Secular dynamics of hierarchical quadruple systems: the case of a triple system orbited by a fourth body

TL;DR

This paper analyzes the long-term gravitational dynamics of hierarchical quadruple systems, revealing how initial conditions influence orbital inclinations and eccentricities, with implications for astrophysical phenomena like mergers and tidal interactions.

Contribution

It develops a Hamiltonian-based framework to describe quadruple systems and identifies key parameters governing their secular evolution, extending understanding beyond previous triple system studies.

Findings

Inclination and eccentricity oscillations depend on initial Kozai-Lidov time-scale ratios.

High eccentricities can occur even with initially coplanar orbits.

Results have implications for stellar mergers, tidal interactions, and gravitational wave events.

Abstract

We study the secular gravitational dynamics of quadruple systems consisting of a hierarchical triple system orbited by a fourth body. These systems can be decomposed into three binary systems with increasing semimajor axes, binaries A, B and C. The Hamiltonian of the system is expanded in ratios of the three binary separations, and orbit-averaged. Subsequently, we numerically solve the equations of motion. We study highly hierarchical systems that are well described by the lowest-order terms in the Hamiltonian. We find that the qualitative behaviour is determined by the ratio $\mathcal{R}_0$ of the initial Kozai-Lidov (KL) time-scales of the binary pairs AB and BC. If $\mathcal{R}_0\ll 1$, binaries AB remain coplanar if this is initially the case, and KL eccentricity oscillations in binary B are efficiently quenched. If $\mathcal{R}_0\gg 1$, binaries AB become inclined, even if initially coplanar. However, there are no induced KL eccentricity oscillations in binary A. Lastly, if $\mathcal{R}_0\sim 1$, complex KL eccentricity oscillations can occur in binary A that are coupled with the KL eccentricity oscillations in B. Even if binaries A and B are initially coplanar, the induced inclination can result in very high eccentricity oscillations in binary A. These extreme eccentricities could have significant implications for strong interactions such as tidal interactions, gravitational wave dissipation, and collisions and mergers of stars and compact objects. As an example, we apply our results to a planet+moon system orbiting a central star, which in turn is orbited by a distant and inclined stellar companion or planet, and to observed stellar quadruples.