Higher-order effects in the dynamics of hierarchical triple systems. III. Astrophysical implications of second-order and dotriacontapole terms

TL;DR

This paper extends the analysis of hierarchical triple systems by including second-order and dotriacontapole effects, revealing their significant impact on orbital dynamics and astrophysical phenomena, especially in black hole systems and planetary orbits.

Contribution

It introduces second-order perturbation terms into the evolution equations of hierarchical triples, improving accuracy over first-order models and revealing new dynamical effects.

Findings

Second-order effects can suppress orbital flips in stellar-mass binaries near supermassive black holes.

Planetary semimajor axes can vary significantly due to second-order effects, unlike first-order predictions.

Enhanced agreement with N-body simulations when including second-order and dotriacontapole terms.

Abstract

We study the long-term evolution of selected hierarchical triple systems in Newtonian gravity. We employ analytic equations derived in Paper II for the evolution of orbit-averaged orbital elements for both inner and outer orbits, which include two classes of contributions. One class consists of linear-order contributions, including quadrupole, octupole, hexadecapole and dotriacontapole orders, the latter scaling as $\epsilon^6$, where $\epsilon = a/A$, the ratio of the semimajor axes of the inner and outer orbits. The second class consists of contributions at {\em second} order in the fundamental perturbation parameter; they contribute at orders $\epsilon^{9/2}$, $\epsilon^{5}$, $\epsilon^{11/2}$ and $\epsilon^{6}$. For well studied triples such as star-planet systems perturbed by a low-mass third body (``hot Jupiters''), second-order and dotriacontapole (SOD) effects induce only small corrections. For stellar-mass binaries orbiting supermassive black holes, SOD corrections can suppress orbital flips that are generated by purely first-order effects. Planets orbiting binary star systems are susceptible to significant variations in the planetary semimajor axis, an effect that does not occur at first perturbative order. SOD effects in triple black hole systems can induce migrations of the eccentricity to significantly larger values than predicted by first-order perturbations, with implications for the gravitational-wave induced inspiral of the inner binary. We also show that in most cases, evolutions using our SOD equations are in better agreement with those from direct integration of the N-body equations of motion than those from first-order perturbations through hexadecapole order.