Oscillations of a polytrope with a toroidal magnetic field

Abstract

The radial and the non-radial (l=2) modes of oscillation of a gaseous polytrope with a toroidal magnetic field are examined using a variational principle. It is found that the frequencies of oscillation of the radial mode and the Kelvin mode (l=2) decrease due to the presence of the magnetic field. The shift in the frequency of the Kelvin mode may be split up into two parts, viz. the shift in frequency due to the magnetic field on the unperturbed sphere [(σ1²)M, say] and the shift in frequency due to the distortion of the structure by the magetic field [(σ1²)s, say]. In the first order calculations using one parameter trial function, it is found that (σ1²)M, is indeed positive but is overweighed by a meters, we find that the general behaviour of (σ1²)M and (σ1²)s is unchanged but that (σ1²)M becomes negative for polytropic indices n ≥ 1.5. In Appendix I we study the effect of a small rotation and toroidal magnetic field on the structure of a polytrope. It is found that the resulting configuration is a prolate spheroid, a sphere or an oblate spheroid according as T ≥≤ q m respectively. Here M denotes the magnetic energy and T the kinetic energy due to rotation and q is a constant which depends on the polytropic index n. The values of q are given in Table I.