Stability of a Riemann S Ellipsoid with a Spheroidal Halo

The stability of Riemann S ellipsoids inside an oblate halo with respect to the second form of oscillations is investigated. It is shown that some ellipsoids with reverse internal circulation of matter, which are stable inside a spherical halo or in its absence, become unstable with respect to second odd forms of oscillation inside an oblate halo. Here there is asymmetry between conjugate ellipsoids from the standpoint of their stability. Only those conjugate ellipsoids that correspond to higher frequencies of reverse circulation of matter than their corresponding angular rotation rates are unstable. The domains of instability of light and heavy conjugate embedded ellipsoids are obtained as a function of the oblateness measure and relative density of the halo.