Secular dynamics of hierarchical multiple systems composed of nested binaries, with an arbitrary number of bodies and arbitrary hierarchical structure. First applications to multiplanet and multistar systems

2015/11/30 by Adrian S. Hamers, Simon Portegies Zwart, Simon F. Portegies Zwart · 4 citations
Mathematics · Physics and Astronomy · #Applied mathematics #Astro and Planetary Science #Astrophysics #Astrophysics and Star Formation Studies #Binary number #Celestial mechanics #Classical mechanics #Computer science #Gravitation #Hamiltonian (control theory) #Mathematics #Pairwise comparison #Physics #Statistical physics #Stellar, planetary, and galactic studies #astro-ph.EP #astro-ph.SR

paper · pdf · doi:10.1093/mnras/stw784

published as 2016, Monthly Notices of the Royal Astronomical Society, Volume 459, Issue 3, p.2827-2874 · Revised to match MNRAS publication. 48 pages, 36 figures

openalex publication_date 2016/04/06 · arxiv created 2016/06/09 · arxiv updated 2016/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 xi of binary separations. At the lowest non-trivial expansion order (quadrupole order, second order in xi), 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 xi, individual terms depend on at most n − 1 binaries. We explicitly derive the Hamiltonian including all terms up and including third order in xi (octupole order), and including the binary pairwise terms up and including fifth order in xi. 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.

Citations

• The Equilibrium Tide Model for Tidal Friction
• Orbital Evolution in Binary and Triple Stars, with an Application to SS Lacertae
• Secular dynamics of hierarchical quadruple systems: the case of a triple system orbited by a fourth body
• New developments for modern celestial mechanics – I. General coplanar three-body systems. Application to exoplanets
• The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies
• Chaotic diffusion in the Solar System
• The Occurrence and Architecture of Exoplanetary Systems
• Existence of collisional trajectories of Mercury, Mars and Venus with the Earth
• Tertiary companions to close spectroscopic binaries

Cited by

• WD 1856 b: a close giant planet around a white dwarf that could have survived a common envelope phase
• White dwarf pollution by planets in stellar binaries
• Multiple Stellar Evolution: a population synthesis algorithm to model the stellar, binary, and dynamical evolution of multiple-star systems
• Quasi-secular evolution of mildly hierarchical triple systems: analytics and applications for GW sources and hot Jupiters