Three-body interactions are ubiquitous in astrophysics. For instance, Kozai–Lidov oscillations in hierarchical triple systems have been studied extensively and applied to a wide range of astrophysical systems. However, mildly hierarchical triples also play an important role, but they are less explored. In this work, we consider the secular dynamics of a test particle in a mildly hierarchical configuration. We find the limit within which the secular approximation is reliable when the outer perturber is in a circular orbit. In addition, we present resonances and chaotic regions using surface-of-section plots, and characterize regions of phase space that allow large eccentricity and inclination variations. Finally, we apply the secular results to the outer Solar System. We focus on the distribution of extreme trans-Neptunian objects (eTNOs) under the perturbation of a possible outer planet (Planet 9), and find that in addition to a low-inclination Planet 9, a polar or a counter-orbiting one could also produce pericenter clustering of eTNOs, while the polar one leads to a wider spread of eTNO inclinations.

Mechanisms have been proposed to enhance the merger rate of stellar-mass black hole binaries, such as the Von Zeipel–Lidov–Kozai mechanism (vZLK). However, high inclinations are required in order to greatly excite the eccentricity and to reduce the merger time through vZLK. Here, we propose a novel pathway through which compact binaries could merge due to eccentricity increase in general, including in a near coplanar configuration. Specifically, a compact binary migrating in an active galactic nucleus disk could be captured in an evection resonance, when the precession rate of the binary equals the orbital period around the supermassive black hole. In our study we include precession due to first-order post-Newtonian precession as well as that due to disk around one or both components of the binary. Eccentricity is excited when the binary sweeps through the resonance, which happens only when it migrates on a timescale 10–100 times the libration timescale of the resonance. Libration timescale decreases as the mass of the disk increases. The eccentricity excitation of the binary can reduce the merger timescale by up to a factor of ∼103−5.

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Recent studies have shown that active galactic nucleus (AGN) disks can host sources of gravitational waves. Compact binaries can form and merge in AGN disks through their interactions with the gas and other compact objects in the disk. It is also possible for the binaries to shorten the merging timescale due to eccentricity excitation caused by perturbations from the supermassive black hole (SMBH). In this paper, we focus on effects due to precession-induced (eviction-like) resonances, where the nodal and apsidal precession rates of the binary are commensurable with the mean motion of the binary around the SMBH. We focus on intermediate-mass black hole (IMBH)–stellar-mass black hole binaries and consider binary orbit inclined from the circum-IMBH disk, which leads to the orbital J 2 precession. We show that if a binary is captured in these resonances and is migrating toward the companion, it can undergo large eccentricity and inclination variations. We derive analytical expressions for the location of fixed points, libration timescale, and width for these resonances and identify two resonances in the near-coplanar regime (the evection and eviction resonances) as well as two resonances in the near-polar regime that can lead to mergers. We also derive analytical expressions for the maximum eccentricity that a migrating binary can achieve for given initial conditions. Specifically, the maximum eccentricity can reach 0.9 when captured in these resonances before orbital decay due to gravitational-wave emission dominates, and the capture is only possible for slow migration (∼10 Myr) two to three orders of magnitude longer than the resonance libration timescale. We also show that capture into multiple resonances is possible and can further excite eccentricities.

The spin-orbit misalignments of stellar-mass black hole binaries (BHB) provide important constraints on the formation channels of merging BHBs. Here, we study the spin evolution of a black-hole component in a BHB around a supermassive BH (SMBH) in an AGN disk. We consider the BH's spin-precession due to the J 2 moment introduced by a circum-BH disk within the warping/breaking radius of the disk. We find that the BH's spin-orbit misalignment (obliquity) can be excited via spin-orbit resonance between the BHB's orbital nodal precession and the BH spin-precession driven by the circum-BH disk. Assuming a 10 7 M SMBH, this typically occurs at a distance of 10 2−4 AU to the SMBH or 10 3−5 GM SMBH /c 2 . In many cases, spin-orbit resonance leads to a high BH obliquity, and a broad distribution of the binary components' obliquities and effective spin parameters.