On Innermost Stable Spherical Orbits near a rotating black hole: A numerical study of the particle motion near the plunging region

According to General Relativity, astrophysical black holes are described by a small number of parameters. Apart from the mass of the black hole (M), among the most interesting characteristics is the spin (a), which determines the degree of rotation, i.e., the angular momentum of the black hole. The latter is observationally constrained by the spectral and timing properties of the radiation signal emerging from an accretion disk of matter orbiting near the event horizon. In the case of planar (standard, equatorial) accretion disk, it is the location of the Innermost Stable Circular Orbit (ISCO) that determines the observable radiation characteristics, and this way allows us to measure the spin. In this paper, we discuss a more general case of the Innermost Stable Spherical Orbits (ISSO) extending above and below the equatorial plane. To this end, we study the non-equatorial geodesic motion of particles following inclined, spherical, relativistically precessing trajectories with the aim of exploring the boundary between the regions of stable (energetically bound) and escaping (energetically unbound) motion. The concept of the ISSO radius should play a role in determining the inner rim of a tilted or geometrically thick accretion flow. We demonstrate that the region of inclined bound orbits has a complicated structure due to enhanced precession near the inner rim. We also explore the fate of particles launched below the radius of the Marginally Bound Spherical Orbit (MBSO): these may either plunge into the event horizon or escape to radial infinity.