Abstract
Energetic events such as solar flares and coronal mass ejections (CMEs), which can lead to solar storms, are driven by the coronal magnetic field (CMF), whose structure and evolution remain not fully understood. When Earth-directed, these storms can trigger auroras - as observed in Portugal in 2024 - but also pose serious risks to radio communications, GPS systems, power grids, and satellite infrastructure.
Under certain conditions, Extreme Ultraviolet (EUV) observations reveal useful information about the 3D geometry of some magnetic field lines, as the emitting plasma is “frozen into” the magnetic field, though they do not provide measurements of the magnetic field. Unlike the solar surface - the photosphere - where the magnetic field can be measured via spectropolarimetry, this is generally not achievable in the typically force-free corona, due to the faintness and thermal broadening of spectral lines. As a result, extrapolation methods are used to infer the coronal magnetic field from routine photospheric measurements.
The most widely used model, the Potential Field Source Surface (PFSS), is fast and computationally efficient, but it calculates a current-free, minimal-energy coronal field using the low measurement uncertainty photospheric radial magnetic field component. This limits its accuracy in active regions, where magnetic free energy - critical for flares and CMEs - is stored. More advanced, state-of-the-art Non-linear Force-Free Field (NLFFF) models allow for electric currents and, therefore, free energy, leading to greater accuracy, but they are computationally intensive and highly sensitive to data quality.
We developed a significantly faster Python code built upon a functional optimization framework previously proposed and implemented by our team. In this new version, we introduce a three-term functional that simultaneously minimizes: (1) the angle between the magnetic field and the tangents to observed EUV loops, (2) the divergence of the magnetic field, and (3) the Lorentz force. Including the Lorentz-force term enables our method to control the degree of force-freeness, an essential physical property typically accessible only to the more computationally demanding NLFFF models.
By minimizing the proposed functional, we derive the perturbations that are iteratively applied to the original PFSS solution. The resulting magnetic field represents a trade-off between alignment with EUV loops, solenoidality (divergence-freeness), and force-freeness, yielding a more physically realistic configuration.
This approach retains the computational efficiency of PFSS while significantly improving the physical consistency of the solution. Validation against EUV observations confirms the method’s ability to produce magnetic field solutions that are more accurate and observationally constrained, providing a new, efficient, and reliable tool for coronal magnetic field studies.