Equipartition field strength on the sunspot boundary

Statistical study

¹ Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165, Ondřejov, Czech Republic
² Institut für Sonnenphysik (KIS), Georges-Köhler-Allee 401a, 79110, Freiburg, Germany

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Received: 18 December 2025
Accepted: 17 February 2026

Abstract

Context. A recent case study of a long-lived sunspot confirmed earlier observational indications that the outer boundary of sunspots is defined by an invariant value of the magnetic field strength. This property is also found in magnetohydrodynamic sunspot simulations, which show that this invariant value corresponds to the equipartition field strength.

Aims. We investigated a large sample of sunspots throughout their evolution to statistically assess the magnetic field strength at their outer boundaries, determine whether it systematically reaches an invariant value, and evaluate possible dependences on sunspot size and evolutionary phase.

Methods. We automatically processed nearly 1000 active regions observed by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory and identified and tracked 312 unique sunspots. The outer boundary of each sunspot was defined using a continuum-intensity threshold. For all evolutionary stages, we computed the mean magnetic field strength along this boundary and its standard deviation.

Results. Across the sample, the mean magnetic field along the boundary decreases during formation, reaches a minimum during the stable phase, and increases again during decay. For sunspots with a fully developed penumbra, this minimum is remarkably consistent, with a value of 605 ± 27 G. The standard deviation of the magnetic field exhibits a similar temporal evolution, and its minimum provides an additional diagnostic of penumbral maturity: fully developed penumbrae correspond to σ_B ⪅ 200 G, whereas higher values indicate either forming or decaying structures. For sunspots with fully developed penumbrae, the mean B value at their boundaries varies weakly with sunspot size; larger sunspots tend to have stronger B.

Conclusions. The results demonstrate that fully developed sunspots systematically reach an invariant magnetic field strength value, namely the equipartition field strength, at their outer boundary. The temporal evolution of the mean field and its spatial variability along the boundary provides a robust diagnostic of sunspot formation, stability, and decay.