Axisymmetric magnetic fields in stars: relative strengths of poloidal and toroidal components

In this third paper in a series on stable magnetic equilibria in stars, I look at the stability of axisymmetric field configurations and, in particular, the relative strengths of the toroidal and poloidal components. Both toroidal and poloidal fields are unstable on their own, and stability is achieved by adding the two together in some ratio. I use Tayler's stability conditions for toroidal fields and other analytic tools to predict the range of stable ratios and then check these predictions by running numerical simulations. If the energy in the poloidal component as a fraction of the total magnetic energy is written as Ep/E, it is found that the stability condition is a(E/U) < Ep/E≲ 0.8 where E/U is the ratio of magnetic to gravitational energy in the star and a is some dimensionless factor whose value is of order 10 in a main-sequence star and of order 103 in a neutron star. In other words, whilst the poloidal component cannot be significantly stronger than the toroidal, the toroidal field can be very much stronger than the poloidal–given that in realistic stars we expect E/U < 10−6. The implications of this result are discussed in various contexts such as the emission of gravitational waves by neutron stars, free precession and a ‘hidden’ energy source for magnetars.