Secular Dynamics in Hierarchical Three-Body Systems

By S. Naoz, W. Farr, Y. Lithwick, F. Rasio, J. Teyssandier. 2011

352 citations recorded by the source corpus.

The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. In this approximation the orbits may change shape and orientation, on time scales longer than the orbital time scales, but the semi major axes are constant. For example, for highly inclined triple systems, the Kozai-Lidov mechanism can produce large-amplitude oscillations of the eccentricities and inclinations. Here we revisit the secular dynamics of hierarchical triple systems. We derive the secular evolution equations to octupole order in Hamiltonian perturbation theory. Our derivation corrects an error in some previous treatments of the problem that implicitly assumed a conservation of the z-component of the angular momentum of the inner orbit (i.e., parallel to the total angular momentum of the system). Already to quadrupole order, our results show new behaviors including the possibility for a system to oscillate from prograde to retrograde orbits. At the octupole order, for an eccentric outer orbit, the inner orbit can reach extremely high eccentricities and undergo chaotic flips in its orientation. We discuss applications to a variety of astrophysical systems, from stellar triples to merging compact binaries and planetary systems. Our results agree with those of previous studies done to quadrupole order only in the limit in which one of the inner two bodies is a massless test particle and the outer orbit is circular;our results agree with previous studies at octupole order for the eccentricity evolution, but not for the inclination evolution.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/52288243b7463ce2393d723daa50a0914f24960e

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In Astronomy and Astrophysics

1. F. Antonini
2. S. Naoz
3. B. Kocsis
4. F. Rasio
5. D. Nesvorný
6. A. Morbidelli
7. Zoltán Haiman
8. Joseph M. Antognini

Citations in the corpus

• Binary Black Hole Mergers from Field Triples: Properties, Rates, and the Impact of Stellar Evolution (2017, methodology): “…the BH triples are evolved forward in time using either the octupole level secular equations of motion of triple systems (e.g., Blaes et al. 2002; Naoz et al. 2013) or a high-precision direct integrator for systems with weaker hierarchies where the secular equations become inaccurate (Mikkola &…”
• BLACK HOLE TRIPLE DYNAMICS: A BREAKDOWN OF THE ORBIT AVERAGE APPROXIMATION AND IMPLICATIONS FOR GRAVITATIONAL WAVE DETECTIONS (2013, background): “However, if the relativistic precession timescales are comparable or lower than the secular Newtonian timescales, then around some critical value of a2 it is possible that the maximum achievable eccentricity is increased with respect to the Newtonian case (Ford et al. 2000; Naoz et al. 2013).”
• Formation and Evolution of Compact-object Binaries in AGN Disks (2019, background): “On the other hand, if the timescale of apsidal precession for either the inner or the outer orbit due to additional mass or general relativity is shorter than the KL timescale, the KL mechanism is typically suppressed (e.g., Fabrycky & Tremaine 2007; Naoz et al. 2013b).”
• BLACK HOLE MERGERS AND BLUE STRAGGLERS FROM HIERARCHICAL TRIPLES FORMED IN GLOBULAR CLUSTERS (2015, methodology): “Lidov-Kozai (LK) oscillations are usually described as a secular phenomenon, occurring on timescales much longer than the orbital period of the outer binary orbit (e.g., Ford et al. 2004; Fabrycky & Tremaine 2007; Naoz et al. 2013).”
• The evolution of hierarchical triple star-systems (2016, background): “It has been shown that this approach is in good agreement with N-body simulations of systems in which the secular approximations are valid (Hamers et al., 2013; Michaely and Perets, 2014; Naoz et al., 2013).”
• Origin and Evolution of Short-period Comets (2017, background): “In the quadrupole approximation, the orbits that start with inclinations i′ < 90◦ with respect to the Galactic plane cannot swap to i′ > 90◦ (e.g., Naoz et al. 2013).”
• Suppression of extreme orbital evolution in triple systems with short-range forces (2014, methodology): “Introducing the total angular momentum Gtot, we can write the mutual inclination of the inner and outer orbits as (Naoz et al. 2013b) cos itot = G2tot − L21(1− e21)− L22(1− e22) 2L1L2 √ 1− e21 √ 1− e22 , (59) where L1 and L2 are given by Equations (15)–(16).”
• The Eccentric Kozai-Lidov Effect and Its Applications (2016, methodology): “…the 1st PN three body Hamiltonian in semi-major axes ratio up to the octupole level of approximation reveals another term which describes www.annualreviews.org • The EKL Effect and Its Applications 25 the general relativity interaction between the inner and outer orbits (Naoz et al. 2013b).”
• Merging Binaries in the Galactic Center: The eccentric Kozai-Lidov mechanism with stellar evolution (2016, methodology): “4 Numerically, this is very similar to the Mardling & Aarseth (2001) nominal triple stars stability criterion, as shown in Naoz et al. (2013b).”
• Greatly Enhanced Merger Rates of Compact-object Binaries in Non-spherical Nuclear Star Clusters (2017, methodology): “…of the potential 10 An equivalent expression is easily found using the Hamiltonian written in terms of orbital elements (see, e.g., Equation 20 in Naoz et al. 2013) after averaging over the fast varying longitude of the ascending nodes. and, therefore, these new LK eccentricity oscillations…”
• RESONANT POST-NEWTONIAN ECCENTRICITY EXCITATION IN HIERARCHICAL THREE-BODY SYSTEMS (2012, methodology): “Replacing H2 from Naoz et al. (2013) Appendix A, with Htot2 = HN2 +H1PN, we find (after following their derivation) the equivalent of their equation (A22) Htot,∗2 (q∗i , p∗i ) = 1 4π2 ∫ 2π 0 dl∗1dl ∗ 2 Htot2 (q∗i , p∗i ) , (B2) where the new momenta and coordinates have a superscript asterix.”
• ECCENTRICITY GROWTH AND ORBIT FLIP IN NEAR-COPLANAR HIERARCHICAL THREE-BODY SYSTEMS (2013, methodology): “When the outer orbit is non-circular, the octupole approximation is needed, inducing variations in eccentricity and inclination on longer timescales, and causes excursions to even higher eccentricities and inclinations above 90◦ (Naoz et al. 2011, 2013a).”
• Timescales of Kozai-Lidov oscillations at quadrupole and octupole order in the test particle limit (2015, background): “…has appeared in the literature, although several studies have asserted that tEKM ∼ tKL/ǫoct is a plausible timescale (e.g., Katz et al. 2011; Naoz et al. 2013b; Li et al. 2015), where ǫoct measures the strength of the octupole order term relative to the quadrupole order term of the…”
• Rapid Eccentricity Oscillations and the Mergers of Compact Objects in Hierarchical Triples (2013, methodology): “This timescale is given by Katz et al. (2011) and Naoz et al. (2013a) as tEKM ∼ tKL ǫoct , where ǫoct is the strength of the octupole-order term in the expansion of the three-body Hamiltonian.”