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

Abstract

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.

Solar-like oscillations and magnetic activity of the slow rotator EK Eridani

“…One may expect a large scale magnetic field to have a stabilizing effect on the overall shape of the star, i.e., resisting deformations caused by the oscillations. As noted by Braithwaite (2009), a dominantly poloidal field tends to align the magnetic axis perpendicular to the rotation axis, thereby contributing to making the star oblate, even for very slow rotation. With this in mind, the surprisingly low oscillation amplitudeper-radial-mode (cf.…”

Relaxed equilibrium configurations to model fossil fields

“…We then focus on its minimum energy eigenmodes for a given mass and helicity, which are derived and applied to modeling relaxed stellar fossil magnetic fields, which are found to be non forcefree. Arguments in favor of the stability of the obtained configurations are finally discussed (Wright 1973;Tayler 1980;Braithwaite 2009;Reisenegger 2009), and we compare their properties with those of relaxed fields obtained in numerical simulations (Braithwaite 2008). The case of general baroclinic equilibrium states will be studied in Paper II (Wright 1969;Moss 1975).…”

“…Furthermore, since the simplest geometrical configurations, such as purely poloidal and purely toroidal fields are known to be unstable (Acheson 1978;Tayler 1973;Markey & Tayler 1973, 1974Goossens & Veugelen 1978;Goossens & Tayler 1980;Goossens et al 1981;Van Assche et al 1982;Spruit 1999;Braithwaite 2006Braithwaite , 2007, the best candidates for stable geometries are mixed poloidal-toroidal fields (Wright 1973;Markey & Tayler 1974;Tayler 1980;Braithwaite 2009).…”

Stably stratified magnetized stars in general relativity

“…The resulting stable magnetic fields have both poloidal and toroidal components with comparable strength and support the conjecture for stability conditions of the magnetized star given by the classical studies mentioned before. By using the numerical magnetohydrodynamics simulation, Braithwaite [42] studied stability conditions for the magnetized stars and obtained a stability condition for his models given in terms of the ratio of the poloidal magnetic energy to the total magnetic energy which is of order unity. Duez et al showed that magnetized stars constructed in Ref.…”