Post-Newtonian secular dynamics of hierarchical triples II. Canonical derivation of the 1PN × quadrupole cross terms and of the quadrupole-squared term
Abstract
We derive the secular Hamiltonian of a hierarchical triple from the 1PN ADM three-body Hamiltonian by a two-stage Lie-Hori-Deprit transformation carried out entirely in canonical variables, and use it to settle the relativistic cross terms left open in Paper I. A bookkeeping rule enumerates all contributions to each order in the 1PN parameter δ and the hierarchy parameter ε. The 1PN monopole interaction reduces to a renormalization of the inner mass by its binding energy, and the order δε3/2, at which Lim and Rodriguez (LR) found a libration term, is empty. The electric and magnetic 1PN × quadrupole cross terms agree exactly with the effective-field-theory result of Kuntz, Serra, and Trincherini; for the magnetic term this is the first independent derivation. We give the next-to-leading order 1PN × quadrupole term in closed form; it depends on the outer argument of pericenter ω2. At quadrupole-squared order we obtain the complete secular Hamiltonian, which generates the secular equations of Will and of Conway and Will; its 2ω2 harmonic is a gauge term, removable even on the commensurability between the ZLK period and the nodal precession, by a generator given in closed form via Tremaine's fictitious time. The EIH and ADM descriptions coincide at 1PN order; the formulations differ only in the definition of the elements, for which we give the osculating-contact map. In secular integrations of LR's triples, the eccentricity enhancement with the octupole is controlled by the ω2-dependent part of the next-to-leading order term and is two to three times that of Paper I, while LR's accelerated mergers remain absent.
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BibTeX
@misc{arakida2026postnewtonian,
title = {{Post-Newtonian secular dynamics of hierarchical triples II. Canonical derivation of the 1PN $\times$ quadrupole cross terms and of the quadrupole-squared term}},
author = {Hideyoshi Arakida},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14505},
eprint = {2610.14505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14505},
}
RIS
TY - UNPB
TI - Post-Newtonian secular dynamics of hierarchical triples II. Canonical derivation of the 1PN × quadrupole cross terms and of the quadrupole-squared term
AU - Hideyoshi Arakida
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.14505
UR - https://arxiv.org/abs/2610.14505
AB - We derive the secular Hamiltonian of a hierarchical triple from the 1PN ADM three-body Hamiltonian by a two-stage Lie-Hori-Deprit transformation carried out entirely in canonical variables, and use it to settle the relativistic cross terms left open in Paper I. A bookkeeping rule enumerates all contributions to each order in the 1PN parameter δ and the hierarchy parameter ε. The 1PN monopole interaction reduces to a renormalization of the inner mass by its binding energy, and the order δε^3/2, at which Lim and Rodriguez (LR) found a libration term, is empty. The electric and magnetic 1PN × quadrupole cross terms agree exactly with the effective-field-theory result of Kuntz, Serra, and Trincherini; for the magnetic term this is the first independent derivation. We give the next-to-leading order 1PN × quadrupole term in closed form; it depends on the outer argument of pericenter ω_2. At quadrupole-squared order we obtain the complete secular Hamiltonian, which generates the secular equations of Will and of Conway and Will; its 2ω_2 harmonic is a gauge term, removable even on the commensurability between the ZLK period and the nodal precession, by a generator given in closed form via Tremaine's fictitious time. The EIH and ADM descriptions coincide at 1PN order; the formulations differ only in the definition of the elements, for which we give the osculating-contact map. In secular integrations of LR's triples, the eccentricity enhancement with the octupole is controlled by the ω_2-dependent part of the next-to-leading order term and is two to three times that of Paper I, while LR's accelerated mergers remain absent.
ER -