On the stability of a self-gravitating inhomogeneous fluid in the form of two confocal ellipsoids carrying Dedekind-type internal currents

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 ellipsoids carrying internal currents of differential vorticity, which allow their equilibrium. These Dedekind-type figures result because some of the members of a series of in-homogeneous rotating spheroids have null frequencies, from which they bifurcate in sequences of fixed ε, the body's relative density. We find that such sequences have each an instability regime, which is wide at low ε, and becomes gradually narrower as ε increases. Instability persists - even for very large ε - at the final portion of the sequences, where the figures whose internal ellipsoid has the most prominent equatorial flattening are located.

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