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Marginally bound circular orbits in the composed black-hole-ring system

The physical and mathematical properties of the non-linearly coupled black-hole-orbiting-ring system are studied analytically to second order in the dimensionless angular velocity $M_{\text{ir}}ω_{\text{H}}$ of the black-hole horizon (here $M_{\text{ir}}$ is the irreducible mass of the slowly rotating central black hole). In particular, we determine analytically, to first order in the dimensionless ring-to-black-hole mass ratio $m/M_{\text{ir}}$, the shift $ΔΩ_{\text{mb}}/Ω_{\text{mb}}$ in the orbital frequency of the {\it marginally bound} circular geodesic that characterizes the composed curved spacetime. Interestingly, our analytical results for the frequency shift $ΔΩ_{\text{mb}}$ in the composed black-hole-orbiting-ring toy model agree qualitatively with the recently published numerical results for the corresponding frequency shift in the physically related (and mathematically much more complex) black-hole-orbiting-particle system. In particular, the present analysis provides evidence that, at order $O(m/M_{\text{ir}})$, the recently observed positive shift in the angular frequency of the marginally bound circular orbit is directly related to the physically intriguing phenomenon of dragging of inertial frames by orbiting masses in general relativity.