Four Which I Knew Not: The Hexadecapole Nearly Cancels the Octupole in Librating Kozai-Lidov Cycles
Abstract
Phenomena requiring close proximity between two bodies, from black-hole mergers to hot Jupiters, may originate in hierarchical triples, in which secular torques from a distant companion drive the inner binary's eccentricity nearly to unity. Population syntheses of these channels commonly integrate the double-averaged (DA) secular equations truncated at octupole order. We show that, for a fixed outer orbit, this truncation fails for the slow evolution on librating Kozai-Lidov cycles, in which the argument of pericenter librates. On these cycles the leading octupole kicks to jz, the component of the inner angular momentum along the outer one, cancel pairwise, and the hexadecapole enters at the order of the surviving octupole torque, with the opposite sign. Strikingly, as the inner mass ratio vanishes the two nearly cancel, and their sum changes sign within the librating window, so octupole-truncated integrations omit part of the leading slow torque on this class. For |jz|≪1, the slow dynamics driven by the hexadecapole is approximately solved analytically as a simple pendulum in the doubled azimuth of the eccentricity vector, with closed-form coefficients on rotating and librating cycles. For a circular outer orbit the hexadecapole-modified cycles are solved exactly. For unequal inner masses the octupole adds to the pendulum's torque on librating cycles, where its alternating kick can also flip orbits, and generally dominates rotating ones. DA integrations follow the predicted excursions, periods and flip criteria. Population consequences depend on the validity of the high-eccentricity reduction and on other processes affecting the cycles.
Related
- Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum, IV: Octupole for Librating Kozai-Lidov Cycles
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- Dyonic black holes supporting nearly-black self-gravitating thin shells
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BibTeX
@misc{klein2026four,
title = {{Four Which I Knew Not: The Hexadecapole Nearly Cancels the Octupole in Librating Kozai-Lidov Cycles}},
author = {Ygal Y. Klein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22487},
eprint = {2610.22487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22487},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Four Which I Knew Not: The Hexadecapole Nearly Cancels the Octupole in Librating Kozai-Lidov Cycles
AU - Ygal Y. Klein
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22487
UR - https://arxiv.org/abs/2610.22487
AB - Phenomena requiring close proximity between two bodies, from black-hole mergers to hot Jupiters, may originate in hierarchical triples, in which secular torques from a distant companion drive the inner binary's eccentricity nearly to unity. Population syntheses of these channels commonly integrate the double-averaged (DA) secular equations truncated at octupole order. We show that, for a fixed outer orbit, this truncation fails for the slow evolution on librating Kozai-Lidov cycles, in which the argument of pericenter librates. On these cycles the leading octupole kicks to j_z, the component of the inner angular momentum along the outer one, cancel pairwise, and the hexadecapole enters at the order of the surviving octupole torque, with the opposite sign. Strikingly, as the inner mass ratio vanishes the two nearly cancel, and their sum changes sign within the librating window, so octupole-truncated integrations omit part of the leading slow torque on this class. For |j_z|≪1, the slow dynamics driven by the hexadecapole is approximately solved analytically as a simple pendulum in the doubled azimuth of the eccentricity vector, with closed-form coefficients on rotating and librating cycles. For a circular outer orbit the hexadecapole-modified cycles are solved exactly. For unequal inner masses the octupole adds to the pendulum's torque on librating cycles, where its alternating kick can also flip orbits, and generally dominates rotating ones. DA integrations follow the predicted excursions, periods and flip criteria. Population consequences depend on the validity of the high-eccentricity reduction and on other processes affecting the cycles.
ER -