Post-Newtonian secular dynamics of hierarchical triples. I. Eccentricity-, inclination- and node-dependence of the multiple-scale formulation, and its canonical consistency

Abstract

We will re-examine the leading post-Newtonian (1PN) cross terms of hierarchical triples obtained by Lim and Rodriguez (LR) with a two-parameter multiple-scale expansion in the hierarchy parameter $\varepsilon=a_1/a_2$ and the 1PN parameter $δ$, which disagree with the effective-field-theory result of Kuntz, Serra, and Trincherini (KST) at the orders $δ\varepsilon^{3/2}$ and $δ\varepsilon^{7/2}$. Reconstructing the LR calculation, we show that the $δ\varepsilon^{3/2}$ ``libration'' term [LR Eq.~(4.5)] originates in the averaging measure: the quadrupole periodic solutions are made mean-free with respect to the outer true anomaly but re-substituted under the time average. The resulting term, evaluated in closed form, reproduces the implemented one exactly; with the time measure required of a canonical generating function it vanishes, in agreement with KST. Equation~(4.5) is one term of the flow of a removable Hamiltonian whose companion terms are absent, which is why it fails the rationality and Hamiltonicity tests. A calculation carried out consistently with a single phase variable yields the complete, gauge-equivalent secular equations, whereas LR's implementation combines a true-anomaly quadrupole solution with an eccentric-anomaly 1PN solution. Two structural defects of the implementation do not contribute to these terms. The $δ\varepsilon^{7/2}$ term of LR has the correct mass dependence only for $m\ll m_3$, and its angular structure disagrees with KST and violates the Hamiltonicity test. Finally, integrations with LR's secular code show that the resonant ZLK modulations reported by LR survive the removal of Eq.~(4.5), whereas their accelerated merger with the octupole disappears.