The Relativistic Roche Problem. I. Equilibrium Theory for a Body in Equatorial, Circular Orbit around a Kerr Black Hole

Abstract

Bodies in orbit around Kerr black holes will experience tidal gravitational forces. A study of the equation of geodesic deviation, referred to reference frames fixed in such bodies, reveals the nature of these forces. We here perform this analysis for all circular, equatorial, geodesic orbits of the Kerr metric whose mass parameter M and angular-momentum parameter Ma satisfy a < M (geometrical units). Free-particle motion, which the deviation equation describes, is simple harmonic in the direction perpendicular to the equatorial plane, and we demonstrate the relation between the angular of this simple harmonic motion and the circular orbit frequency. The equation for free-particle motion leads to a generalized Euler equation that describes fluid flow in the orbiting reference frames. We employ the latter to study the conditions for equilibrium of infinitesimal, incompressible, homogeneous, self-gravitating fluid bodies in circular, equatorial, geodesic motion. Because the potential in the equation depends only quadratically on the local coordinates, ellipsoidal figures of equilibrium are possible. For each possible orbit determined by the Kerr angular momentum Ma and the orbit radius r, a family of ellipsoids parametrized by the fluid density p is in equilibrium. A minimum possible density characterizes each family; this is the Roche-limiting density for the orbit. For fluid bodies in stable circular orbits, the Roche limit is qualitatively like that in Newtonian situations; at the last stable, circular, equatorial orbit of each Kerr metric, where r is a specified function of M and of a, equilibrium bodies must satisfy [M/(irpr3)] < 0.0664. For fluid bodies in the highly energetic, unstable circular orbits near the photon orbits, the fluid density required for the existence of a body in equilibrium is magnified by the square of its energy-at-infinity per unit mass. Subject headings: binaries - black holes - gravitation - hydrodynamics - relativity