The location of the last stable orbit in Kerr spacetime

2019/12/16 by Leo C. Stein, Niels Warburton · 1 voice · 80 citations
Mathematics · Physics and Astronomy · #Angular momentum #Astrophysical Phenomena and Observations #Black Holes and Theoretical Physics #Classical mechanics #Computer science #Geometry #Mathematics #Orbit (dynamics) #Parameter space #Perturbation (astronomy) #Physics #Plasma #Polar #Pulsars and Gravitational Waves Research #Quantum mechanics #Rotating black hole #Separatrix #Space (punctuation) #Spacetime #Speedup #astro-ph.HE #gr-qc #hep-th

paper · pdf · doi:10.1103/physrevd.101.064007

published in Physical review. D/Physical review. D. 101(6) (American Physical Society) · 11+4 pages, 6 figures, 1 ancillary Mathematica file. v2: Minor edits, matched version published in PRD. Code available at https://bhptoolkit.org/

arxiv created 2020/03/05 · openalex publication_date 2020/03/05 · arxiv updated 2020/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Black hole spacetimes, like the Kerr spacetime, admit both stable and plunging orbits, separated in parameter space by the separatrix. Determining the location of the separatrix is of fundamental interest in understanding black holes, and is of crucial importance for modeling extreme mass-ratio inspirals. Previous numerical approaches to locating the Kerr separatrix were not always efficient or stable across all of parameter space. In this paper we show that the Kerr separatrix is the zero set of a single polynomial in parameter space. This gives two main results. First, we thoroughly analyze special cases (extreme Kerr, polar orbits, etc.), finding strict bounds on the limits of roots, and unifying a number of results in the literature. Second, we pose a stable numerical method which is guaranteed to quickly and robustly converge to the separatrix. This new approach is implemented in the Black Hole Perturbation Toolkit, and results in a ~45x speedup over the prior robust approach.

Cited by

• Actions of spinning compact binaries: Spinning particle in Kerr matched to dynamics at 1.5 post-Newtonian order
• Spherical orbits around a Kerr black hole
• Particles with precessing spin in Kerr spacetime: analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane
• Deformed Schwarzschild horizons in second-order perturbation theory
• Eccentricity distribution of extreme mass ratio inspirals
• Post-Newtonian expansion of gravitational energy and angular momentum fluxes: Inclined spherical orbits about a Kerr black hole
• Gravitational Wave Signatures of Quasi-Periodic Eruptions: LISA Detection Prospects for RX J1301.9+2747
• Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit
• Spherical inspirals of spinning bodies into Kerr black holes
• Radii of spherical timelike geodesics in Kerr-Newman black holes
• Searching for extreme mass ratio inspirals in LISA: from identification to parameter estimation
• Circular equatorial orbits of extended bodies with spin-induced quadrupole around a Kerr black hole: Comparing spin-supplementary conditions
• Rapid determination of LISA sensitivity to extreme mass ratio inspirals with machine learning
• Assessment of effective-one-body radiation reactions for generic planar orbits

Discussions

• If you see this, post a random fact about yourself: I discovered the Kerr separatrix polynomial arxiv.org/abs/1912.07609 [bsky, 43 points, 2 comments]

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