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The physical and mathematical properties of the non-linearly coupledblack-hole-orbiting-ring system are studied analytically to second order in thedimensionless angular velocity $M_{\text{ir}}\omega_{\text{H}}$ of theblack-hole horizon (here $M_{\text{ir}}$ is the irreducible mass of the slowlyrotating central black hole). In particular, we determine analytically, tofirst order in the dimensionless ring-to-black-hole mass ratio$m/M_{\text{ir}}$, the shift $\Delta\Omega_{\text{mb}}/\Omega_{\text{mb}}$ inthe orbital frequency of the {\it marginally bound} circular geodesic thatcharacterizes the composed curved spacetime. Interestingly, our analyticalresults for the frequency shift $\Delta\Omega_{\text{mb}}$ in the composedblack-hole-orbiting-ring toy model agree qualitatively with the recentlypublished numerical results for the corresponding frequency shift in thephysically related (and mathematically much more complex)black-hole-orbiting-particle system. In particular, the present analysisprovides evidence that, at order $O(m/M_{\text{ir}})$, the recently observedpositive shift in the angular frequency of the marginally bound circular orbitis directly related to the physically intriguing phenomenon of dragging ofinertial frames by orbiting masses in general relativity.

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