Revisiting the Flowers-Ruderman instability of magnetic stars

Abstract

In 1977, Flowers & Ruderman described a perturbation that destabilizes a purely dipolar magnetic field in a fluid star. They considered the effect of cutting the star in half along a plane containing the symmetry axis and rotating each half by 90° in opposite directions, which would cause the energy of the magnetic field in the exterior of the star to be greatly reduced, just as it happens with a pair of aligned magnets. We formally solve for the energy of the external magnetic field and check that it decreases monotonically along the entire rotation. We also describe the instability using perturbation theory, and show that it happens due to the work done by the interaction of the magnetic field with surface currents. Finally, we consider the stabilizing effect of adding a toroidal field by studying the potential energy perturbation when the rotation is not done along a sharp cut, but with a continuous displacement field that switches the direction of rotation across a region of small but finite width. Using these results, we estimate the relative strengths of the toroidal and poloidal fields needed to make the star stable to this displacement and show that the energy of the toroidal field required for stabilization is much smaller than the energy of the poloidal field. We also show that, contrary to a common argument, the Flowers–Ruderman instability cannot be applied many times in a row to reduce the external magnetic energy indefinitely.