Non-radial oscillations and convective instability of gaseous masses

Abstract

Modes of non-radial oscillation of gaseous masses belonging to spherical harmonics of orders l = 1 and 3 are considered on the basis of the first- and the third-order virial equations. For an assumed Lagrangian displacement ξ of the form ξ_i = (L_i;jkx_jx_k + L_i)e^iσt (where L_i;jk and L_i, represent a total of twenty-one unspecified constants and σ is the characteristic frequency to be determined), the theory predicts the occurrence of modes of oscillation of two different types: modes (belonging to 1 = 3) which are analogous to the Kelvin modes of an incompressible sphere and modes (belonging to 1 = 1) which are analogous to those discovered by Pekeris for a homogeneous compressible sphere and which exhibit its convective instability. For the latter modes, the virial equations lead to a characteristic equation for σ² of degree 2 whose coefficients are integrals over the variables of the unperturbed configuration, including its superpotential. The theory is applied to the polytropic gas spheres, and it is shown that they are convectively unstable (for the modes belonging to 1 = 1) if the ratio of the specific heats γ is less than a certain critical value The critical values of γ predicted by the (approximate) theory differ from 1 + l/n (where n is the polytropic index) by less than 1 per cent over the range of n (≤3.5) considered; the extent of this agreement is a measure of the accuracy of the method based on the virial equations and the assumed form of the Lagrangian displacement.