Going into a tailspin near the abyss: analytic solutions for spinning particles on near equatorial, plunging orbits in Kerr spacetime

Abstract

This work presents, the first time, analytic solutions for the nearly equatorial, plunging motion of a spinning test-particle in Kerr spacetime. The equations of motion are solved at first-order in the small-body spin for all classes of plunging orbits with energy $E < 1$. The solutions incorporate the small precession of the orbital plane caused by the precession of the particle's spin. Additionally, we present the correction to the radius of the innermost bound circular orbit in closed form, and introduce a novel, Keplerian-like parametrization for generic plunging orbits. Our solutions will be useful in the modelling of inspiral-merger-ringdown waveforms with self-force methods and black hole perturbation theory.

Figures (9)

• Figure 1: Spin correction to the IBCO radius $\delta r_\text{ibco}$ for different primary spins.
• Figure 2: Spin corrections to radial (left panel), coordinate time (middle panel) and azimuthal trajectories (right panel) for a prograde homoclinic orbit and prograde critical plunge for a spinning binary with $a =0.9$, $\chi_\parallel = 1/2$, $\chi_\perp =\sqrt{3}/2$. The underlying fiducial geodesic has eccentricity $e_{\rm g} =0.32$. In the middle and right panel, the dashed lines corresponds, from left to right, to the event horizons $r_-$ and $r_+$, and the geodesic double root $r_{2\rm g}$. In all three panels, the shift to homoclinic orbits starts at $r_{1 \rm g}$ and end at $r_{2 \rm g}$.
• Figure 3: Homoclinic orbits and critical plunges 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_critical_plunge']}, while the mass-ratio is fixed to $\epsilon = 1/20$. The two black rings corresponds to the primaryouter horizon $r_+ = 1+\sqrt{1 -a^2}$ and inner 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 4: Spin corrections to radial (left panel), coordinate time (middle panel) and azimuthal trajectories (right panel) for a prograde ISCO plunge for a spinning binary with $a =0.9$, $\chi_\parallel = 1/2$, $\chi_\perp = \sqrt{3}/2$. In the middle and right panel, the dashed lines corresponds, from left to right, to the event horizons $r_-$ and $r_+$, and the geodesic ISCO $r_\text{g,isco}$.
• Figure 5: Plots of ISCO plunges 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_isco_plunge']}, while the mass-ratio is fixed to $\epsilon = 1/20$. The two black rings corresponds to the primaryouter horizon $r_+ = 1+\sqrt{1 -a^2}$ and inner 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.