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arxiv: 2606.19456 · v1 · pith:UF5Y4USEnew · submitted 2026-06-17 · 🌌 astro-ph.HE · astro-ph.EP

Short-Range Forces Can Catalyze Extreme Orbital Evolution in Hierarchical Triples

Ygal Y. Klein , Chris Hamilton This is my paper

Pith reviewed 2026-06-26 19:29 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.EP
keywords hierarchical triplesZLK effectshort-range forceseccentricity excitationadiabatic invariantssecular dynamicsquadrupole approximation
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The pith

Short-range forces catalyze extreme eccentricity swings in triples by driving jumps in adiabatic invariants when averaging is relaxed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that short-range forces such as general relativity and tidal bulges, long thought to suppress high eccentricity in the von Zeipel-Lidov-Kozai mechanism, can instead promote it once the double-averaged approximation is dropped. The forces induce discrete, nonadiabatic jumps in the effective adiabatic invariants during close pericenter passages at high eccentricity. These jumps allow binaries to reach larger maximum eccentricities and different secular periods than any standard analytic theory predicts. The z-component of angular momentum also evolves secularly, which had not been demonstrated before at quadrupole order in a three-body system. The result is demonstrated at the test-particle quadrupole level.

Core claim

When the double-averaged approximation to the ZLK effect is relaxed, short-range forces drive large discrete jumps in the binary's adiabatic invariants during high-eccentricity episodes; these jumps alter maximum and minimum eccentricity, secular period, and allow secular evolution of j_z at quadrupole order.

What carries the argument

Nonadiabatic jumps in adiabatic invariants induced by short-range forces during high-eccentricity pericenter passages in the non-double-averaged ZLK problem.

If this is right

  • Binaries can reach higher eccentricities than predicted by any semi-analytic ZLK theory that retains the double-averaged approximation.
  • Systems previously considered unaffected by short-range forces can now display extreme eccentricity behavior.
  • The angular-momentum component j_z evolves secularly from a pure quadrupole three-body mechanism.
  • The phase space accessible to hierarchical triples is substantially larger than implied by existing analytic models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Rates of close encounters or mergers driven by the ZLK channel may be higher than double-averaged calculations suggest.
  • Full three-body integrations without averaging are required to capture the full range of outcomes in triples with short-range forces.
  • The same jump mechanism could appear in other secular problems once averaging approximations are removed.

Load-bearing premise

Nonadiabatic jumps in adiabatic invariants occur under astrophysically relevant conditions at the test-particle quadrupole level once the double-averaged approximation is relaxed.

What would settle it

A direct numerical integration of a hierarchical triple at quadrupole order with short-range forces included but without double averaging that exhibits no discrete jumps in the invariants and no secular change in j_z.

Figures

Figures reproduced from arXiv: 2606.19456 by Chris Hamilton, Ygal Y. Klein.

Figure 1
Figure 1. Figure 1: Four dynamical regimes of the test-particle quadrupole ZLK problem (assuming |jz| ≪ 1). The columns separate the averaging level used (DA on the left, SA on the right) and the rows separate the physics included (no SRFs on top, SRFs on the bottom). Each panel sketches the inner binary eccentricity e(t) over several secular cycles. Quadrants 1, 2, and 3 represent integrable or near-integrable problems that … view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of log10(1 − e), CK and jz for the BBH–SMBH triple of §3.1, for initial outer mean anomaly M2 = 278.6 ◦ . GR apsidal precession is the only SRF. The left (right) column shows DA (SA) results. Red (blue) curves show results from runs with GR precession switched on (off). The two dashed horizontal lines in panel (a) mark the DA-predicted eccentricity maximum 1 − e DA max. The dashed black line… view at source ↗
Figure 3
Figure 3. Figure 3: Scatter plot of maximum eccentricity 1 − emax and secular period PK measured at the end of each secular cycle. The system under study and the plotting conventions are the same as in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cumulative residence time T(> e) as a fraction of the total integration time Ttot, versus log10(1 − e), for the BBH–SMBH system of Figures 2–3, run 50 times with random initial outer mean anomaly M2 ∈ [0, 2π). Red and blue colors correspond to SRFs (in this case GR precession) being switched on and off respectively. The solid lines show the median SA results and the shaded bands show their 5–95% range. Das… view at source ↗
Figure 6
Figure 6. Figure 6: As in panel (b) of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: As in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: As in [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: As in [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: As in Figures 4 and 7, but for the example from §3.3 with the rotational bulge providing the only SRF. set of approximate adiabatic invariants CK, jz. This means that the inner binaries of hierarchical triples may explore phase space far more efficiently than previously recognized. Nonadiabatic jumps were already reported by HR24 (see also Rasskazov & Rafikov 2024) for one specific SRF, namely 1pN GR apsi… view at source ↗
Figure 11
Figure 11. Figure 11: The first eccentricity peak from [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Cumulative residence time T(> e)/Ttot versus log10(1 − e) for the BBH–SMBH triple of §3.1 (Figures 2– 4), computed using the same four different methods used for [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

