Magnetic Energy and Helicity Budgets in the Active Region Solar Corona. I. Linear Force-Free Approximation

Expanding on an earlier work that relied on linear force-free (LFF) magnetic fields, we self-consistently derive the instantaneous free magnetic energy and relative magnetic helicity budgets of an unknown three-dimensional nonlinear force-free (NLFF) magnetic structure extending above a single known lower-boundary magnetic field vector. The proposed method does not rely on the detailed knowledge of the three-dimensional field configuration but is general enough to employ only a magnetic connectivity matrix on the lower boundary. The calculation yields a minimum free magnetic energy and a relative magnetic helicity consistent with this free magnetic energy. The method is directly applicable to photospheric or chromospheric vector magnetograms of solar active regions. Upon validation, it basically reproduces magnetic energies and helicities obtained by well known, but computationally more intensive and non-unique, methods relying on the extrapolated three-dimensional magnetic field vector. We apply the method to three active regions, calculating the photospheric connectivity matrices by means of simulated annealing, rather than a model-dependent NLFF extrapolation. For two of these regions we correct for the inherent LFF overestimation in free energy and relative helicity that is larger for larger, more eruptive, active regions. In the third region studied, our calculation can lead to a physical interpretation of observed eruptive manifestations. We conclude that the proposed method, including the proposed inference of the magnetic connectivity matrix, is practical enough to contribute to a physical interpretation of the dynamical evolution of solar active regions.