# Rigidly rotating, incompressible spheroid-ring systems: new bifurcations, critical rotations and degenerate states

## TL;DR

This paper explores the equilibrium configurations of incompressible spheroid-ring systems under rigid rotation, revealing new bifurcations, critical rotation points, and degenerate states through numerical analysis.

## Contribution

It introduces a detailed numerical investigation of spheroid-ring systems, uncovering novel bifurcation sequences, degeneracy bands, and the absence of equilibrium beyond certain rotation thresholds.

## Key findings

- Only detached binary configurations are found, ending in contact states.
- No equilibrium exists for rotation frequencies above a critical value.
- A continuum of bifurcations involves expanding, initially massless loops.

## Abstract

The equilibrium of incompressible spheroid-ring systems in rigid rotation is investigated by numerical means for a unity density contrast. A great diversity of binary configurations is obtained, with no limit neither in the mass ratio, nor in the orbital separation. We found only detached binaries, meaning that the end-point of the $\epsilon_2$-sequence is the single binary state in strict contact, easily prone to mass-exchange. The solutions show a remarkable confinement in the rotation frequency-angular momentum diagram, with a total absence of equilibrium for $\Omega^2/ \pi G \rho \gtrsim 0.21$. A short band of degeneracy is present next to the one-ring sequence. We unveil a continuum of bifurcations all along the ascending side of the Maclaurin sequence for eccentricities of the ellipsoid less than $\approx 0.612$ and which involves a gradually expanding, initially massless loop.