Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits

Abstract

Very large mass ratio binary black hole systems are of interest both as a clean limit of the two-body problem in general relativity, as well as for their importance as sources of low-frequency gravitational waves. At lowest order, the smaller body moves along a geodesic of the larger black hole's spacetime. Accurate models of such systems require post-geodesic corrections to this motion. Post-geodesic effects that drive the small body away from the geodesic include the gravitational self force, which incorporates the backreaction of gravitational-wave emission, and the spin-curvature force, which arises from coupling of the small body's spin to the black hole's spacetime curvature. In this paper, we describe a method for precisely computing bound orbits of spinning bodies about black holes. Our analysis builds off of pioneering work by Witzany which demonstrated how to describe the motion of a spinning body to linear order in the small body's spin. Exploiting the fact that in the large mass-ratio limit spinning-body orbits are close to geodesics (in a sense that can be made precise) and using closed-form results due to van de Meent describing precession of the small body's spin along black hole orbits, we develop a frequency-domain formulation of the motion which can be solved very precisely. We examine a range of orbits with this formulation, focusing in this paper on orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is O(S) out of the equatorial plane), but for which the small body's spin is arbitrarily oriented. We discuss generic orbits with general small-body spin orientation in a companion paper. We characterize the behavior of these orbits, contrasting them with geodesics, and show how the small body's spin shifts the frequencies Ωr and Ω φ which affect orbital motion. These frequency shifts change accumulated phases which are direct gravitational-wave observables, illustrating the importance of precisely characterizing these quantities for gravitational-wave observations.

Tidally-induced nonlinear resonances in EMRIs with an analogue model

“…Generic EMRI orbits, which to zeroth order are bound Kerr geodesics, can be highly eccentric and inclined, and therefore the emitted GWs are expected to encode a rich phenomenology [4]. Geodesic motion in Kerr is integrable [5], so a bound Kerr orbit lies on a phase-space torus characterized by three frequencies: one each associated with the radial, polar, and axial motion [6,7].…”

Motion of a spinning particle under the conservative piece of the self-force is Hamiltonian to first order in mass and spin

“…The dynamics to leading order in spin is obtained by dropping all of the terms quadratic in spin in Eqs. (5), and keeping only the term in the momentum evolution equation (5b) which is linear in spin [2,40]. This term corresponds to the leading spin-curvature coupling effect listed in Table I.…”

“…However, it may yield useful information for the effective one body framework [20,21] (since our calculation validates the Hamiltonian dynamics assumption) and thereby aid waveform modeling for comparable mass binary systems for LIGO. We also note that our anal- First order conservative spin-induced self-force and self-torque 30,39,40] TABLE I. A summary of different conservative effects in the dynamics of binary systems with spinning components in general relativity, in the small mass ratio limit µ ≪ M, where M is the mass of the primary and µ the mass of the secondary.…”

Particle Motion under the Conservative Piece of the Self-Force is Hamiltonian

“…The explicit Hamiltonian we derive may be useful for computations of waveforms for extreme mass ratio inspirals by LISA. In that context, incorporating the spin of the small body will be necessary to obtain accurate waveforms [30][31][32][33]. Spin effects will be comparable to effects that arise from the subleading point particle self-force, assuming that S /µ 2 ∼ 1 for typical compact object sources.…”