Separable solutions of force-free spheres and applications to solar active regions

Abstract

In this paper, we present a systematic study of the force-free field equation for simple axisymmetric configurations in spherical geometry and apply it to the solar active regions. The condition of separability of solutions in the radial and angular variables leads to two classes of solutions: linear and non-linear force-free fields. We have studied these linear solutions Chandrasekhar (1956) and extended the non-linear solutions given in Low & Lou (1990) for the radial power law index to the irreducible rational form $n= p/q$, which is allowed for all cases of odd $p$ and cases of $q>p$ for even $p$ (the poloidal flux $\psi\propto1/r^n$ and field $\mathbf{B}\propto 1/r^{n+2}$). We apply these solutions to simulate photospheric vector magnetograms obtained using the spectro-polarimeter onboard Hinode. The effectiveness of our search strategy is first demonstrated on test inputs of dipolar, axisymmetric and non-axisymmetric linear force-free fields. Using the best fit to these magnetograms, we build 3D axisymmetric field configurations and calculate the energy and relative helicity with two independent methods which are in agreement. We have analyzed 5 magnetograms for AR 10930 spanning a period of 3 days during which two X class flares occurred which allowed us to find the free energy and relative helicity of the active region before and after the flare; our analysis indicates a peak in these quantities before the flare events which is consistent with the results mentioned in literature. We also analyzed single polarity regions AR 10923 and 10933 which showed very good fits with potential fields. This method can provide useful reconstruction of the non-linear force-free (NLFF) fields as well as reasonably good input fields for other numerical techniques.