Particles with precessing spin in Kerr spacetime: analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane

TL;DR

This work provides analytic, first-order-in-spin solutions for the motion of a spinning test particle in Kerr spacetime, focusing on nearly equatorial orbits and separating the dynamics into planar geodesic motion plus small polar corrections. By exploiting spin gauges, it derives closed-form or quadrature expressions for both periodic and homoclinic trajectories, with a novel fixed eccentricity (FE) gauge yielding finite spin corrections at the separatrix. The FE gauge enables the first closed-form spin corrections to homoclinic orbits and a clean determination of the separatrix shift, aligning with and extending previous numerical results. These results have practical implications for modeling inspiral and transition-to-plunge phases in spinning EMRI binaries within a two-time-scale framework, and offer a pathway to quasi-spherical extensions and quadrupole effects.

Abstract

We present a family of analytic solutions for the nearly-equatorial motion of a test particle with precessing spin in Kerr spacetime. We solve the equations of motion up to linear order in the small body's spin for periodic and homoclinic orbits. At zero order, the particle moves along equatorial geodesics. The spin-curvature force introduces post-geodesic corrections which, for generic spin orientations, cause the precession of the orbital plane. We derive the solutions for eccentric orbits in terms of Legendre elliptic integrals and Jacobi elliptic functions for generic referential geodesics (known as ``spin gauges"). Our analytical solutions perfectly match the numerical trajectories obtained by Drummond and Hughes in Phys. Rev. D 105, 124041 (2022), and Piovano et al. in Phys. Rev. D 111, 044009 (2025). Furthermore, we present, for the first time, the solutions for homoclinic orbits for a spinning particle in Kerr spacetime, and the spin-corrections to the location of the separatrix. The homoclinic trajectories are described in closed form using elementary functions. Finally, we introduce a novel parametrization for the motion of a spinning particle, called ``fixed eccentricity spin gauge". This is the only spin gauge in which the corrections to periodic orbits are finite at the geodesic separatrix, and continuously reduce to the last stable orbits under appropriate limits. Our results will be useful for modeling the inspiral and transition-to-plunge phases of asymmetric mass binaries within the two-time-scale framework.

Figures (5)

• Figure 1: Spin corrections to radial (left panel), coordinate time (middle panel) and azimuthal trajectories (right panel) for eccentric, prograde orbits for a spinning binary with $a =0.9$, $\chi_\parallel = 1/2$, $\chi_\perp = \sqrt{3}/2$. All shifts to the trajectories are computed in the FE gauge. Orbital parameters of the fiducial geodesic: $e_{\rm g} =0.7$, $p_{\rm g} = p^*_{\rm g} + 1/10$, with $p^*_{\rm g}$ the location of the geodesic separatrix at $e_{\rm g} = 0.7$.
• Figure 2: Spin corrections to the location of the separatrix $\delta p^*$ for different orbital parameters. Top panel: $\delta p^*$ as a function of the primary spins $a$ for fixed geodesic eccentricity $e_{\rm g} =0.7$. Bottom panel: $\delta p^*$ for different values of geodesic eccentricities $e_{\rm g}$ and primary spin $a=0.9$.
• Figure 3: Spin corrections to radial (left panel), coordinate time (middle panel) and azimuthal trajectories (right panel) for homoclinic, prograde orbits for a spinning binary with $a =0.9$, $\chi_\parallel = 1/2$, $\chi_\perp =\sqrt{3}/2$. All shifts to the trajectories are computed in the FE gauge. The underlying fiducial geodesic has eccentricity $e_{\rm g} =0.25$.
• Figure 4: Plots of zoom-whirl orbits Glampedakis:2002ya for a spinning particle (red-line) and its underlying fiducial geodesic (purple, dashed line). The orbital parameters and the linear-in-spin corrections are the same of Fig. \ref{['fig:shift_periodic_orbit']}, while the mass-ratio is fixed to $q = 1/100$. The radius of the black disc corresponds to the SMBH outer horizon $r_+ = 1+\sqrt{1 -a^2}$. Top panel: projection of the orbits onto the equatorial plane. Bottom panel: orthogonal projection of the orbits onto a co-rotating polar plane.
• Figure 5: Homoclinic orbits for a spinning particle (red-line) and its underlying fiducial geodesic (purple, dashed line). The orbital parameters and the linear-in-spin corrections are the same of Fig. \ref{['fig:shift_homoclinic_orbit']}, while the mass-ratio is fixed to $q = 1/10$. The radius of the black disc corresponds to the SMBH outer horizon $r_+ = 1+\sqrt{1 -a^2}$. Top panel: projection of the orbits onto the equatorial plane. Bottom panel: orthogonal projection of the orbits onto a co-rotating polar plane.