The role of the Lorentz force in sunspot equilibrium

¹ Institut für Sonnenphysik, Georges-Köhler-Allee 401A, D-79110 Freiburg, Germany
² Institute for Solar Physics, Department of Astronomy, Stockholm University, AlbaNova University Centre, 10691 Stockholm, Sweden
³ High Altitude Observatory, NSF National Center for Atmospheric Research, 3080 Center Green Dr., Boulder 80301, USA
⁴ Max Planck Institute for Solar System Research, Justus-von-Liebig 3, D-37077 Göttingen, Germany
⁵ Instituto de Astrofísica de Canarias, Avd. Vía Láctea s/n, E-38205 La Laguna, Spain
⁶ Departamento de Astrofísica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, Spain

^⋆ Corresponding author.

Received: 24 February 2025
Accepted: 27 May 2025

Abstract

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.