Solar Phys (2011) 272:243–255 DOI 10.1007/s11207-011-9826-2 Estimating the Relative Helicity of Coronal Magnetic Fields J.K. Thalmann · B. Inhester · T. Wiegelmann Received: 13 April 2011 / Accepted: 16 July 2011 / Published online: 1 September 2011 © Springer Science+Business Media B.V. 2011 Abstract To quantify changes of the solar coronal field connectivity during eruptive events, one can use magnetic helicity, which is a measure of the shear or twist of a current-carrying (non-potential) field. To find a physically meaningful quantity, a relative measure, giving the helicity of a current-carrying field with respect to a reference (potential) field, is often evaluated. This requires a knowledge of the three-dimensional vector potential. We present a method to calculate the vector potential for a solenoidal magnetic field as the sum of a Laplacian part and a current-carrying part. The only requirements are the divergence free- ness of the Laplacian and current-carrying magnetic field and the sameness of their normal field component on the bounding surface of the considered volume. Keywords Helicity, magnetic · Magnetic fields, corona 1. Introduction During solar eruptions like flares or coronal mass ejections, part of the previously stored magnetic energy is transformed into other forms of energy, such as kinetic energy or heat. Along with this energy release, the corona often undergoes changes in the field line con- nectivity. A favorable way of quantifying these topological changes of magnetic fields is to investigate the magnetic helicity, which is a measure of the shear or twist of a current- carrying (non-potential) field. The magnetic helicity is (almost) conserved in (resistive) ideal magnetohydrodynamics since its dissipation time is long compared to the time during which energy is dissipated. The major change of coronal helicity is assumed to be due to upward convection from below the solar surface and advection into interplanetary space by coronal mass ejections. Hence, the coronal helicity content may be completely determined by the flow of helicity through the photosphere and its loss rate through the solar wind. J.K. Thalmann ( ) · B. Inhester · T. Wiegelmann Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany e-mail: thalmann@mps.mpg.de