[Paper Review] Precisely computing bound orbits of spinning bodies around black holes II: Generic orbits

Lisa V. Drummond, Scott A. Hughes|arXiv (Cornell University)|Jan 31, 2022

Pulsars and Gravitational Waves ResearchPhysics and Astronomy56 references39 citations

TL;DR

This paper presents a frequency-domain method to precisely compute bound orbits of spinning test bodies around Kerr black holes in fully generic configurations—eccentric, inclined, and with arbitrarily oriented spins. By extending a geodesic-based framework to include spin-curvature forces, it computes shifts in orbital frequencies (Ωr, Ωθ, Ωφ) due to spin, enabling accurate modeling of post-geodesic effects critical for extreme mass-ratio inspiral (EMRI) gravitational waveforms.

ABSTRACT

In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Non-spinning test bodies follow geodesics of the spacetime in which they move. A test body's spin couples to the curvature of that spacetime, introducing a "spin-curvature force" which pushes the body's worldline away from a geodesic trajectory. The spin-curvature force is an important example of a post-geodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such effects in models of gravitational waves produced from the inspiral of stellar mass bodies into massive black holes. In this paper's predecessor, we describe a technique for computing bound orbits of spinning bodies around black holes with a frequency-domain description which can be solved very precisely. In that paper, we present an overview of our methods, as well as present results for orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is no more than $\mathcal{O}(S)$ out of the equatorial plane). In this paper, we apply this formulation to the fully generic case -- orbits which are inclined and eccentric, with the small body's spin arbitrarily oriented. We compute the trajectories which such orbits follow, and compute how the small body's spin affects important quantities such as the observable orbital frequencies $\Omega_r$, $\Omega_ heta$ and $\Omega_\phi$.

Motivation & Objective

• To model the motion of spinning test bodies in generic orbits around Kerr black holes, including eccentric, inclined, and non-equatorial trajectories.
• To quantify how spin-curvature forces modify orbital frequencies Ωr, Ωθ, and Ωφ relative to geodesic orbits.
• To develop a frequency-domain framework that incorporates spin precession and orbital coupling for arbitrary spin orientations.
• To enable high-precision modeling of post-geodesic effects essential for accurate gravitational waveforms in extreme mass-ratio inspirals (EMRIs).
• To provide a reference geodesic framework with adjustable turning points to isolate spin-induced orbital shifts.

Proposed method

• Uses a frequency-domain expansion of orbital functions with harmonics in radial (Ωr), polar (Ωθ), and spin precession (Ωs) frequencies.
• Applies a Fourier decomposition of the spin-curvature force f^α_S in terms of Ωr = ˆΩr + ΩS_r, Ωθ = ˆΩθ + ΩS_θ, and Ωs.
• Defines a reference geodesic with fixed semi-latus rectum p, eccentricity e, and inclination I to anchor orbital parameters.
• Computes shifts in orbital frequencies (ΩS_r, ΩS_θ) relative to the reference geodesic to isolate spin effects.
• Solves the Mathisson-Papapetrou equations in the frequency domain using a perturbative approach in the small mass ratio ε = µ/M.
• Utilizes a parameterization of radial and polar motion via relativistic true anomaly angles ˆχr and ˆχθ for geodesic reference orbits.

Experimental results

Research questions

• RQ1 How do spin-curvature forces modify the fundamental orbital frequencies Ωr, Ωθ, and Ωφ in generic, non-equatorial, eccentric orbits?
• RQ2 What is the structure of the orbital trajectory when the small body’s spin is misaligned with the orbital plane or black hole spin axis?
• RQ3 How do the radial and polar turning points of spinning-body orbits compare to those of a reference geodesic with the same p and e?
• RQ4 To what extent does spin precession introduce new harmonic content (via Ωs) in the orbital dynamics?
• RQ5 How can a frequency-domain framework be systematically extended to include spin effects beyond the equatorial limit?

Key findings

• The method successfully computes bound orbits of spinning bodies in fully generic configurations, including inclined and eccentric orbits with arbitrary spin orientation.
• Spin-curvature forces induce measurable shifts in orbital frequencies: ΩS_r and ΩS_θ are computed relative to a reference geodesic with the same p and e.
• The radial and polar motions of spinning bodies remain bounded between the same turning points as the reference geodesic (pM/(1±e) and ±sin I), despite orbital deformation.
• The spin precession frequency Ωs introduces a new harmonic structure in the orbital Fourier expansion, with j ∈ {−1, 0, 1} for spin harmonics.
• The framework enables high-precision computation of post-geodesic corrections essential for modeling EMRI waveforms with sub-cyclical accuracy.
• The approach generalizes prior work on nearly equatorial orbits to the full generic case, providing a foundation for waveform modeling in LISA-era data analysis.