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We study the secular gravitational dynamics of quadruple systems consistingof a hierarchical triple system orbited by a fourth body. These systems can bedecomposed into three binary systems with increasing semimajor axes, binariesA, B and C. The Hamiltonian of the system is expanded in ratios of the threebinary separations, and orbit-averaged. Subsequently, we numerically solve theequations of motion. We study highly hierarchical systems that are welldescribed by the lowest-order terms in the Hamiltonian. We find that thequalitative behaviour is determined by the ratio $\mathcal{R}_0$ of the initialKozai-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 thecase, 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 inbinary A that are coupled with the KL eccentricity oscillations in B. Even ifbinaries A and B are initially coplanar, the induced inclination can result invery high eccentricity oscillations in binary A. These extreme eccentricitiescould have significant implications for strong interactions such as tidalinteractions, gravitational wave dissipation, and collisions and mergers ofstars and compact objects. As an example, we apply our results to a planet+moonsystem orbiting a central star, which in turn is orbited by a distant andinclined stellar companion or planet, and to observed stellar quadruples.

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