## Dynamics of hierarchical quadruple systems: the case of a triple system orbited by a fourth body

### Adrian Hamers, Leiden Observatory

Advisors: Hagai Perets, Fabio Antonini

**Abstract:**We study the secular 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 for the system is derived and expanded in the ratios of binary separations r

_{A}/r

_{B}, r

_{B}/r

_{C}and r

_{A}/r

_{C}, to up and including combined fourth order. At each order we find three terms that are each mathematically equivalent to the corresponding terms that appear in the hierarchical three-body problem, and that depend on the properties of only two binaries. In addition to these terms, for octupole and higher orders we find 'cross terms' that depend on properties of all three binaries. Subsequently we orbit-average the Hamiltonian and numerically solve the equations of motion. We study the general dynamics in the case of widely separated binaries, which are well described in terms of the lowest order terms in the Hamiltonian expansion. We find that the qualitative behaviour is determined by the ratio R0 of the Kozai-Lidov (KL) time-scales of the binary combinations AB and BC. If R0 << 1, AB remains coplanar if this is initially the case, and KL eccentricity oscillations in B are efficiently quenched. If R0 >> 1, AB becomes inclined, even if initially coplanar. There are no induced KL eccentricity oscillations in binary A, however. Lastly, if R0 ~ 1, complex KL eccentricity oscillations can occur in A, and 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. We also include the effects of general relativity, and we find an interesting mechanism to overcome the well-known quenching of KL cycles by relativistic precession in binary A for a certain regime in parameter space. This is due to enhanced eccentricity of B, which can decrease the AB KL time-scale to below the relativistic precession time-scale in binary A. We briefly 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. We find that there are regions in parameter space where an initially coplanar planet+moon system with respect to the central star can become inclined and the eccentricity in the planet+moon system can be excited. We also apply our results to binaries around a supermassive black hole (SBH). Assuming an external torque due to either a circumnuclear disk or a hypothetical intermediate mass black hole, we expect that mass precession quenches any potential oscillations in the orbit of the binary around the SBH, thereby avoiding any potentially interesting effects of the latter orbit on the binary orbit.