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

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.