The practical application of the magnetic virial theorem

The magnetic virial theorem states that the magnetic energy contained in a coronal force-free magnetic field is given by a surface integral at the photospheric boundary involving the three vector magnetic field components. It is possible to use this theorem together with vector magnetograph data to compute the magnetic energy of solar active regions. In particular, one might compute the energy of an active region both before and after a flare occurs to determine whether the energy observed to be released during the flare is actually extracted from the magnetic field, as is commonly believed. In order to attach any significance to such a determination, one must understand how errors in the vector magnetograph measurements produce errors in the virial theorem energy. We have therefore numerically simulated the effects of realistic errors on known magnetic fields. These include errors due to random polarization noise, crosstalk between different polarization signals, systematic polarization bias, and seeing-induced crosstalk. We have also derived analytical expressions for the energy errors which apply under certain idealized conditions. Our results serve as a useful tool for evaluating the ability of vector magnetographs to provide suitable data for the accurate determination of magnetic energies using the virial theorem.