2016MNRAS.459.2827H

We present a method for studying the secular gravitational dynamics of hierarchical multiple systems consisting of nested binaries, which is valid for an arbitrary number of bodies and arbitrary hierarchical structure. We derive the Hamiltonian of the system and expand it in terms of the - assumed to be - small ratios x_i of binary separations. At the lowest non-trivial expansion order (quadrupole order, second order in x_i), the Hamiltonian consists of terms which, individually, depend on binary pairs. At higher orders, in addition to terms depending on binary pairs, we also find terms which, individually, depend on more than two binaries. In general, at order n in x_i, individual terms depend on at most n - 1 binaries. We explicitly derive the Hamiltonian including all terms up and including third order in x_i (octupole order), and including the binary pairwise terms up and including fifth order in x_i. These terms are orbit averaged, and we present a new algorithm for efficiently solving the equations of motion. This algorithm is highly suitable for studying the secular evolution of hierarchical systems with complex hierarchies, making long-term integrations of such systems feasible. We show that accurate results are obtained for multiplanet systems with semimajor axis ratios as large as ≈0.4, provided that high-order terms are included. In addition to multiplanet systems with a single star, we apply our results to multistar systems with multiple planets.