The Virial Tensor and its Application to SelfGravitating Fluids.
Abstract
The we4lknown conditions for the Maclaurin spheroids and the Jacobi ellipsoids are found as consequences of the tensor virial equations. The perturbation forms of the tensor virial equations are derived and applied to the stability of the selfgravitating sphere, the Maclaurin spheroids, and the Jacobi ellipsoids. The effect of rotation on the oscillation frequencies of a selfgravitating sphere is found in detail for secondorder harmonic deformations of the equilibrium shape. To first order in the angular speed , the five oscillation frequencies associated with the five secondorder harmonics are (y  , (To  o, (To + 31 and oO + , where (To is the Kelvin frequency of the nonrotating, spherical liquid mass. Two cn,tical values of the eccentricity are found for the Maclaurin opheroids. The first, e = 0.8127, marks the point where the series of Maclaurin opheroids has a member in common with the Jacobi ellipsoids. The second, e = 0.9529, is the place where the Maclaurin opheroids become unstable. The effect of certain special perturbations of the Maclaurin spheroids with these values of the eccentricity is discussed.
 Publication:

The Astrophysical Journal
 Pub Date:
 September 1961
 DOI:
 10.1086/147175
 Bibcode:
 1961ApJ...134..500L