One of the most important geodesics in a black hole spacetime is the marginally bound spherical orbit. This critical geodesic represents the innermost spherical orbit which is bound to the central black hole. The radii $r_{\mathrm{mb}}(\overline{a})$ of the marginally bound equatorial circular geodesics of rotating Kerr black holes were found analytically by Bardeen et al. more than four decades ago (here $\overline{a}\equiv J/M^2$ is the dimensionless angular momentum of the black hole). On the other hand, no closed-form formula exists in the literature for the radii of generic (nonequatorial) marginally bound geodesics of the rotating Kerr spacetime. In the present study, we analyze the critical (marginally bound) orbits of rapidly rotating Kerr black holes. In particular, we derive a simple analytical formula for the radii $r_{\mathrm{mb}}(\overline{a}\simeq 1;\cos i)$ of the marginally bound spherical orbits, where $\cos i$ is an effective inclination angle (with respect to the black hole equatorial plane) of the geodesic. We find that the marginally bound spherical orbits of rapidly rotating black holes are characterized by a critical inclination angle, $\cos i=\sqrt{2/3}$, above which the coordinate radii of the geodesics approach the black hole radius in the extremal $\overline{a}\to 1$ limit. It is shown that this critical inclination angle signals a transition in the physical properties of the orbits: in particular, it separates marginally bound spherical geodesics which lie a finite proper distance from the black hole horizon from marginally bound geodesics which lie an infinite proper distance from the horizon.

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