Chromospheric and photospheric properties of sunspots as inferred from Stokes inversions under magnetohydrostatic and non-local thermodynamic equilibrium

¹ Institut für Sonnenphysik, Georges-Köhler-Allee 401A, 79110 Freiburg, Germany
² Institute for Solar Physics, Department of Astronomy, Stockholm University, AlbaNova University center, 10691 Stockholm, Sweden
³ Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
⁴ ETH Zürich, Institute for Particle Physics and Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
⁵ Istituto ricerche solari Aldo e Cele Daccò (IRSOL), Faculty of Informatics, Università della Svizzera italiana, CH-6605 Locarno, Switzerland

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 2 February 2026
Accepted: 9 March 2026

Abstract

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.