On the stability of a self-gravitating inhomogeneous fluid in the form of two confocal spheroids rotating with different angular velocities

The second order virial equations are employed to analyze in a first approximation the stability of a self-gravitating fluid made up of two confocal spheroids with solid-body differential rotation. The equilibrium solution for such a model is known to consist of a series which in the present study is shown to be, almost throughout, stable. On the other hand, the series contains some spheroids of zero frequency, an indication that secondary sequences of tri-axial figures could branch off from them. Such is indeed the case, even if the bifurcated sequences consist exclusively of Dedekind-type figures, as they are static with internal motions of differential vorticity. Anyway, Jacobi-type figures could not beforehand be expected to give out from the model, since it would be inconsistent with Hamy's theorem.

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