Models of force-free spheres and applications to solar active regions

Abstract

Here we present a systematic study of force-free field equation for simple axisymmetric configurations in spherical geometry. The condition of separability of solutions in 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 nonlinear solutions given in Low & Lou (1990) to the irreducible rational form n = p/q, which is allowed for all cases of odd p and to cases of q > p for even p. We have further calculated their energies in shell geometries using the virial theorem and the energy density integral. The relative helicities are computed independently using the Finn Antonesen formula (Finn & Antonsen 1985) and the Berger formula (Berger 1985) which are analyitically equivalent. We demonstrate here a method here to be used to fit observed magnetograms as well as to provide good exact input fields for testing other numerical codes used in reconstruction on the non-linear force-free fields.