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 _ { \text { mb } } ( { \bar { 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 \bar { 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 ( non -equatorial ) 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 _ { \text { mb } } ( \bar { 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 \bar { 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 .