that reproduce the spatial resolution we will obtain with future observations made by the 4 m class telescopes DKIST and EST. We first study multiple inversion configurations using the SIR code and the Fe i transitions at 630 nm until we obtain minor differences between the input and the inferred atmosphere in a wide range of heights. Also, we examine how the inversion accuracy depends on the noise level of the Stokes profiles. The results indicate that when the majority of the inverted pixels come from strongly magnetised areas, there are almost no restrictions in terms of the noise, obtaining good results for noise amplitudes up to 1 x 10(-3) of I-c. At the same time, the situation is different for observations where the dominant magnetic structures are weak, and noise restraints are more demanding. Moreover, we find that the accuracy of the fits is almost the same as that obtained without noise when the noise levels are on the order of 1 x 10(-4) of I-c. We, therefore, advise aiming for noise values on the order of or lower than 5 x 10(-4) of I-c if observers seek reliable interpretations of the results for the magnetic field vector reliably. We expect those noise levels to be achievable by next-generation 4m class telescopes thanks to an optimised polarisation calibration and the large collecting area of the primary mirror.

Context. Electric currents play an important role in the energy balance of the plasma in the solar atmosphere. They are also indicative of non-potential magnetic fields and magnetic reconnection. Unfortunately, the direct measuring of electric currents has traditionally been riddled with inaccuracies.

Aims. We study how accurately we can infer electric currents under different scenarios.

Methods. We carry out increasingly complex inversions of the radiative transfer equation for polarized light applied to Stokes profiles synthesized from radiative three-dimensional magnetohydrodynamic (MHD) simulations. The inversion yields the magnetic field vector, B, from which the electric current density, j, is derived by applying Ampere’s law.

Results. We find that the retrieval of the electric current density is only slightly affected by photon noise or spectral resolution. However, the retrieval steadily improves as the Stokes inversion becomes increasingly elaborated. In the least complex case (a Milne-Eddington-like inversion applied to a single spectral region), it is possible to determine the individual components of the electric current density (j_x, j_y, j_z) with an accuracy of σ = 0.90 − 1.00 dex, whereas the modulus (∥j∥) can only be determined with σ = 0.75 dex. In the most complicated case (with multiple spectral regions, a large number of nodes, Tikhonov vertical regularization, and magnetohydrostatic equilibrium), these numbers improve to σ = 0.70 − 0.75 dex for the individual components and σ = 0.5 dex for the modulus. Moreover, in regions where the magnetic field is above 300 gauss, ∥j∥ can be inferred with an accuracy of σ = 0.3 dex. In general, the x and y components of the electric current density are retrieved slightly better than the z component. In addition, the modulus of the electric current density is the best retrieved parameter of all, and thus it can potentially be used to detect regions of enhanced Joule heating.

Conclusions. The fact that the accuracy does not worsen with decreasing spectral resolution or increasing photon noise, and instead increases as the Stokes inversion complexity grows, suggests that the main source of errors in the determination of electric currents is the lack of realism in the inversion model employed to determine variations in the magnetic field along the line of sight at scales smaller than the photon mean-free path, along with the intrinsic limitations of the model due to radiative transfer effects.