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

2014/12/31 by Adrian S. Hamers, Hagai B. Perets, Fabio Antonini +2 · 5 citations
Chemistry · Physics and Astronomy · #Astro and Planetary Science #Astronomy #Astrophysics #Dynamics (music) #Geometry #Molecular spectroscopy and chirality #Physics #Quantum chaos and dynamical systems #Statistical physics #Triple system #astro-ph.SR

paper · pdf · doi:10.1093/mnras/stv452

published as 2015, Monthly Notices of the Royal Astronomical Society, Volume 449, Issue 4, p.4221-4245 · Revised to match MNRAS publication. 27 pages, 16 figures

openalex publication_date 2015/04/13 · arxiv created 2016/06/09 · arxiv updated 2016/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the secular gravitational 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 of the system is expanded in ratios of the three binary separations, and orbit averaged. Subsequently, we numerically solve the equations of motion. We study highly hierarchical systems that are well described by the lowest order terms in the Hamiltonian. We find that the qualitative behaviour is determined by the ratio |\mathcal R₀| of the initial Kozai–Lidov (KL) time-scales of the binary pairs AB and BC. If |\mathcal R₀≪ 1|⁠, binaries AB remain coplanar if this is initially the case, and KL eccentricity oscillations in binary B are efficiently quenched. If |\mathcal R₀≫ 1|⁠, binaries AB become inclined, even if initially coplanar. However, there are no induced KL eccentricity oscillations in binary A. Lastly, if |\mathcal R₀∼ 1|⁠, complex KL eccentricity oscillations can occur in binary A 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. These extreme eccentricities could have significant implications for strong interactions such as tidal interactions, gravitational wave dissipation, and collisions and mergers of stars and compact objects. As an example, we 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, and to observed stellar quadruples.

Citations

• Table for Estimating the Goodness of Fit of Empirical Distributions
• Orbital Evolution in Binary and Triple Stars, with an Application to SS Lacertae
• The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies
• Prehistory of the ’’Runge–Lenz’’ vector
• On WD-WD Mergers in Triple Systems: The Role of Kozai Resonance with Tidal Friction
• More on the prehistory of the Laplace or Runge–Lenz vector
• A universal instability of many-dimensional oscillator systems

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• Greatly Enhanced Merger Rates of Compact-object Binaries in Non-spherical Nuclear Star Clusters
• Stability of exomoons around the Kepler transiting circumbinary planets