The Effect of A Toroidal Magnetic Field on the Radial Oscillations of Stars

Abstract

The internal structure of a polytrope n = 3 containing a toroidal magnetic field is investigated. For static equilibrium configurations the general solution for the structure of the field is given and a particular solution $$H_t\propto r\rho \enspace\text{sin}\enspace\theta$$ is investigated in detail. The linearized equations for small radial motion about the equilibrium configuration are presented and with a time dependence $$\propto e^{i\sigma t}$$ these equations reduce to an eigenvalue equation for σ ² . A variational principle for determining σ is derived and σ ² is estimated using this principle as well as by direct numerical iteration, for values of the ratio of specific heats of the gas Γ = 4/3, 4/3 + 0.01, and 5/3. Results are given for different field strengths. For Γ = 4/3 the star is neutrally stable whether or not there is a magnetic field, whereas for the other values of Γ the magnetic field decreases the value of σ as compared to the non-magnetic values.