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The SecularMultiple code, presented in two previous papers of this series,integrates the long-term dynamical evolution of multiple systems with anynumber of bodies and hierarchical structure, provided that the system iscomposed of nested binaries. In the formalism underlying SecularMultiple, wepreviously averaged over all orbits in the system. This approximationsignificantly speeds up numerical integration of the equations of motion,making large population synthesis studies possible. However, the orbitaveraging approximation can break down when the secular evolution timescale ofthe system is comparable to or shorter than any of the orbital periods in thesystem. Here, we present an update to SecularMultiple in which we incorporatehybrid integration techniques, and orbit-averaging corrections. With thisupdate, the user can specify which orbits should be integrated directly(without averaging), or assuming averaged orbits. For orbits that areintegrated directly, we implemented two integration techniques, one which isbased on the regularised Kustaanheimo-Stiefel equations of motion in elementform. We also implemented analytical orbit-averaging corrections for pairwiseinteractions to quadrupole order. The updates presented here provide moreflexibility for integrating the long-term dynamical evolution of hierarchicalmultiple systems. By effectively combining direct integration and orbitaveraging the long-term evolution can be accurately computed, but withsignificantly lower computational cost compared to existing direct N-bodycodes. We give a number of examples in which the new features are beneficial.Our updated code, which is written in C++ supplemented with a user-friendlyinterface in Python, is freely available.

monthly notices of the royal astronomical society

Secular dynamics of hierarchical multiple systems composed of nested  binaries, with an arbitrary number of bodies and arbitrary hierarchical  structure. III. Suborbital effects: hybrid integration techniques and  orbit-averaging corrections