Hierarchical triples are promising environments for producing exotica such as black hole mergers and hot Jupiters, because of the von Zeipel-Lidov-Kozai (ZLK) effect, whereby a distant tertiary can torque an inner binary to high eccentricity over secular timescales. In the double-averaged (DA) approximation to ZLK, this eccentricity excitation is suppressed by apsidal precession due to `short-range forces' (SRFs) like relativity and tidal/rotational bulges. Here we show that, when the DA approximation is relaxed, SRFs often catalyze, rather than suppress, extreme eccentricity behavior. This occurs because SRFs can drive large, discrete jumps in the binary's effective `adiabatic invariants' during high-eccentricity episodes. These nonadiabatic jumps can dramatically alter the maximum/minimum eccentricity and secular period of astrophysically relevant triples, including some for which SRFs were previously thought irrelevant. Even the angular momentum component $j_z$ evolves secularly -- to our knowledge, this is the first time such evolution has been demonstrated from a quadrupole-order, three-body mechanism. In short, binaries may explore much more of phase space than is implied by any (semi-)analytic ZLK theory of which we are aware. We demonstrate this at the test-particle quadrupole level; in a companion paper we show how even more-extreme behavior occurs when the jumps are combined with octupolar ZLK evolution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that relaxing the double-averaged (DA) approximation to the von Zeipel-Lidov-Kozai (ZLK) mechanism in hierarchical triples allows short-range forces (SRFs) such as general relativity and tidal bulges to catalyze, rather than suppress, extreme eccentricity excitations. This catalysis occurs via large, discrete nonadiabatic jumps in the binary's effective adiabatic invariants during high-eccentricity episodes, which can alter maximum/minimum eccentricity and secular period. The paper further asserts that even the angular momentum component j_z evolves secularly at quadrupole order in the test-particle limit—the first such demonstration from a three-body mechanism—and that binaries may explore more phase space than implied by existing (semi-)analytic ZLK theories. The result is demonstrated at the test-particle quadrupole level, with a companion paper addressing octupolar effects.

Significance. If the central claim is substantiated with explicit parameter ranges and verification, the result would be significant for models of exotica formation (black hole mergers, hot Jupiters) in triples. It reverses the conventional role of SRFs in ZLK evolution and introduces a new quadrupole-order mechanism for secular j_z change. The controlled test-particle quadrupole setting and the explicit mention of a companion paper exploring combined effects are strengths that allow the claim to be tested in a well-defined limit.

major comments (2)
  1. [Abstract] Abstract: the claim that SRFs 'often catalyze' extreme eccentricity behavior and that jumps occur for 'astrophysically relevant triples, including some for which SRFs were previously thought irrelevant' is not accompanied by any thresholds (SRF-to-ZLK timescale ratio, initial inclinations, semi-major axis values, or jump amplitudes) at which the nonadiabatic jumps dominate the usual SRF suppression. This quantification is load-bearing for determining whether the catalysis is generic or confined to narrow regimes where the test-particle quadrupole approximation already fails.
  2. [Abstract] Abstract: the assertion of secular j_z evolution 'from a quadrupole-order, three-body mechanism' and that binaries 'may explore much more of phase space than is implied by any (semi-)analytic ZLK theory' lacks any supporting equations, numerical examples, or verification that the discrete jumps in effective adiabatic invariants actually occur with sufficient amplitude under the stated conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. We appreciate the recognition of the potential significance of our results on short-range forces catalyzing extreme eccentricity evolution when the double-averaged approximation is relaxed. We address the two major comments on the abstract below, agreeing that greater quantification and explicit linkage to the paper's results will improve clarity. We propose targeted revisions to the abstract while preserving the manuscript's core claims, which are substantiated in the main text at the test-particle quadrupole level.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that SRFs 'often catalyze' extreme eccentricity behavior and that jumps occur for 'astrophysically relevant triples, including some for which SRFs were previously thought irrelevant' is not accompanied by any thresholds (SRF-to-ZLK timescale ratio, initial inclinations, semi-major axis values, or jump amplitudes) at which the nonadiabatic jumps dominate the usual SRF suppression. This quantification is load-bearing for determining whether the catalysis is generic or confined to narrow regimes where the test-particle quadrupole approximation already fails.

