On the oscillations and the stability of rotating gaseous masses

Abstract

In this paper the oscillations and the stability of a rotating gaseous mass are considered on the basis of an appropriate tensor form of the virial theorem, On the assumption that the Lagrangian displacement ξ can be expressed in the form ξ_j = X_jrx_r e^λt (X_jr constants), a characteristic equation for λ (of order eighteen) is derived from the nine integral relations provided by the virial theorem, An examination of the roots of this characteristic equation enables the enumeration of the properties of all the natural modes of oscillation belonging essentially to harmonics not higher than the second. It is shown that there are three principal groups among these modes: a group of three modes, each of which exhibits a doublet character; a group of two modes, one of which becomes neutral at a point where the condition for the occurrence of a point of bifurcation is satisfied and both of which become over stable at a higher angular velocity; and a group which represents the coupling of two modes, one of which is purely radial and the other of which is purely non-radial in the absence of rotation. In addition to these modes, there are two "trivial" modes, one of which is neutral and the other of which has a characteristic frequency equal to the angular velocity.