The adiabatic stability of stars containing magnetic fields - VI. The influence of rotation

Abstract

Conditions for the stability against adiabatic perturbations of a star containing a purely toroidal magnetic field are established. These conditions are necessary and sufficient for stability, if the change in gravitational field produced by the perturbation is neglected. They remain necessary for stability if the change in gravitational field is included. The criteria are given in closed algebraic form so that they can be applied readily, if an equilibrium model of a star with a toroidal magnetic field is proposed. Because the criteria are complicated, it is difficult to draw general conclusions from them. It is however possible to show that instability must occur close to the axis of symmetry of the star, if there is a non-zero electric current density on the axis. This proves that a large class of configurations are unstable. The occurrence of this instability depends only on the shape of the field and not on its strength, whereas in general the field strength will be important in determining instability. It is not easy to predict the ultimate effect of such instabilities. They have very short growth times compared to characteristic times of stellar evolution and, even if the instability is initially contained as an oscillation of finite amplitude, it may lead to a considerably enhanced decay of the magnetic field. It is unlikely that stars exist with a purely toroidal field, although it may be stronger than the poloidal component. However, models have previously been obtained of stars with only a toroidal field and their oscillation properties have been discussed. Because these discussions by other authors have involved integrals over the whole star, the localized instabilities found in this paper have not been discovered.