Homoclinic Orbits around Spinning Black Holes I: Exact Solution for the Kerr Separatrix

Under the dissipative effects of gravitational radiation, black hole binaries will transition from an inspiral to a plunge. The separatrix between bound and plunging orbits features prominently in the transition. For equatorial Kerr orbits, we show that the separatrix is a homoclinic orbit in one-to-one correspondence with an energetically-bound, unstable circular orbit. After providing a definition of homoclinic orbits, we exploit their correspondence with circular orbits and derive exact solutions for them. This paper focuses on homoclinic behavior in physical space, while in a companion paper we paint the complementary phase space portrait. The exact results for the Kerr separatrix could be useful for analytic or numerical studies of the transition from inspiral to plunge.

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