Bifurcation and stability of uniformly rotating homogeneous ellipsoids surrounded by a massive thin ring

TL;DR

This paper analyzes how a massive ring influences the equilibrium, bifurcation, and stability of rotating ellipsoids, revealing that the ring's gravity can stabilize the spheroid against certain deformations.

Contribution

It introduces a detailed study of equilibrium sequences and bifurcation points of rotating ellipsoids influenced by a surrounding ring, highlighting the stabilizing effect of the ring's gravity.

Findings

Bifurcation points depend on the ring's gravitational parameter.

The ring's gravity can prevent the classical bifurcation at zero angular frequency.

The ring stabilizes the spheroid against bar-shaped deformations.

Abstract

We examine the effects of a massive concentric ring around a spheroid or an ellipsoid with uniform density and uniform rotation. Equilibrium sequences of axisymmetric Maclaurin-like spheroid and triaxial Jacobi-like ellipsoids are obtained. Due to the gravitational field of the ring, Maclaurin-like spheroid does not have a spherical limit when the object's angular frequency vanishes. At a critical value of the eccentricity of the spheroid's meridional section, a triaxial Jacobi-like ellipsoid bifurcates. When a parameter characterizing the gravitational field of the ring is smaller than a threshold, the bifurcation points of Maclaurin-like and Jacobi-like ellipsoids exist and the critical eccentricity is slightly larger than that of the classical Maclaurin-to-Jacobi bifurcation. When the parameter exceeds the threshold, the Maclaurin-like spheroid does not have the bifurcation point and the Jacobi-like ellipsoid appears at the lower eccentricity than the Maclaurin-like spheroid. By comparisons of the energy of the ellipsoids with the same angular momentum, it is shown that the critical point of bifurcation does not correspond to the onset of the secular instability of Maclaurin-like spheroid. It is concluded that the gravitational field of a massive ring surrounding a uniformly rotating spheroid stabilizes it against a bar-shaped deformation due to viscous dissipations.