IX - Rotating Fluid Masses

Summary

I PROPOSE in this chapter to sketch in as non-technical language as possible the classical subject of the forms of equilibrium of rotating, gravitating fluid masses and their stability, as it was when Jeans began to make contributions to it, and the nature of Jeans's contributions.

Suppose we consider a mass of incompressible liquid, spinning about an axis and isolated in space. What form will it assume, and how will the form change, if at all, as the mass shrinks? By the theorem of the conservation of angular momentum, as such a mass shrinks, and its moment of inertia consequently decreases, its angular velocity will increase. The problem can therefore be reduced to that of the forms of equilibrium of a rotating homogeneous mass as its angular velocity increases from zero upwards. When the angular velocity is zero, the form is evidently that of a sphere (though this is by no means as easy to prove as it looks). It was shown by Newton, and more particularly by Maclaurin, that as angular velocity sets in, the form is initially that of an ellipsoid of revolution, an oblate spheroid in fact, with the shorter (polar) axis lying along the axis of rotation. As the angular velocity increases, the polar axis shortens and the equatorial axes lengthen. This process goes on until the angular velocity w reaches a certain maximum (given by ω²/2πGρ= 0.2247, corresponding to an eccentricity of meridian section equal to 0.93). For higher values of the angular velocity, no forms of equilibrium of the type of a spheroid are possible. But a second set of spheroidal figures are possible, corresponding to decreasing angular velocity but still increasing eccentricity, until in the limit the form assumed is that of a flat disk of very large radius and zero thickness, rotating very slowly about an axis through its centre normal to its plane.

Not all these configurations of relative equilibrium, however, are stable. The configurations for which ω²/2πGρ has diminished again beyond the value of 0.2247 are all unstable, in the ordinary sense of that word. Thus all the disk-like configurations are unstable. But some of the configurations on the ascending branch of angular velocity are unstable in another sense. This type of instability occurs when dissipative forces are present. It arises in the following way.