A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. III. The Free Energy

摘要

An analytical approximation of a nonlinear force-free magnetic field (NLFFF) solution was developed in Paper I, while a numerical code that performs fast forward-fitting of this NLFFF approximation to a line-of-sight magnetogram and coronal 3D loops has been described and tested in Paper II. Here we calculate the free magnetic energy E _(free)=E _N-E _P, i.e., the difference of the magnetic energies between the non-potential field and the potential field. A second method to estimate the free energy is obtained from the mean misalignment angle change Δμ=μ _P-μ N between the potential and non-potential field, which scales as E _(free)/E _P≈tan~2(Δμ). For the four active regions observed with STEREO in 2007 we find free energies in the range of q _(free)=(E _(free)/E _P)≈1 % - 10 %, with an uncertainty of less than ± 2 % between the two methods, while the free energies obtained from 11 other NLFFF codes exhibit a larger scatter of about ± 10 %. We also find a correlation between the free magnetic energy and the GOES flux of the largest flare that occurred during the observing period, which can be quantified by an exponential relationship, F _(GOES)∝exp(q _(free)/0.015), implying an exponentiation of the dissipated currents.