This paper gives a complete characterization of the location of resonant orbits in a Kerr spacetime for all possible black hole spins and orbital parameter values. A resonant orbit in this work is defined as a geodesic for which the longitudinal and radial orbital frequencies are commensurate. Our analysis is based on expressing the resonance condition in its most symmetric form using Carlson’s elliptic integrals, which enable us to provide exact results together with a number of concise formulas characterizing the explicit dependence on the system parameters. The locations of resonant orbits identify regions where intriguing observable phenomena could occur in astrophysical situations when various sources of perturbation act on the binary system. Resonant effects may have observable implications for the inspirals of compact objects into a supermassive black hole. During a generic inspiral the slowly evolving orbital frequencies will pass through a series of low-order resonances where the ratio of orbital frequencies is equal to the ratio of two small integers. At these locations rapid changes in the orbital parameters could produce a measurable phase shift in the emitted gravitational and electromagnetic radiation. Resonant orbits may also capture gas or larger objects leading to further observable characteristic electromagnetic emission. According to the Kolmogorov-Arnold-Moser theorem, low-order resonant orbits demarcate the regions where the onset of geodesic chaos could occur when the Kerr Hamiltonian is perturbed. Perturbations are induced for example if the spacetime of the central object is non-Kerr, if gravity is modified, if the orbiting particle has large multipole moments, or if additional masses are nearby. We find that the $1/2$ and $2/3$ resonances occur at approximately 4 and 5.4 Schwarzschild radii ($R_s$) from the black hole’s event horizon. For compact object inspirals into supermassive black holes ($\sim {10}^6{M}_{\odot }$) this region lies within the sensitivity band of space-based gravitational wave detectors such as eLISA. When interpreted within the context of the supermassive black hole at the Galactic center, Sgr A*, this implies that characteristic length scales of $41\mu \mathrm{as}$ and $55\mu \mathrm{as}$ and time scales of 50 min and 79 min respectively should be associated with resonant effects if Sgr A* is nonspinning, while spin decreases these values by up to $\sim 32\%$ and $\sim 28\%$. These length scales are potentially resolvable with radio very-long-baseline interferometry measurements using the Event Horizon Telescope. We find that all low-order resonances are localized to the strong field region. In particular, for distances $r>50R_s$ from the black hole, the order of the resonances is sufficiently large that resonant effects of generic perturbations are not expected to lead to drastic changes in the dynamics. This fact guarantees the validity of using approximations based on averaging to model the orbital trajectory and frequency evolution of a test object in this region. Observing orbital motion in the intermediate region $50R_s<r<1000R_s$ is thus a “sweet spot” for systematically extracting the multipole moments of the central object by observing the orbit of a pulsar—since the object is close enough to be sensitive to the quadruple moment of the central object but far enough away not to be subjected to resonant effects.

• Received 5 February 2015

DOI:https://doi.org/10.1103/PhysRevD.91.083001