Revisiting the Flowers-Ruderman instability of magnetic stars

TL;DR

This paper revisits the Flowers-Ruderman instability in magnetic stars, providing a formal energy analysis, exploring stabilization by toroidal fields, and clarifying the instability's limitations through perturbation theory.

Contribution

It offers a detailed energy-based analysis of the Flowers-Ruderman instability, including the stabilizing role of toroidal fields and the invalidity of repeated application of the instability.

Findings

The external magnetic energy decreases monotonically during the instability.

A small toroidal field can stabilize the star against the instability.

The instability cannot be applied repeatedly to indefinitely reduce magnetic energy.

Abstract

In 1977, Flowers and Ruderman described a perturbation that destabilises 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 $90\degr$ 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 monotonously along the entire rotation. We also describe the instability using perturbation theory, and see that it happens due to the work done by the interaction of the magnetic field with surface currents. Finally, we consider the stabilising 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 field needed to make the star stable to this displacement and see that the energy of the toroidal field required for stabilisation 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.