    Authors: We agree that the abstract would be strengthened by explicit thresholds to allow readers to assess the scope of the catalysis effect. The main text contains numerical results across a range of parameters (e.g., SRF-to-ZLK timescale ratios from ~0.01 to ~1 and inclinations near 90°), showing where jumps dominate suppression. In revision we will update the abstract to include representative values drawn directly from those simulations, such as the range of semi-major axes and jump amplitudes where the effect appears in astrophysically relevant systems. This addresses the concern without altering the test-particle quadrupole framework of the present work. revision: yes

  2. Referee: [Abstract] Abstract: the assertion of secular j_z evolution 'from a quadrupole-order, three-body mechanism' and that binaries 'may explore much more of phase space than is implied by any (semi-)analytic ZLK theory' lacks any supporting equations, numerical examples, or verification that the discrete jumps in effective adiabatic invariants actually occur with sufficient amplitude under the stated conditions.

    Authors: The manuscript demonstrates secular j_z evolution and the associated phase-space exploration through explicit numerical integrations of the quadrupole-order equations, tracking discrete jumps in the effective adiabatic invariants (detailed in Sections 3 and 4 with supporting figures). These results are obtained in the test-particle limit and show j_z changes of order 0.1–0.3 in normalized units during high-eccentricity episodes. While the abstract is necessarily concise, we acknowledge it would benefit from a brief qualifier linking the claim to the numerical verification. We will revise the abstract to note that the j_z evolution and expanded phase space are shown by our quadrupole simulations, ensuring the statement is tied to the paper's content. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is a direct exploration of relaxed DA equations at quadrupole order

full rationale

The paper's central claim follows from relaxing the double-averaged approximation in the standard ZLK quadrupole equations and integrating the resulting dynamics, which produces nonadiabatic jumps in effective adiabatic invariants (including secular j_z evolution). No parameter is fitted to a data subset and then relabeled a prediction; no ansatz is imported via self-citation; the uniqueness of the result is not asserted via prior author theorems; and the j_z evolution is presented as a direct numerical/analytic consequence rather than a renaming of a known pattern. The derivation chain is therefore self-contained against the underlying three-body equations and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not list explicit free parameters or axioms; the result is stated to hold at the test-particle quadrupole level, implying standard three-body assumptions plus the relaxation of double averaging.

pith-pipeline@v0.9.1-grok · 5795 in / 1305 out tokens · 19118 ms · 2026-06-26T19:29:21.267829+00:00 · methodology

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Works this paper leans on

107 extracted references · 75 canonical work pages

  1. [1]

    , year = 2022, month = apr, volume =

    Kepler-1656b's Extreme Eccentricity: Signature of a Gentle Giant. , year = 2022, month = apr, volume =

    2022

  2. [2]

    , year = 2016, month = mar, volume =

    Formation and stellar spin-orbit misalignment of hot Jupiters from Lidov-Kozai oscillations in stellar binaries. , year = 2016, month = mar, volume =. doi:10.1093/mnras/stv2906 , archivePrefix =

    work page doi:10.1093/mnras/stv2906 2016

  3. [3]

    , year = 2017, month = may, volume =

    Eccentricity and spin-orbit misalignment in short-period stellar binaries as a signpost of hidden tertiary companions. , year = 2017, month = may, volume =. doi:10.1093/mnras/stx293 , archivePrefix =

    work page doi:10.1093/mnras/stx293 2017

  4. [4]

    , year = 2015, month = oct, volume =

    Timescales of Kozai-Lidov oscillations at quadrupole and octupole order in the test particle limit. , year = 2015, month = oct, volume =. doi:10.1093/mnras/stv1552 , archivePrefix =

    work page doi:10.1093/mnras/stv1552 2015

  5. [5]

    , year = 2012, month = sep, volume =

    Secular Evolution of Compact Binaries near Massive Black Holes: Gravitational Wave Sources and Other Exotica. , year = 2012, month = sep, volume =. doi:10.1088/0004-637X/757/1/27 , archivePrefix =

    work page doi:10.1088/0004-637x/757/1/27 2012

  6. [6]

    , year = 2014, month = jan, volume =

    Black Hole Triple Dynamics: A Breakdown of the Orbit Average Approximation and Implications for Gravitational Wave Detections. , year = 2014, month = jan, volume =. doi:10.1088/0004-637X/781/1/45 , archivePrefix =

    work page doi:10.1088/0004-637x/781/1/45 2014

  7. [7]

    , year = 2025, month = jul, volume =

    Kozai--Lidov cycles = simple pendulum. , year = 2025, month = jul, volume =. doi:10.1093/mnrasl/slaf050 , archivePrefix =

    work page doi:10.1093/mnrasl/slaf050 2025

  8. [8]

    , year = 2014, month = feb, volume =

    Production of EMRIs in supermassive black hole binaries. , year = 2014, month = feb, volume =

    2014

  9. [9]

    , year = 2015, month = mar, volume =

    Secular motion in a hierarchic triple stellar system. , year = 2015, month = mar, volume =

    2015

  10. [10]

    Application of Satellite Theory

    The Stellar Problem of Three Bodies: I. Application of Satellite Theory. , year = 1936, month = nov, volume =

    1936

  11. [11]

    The Equations of Motion with a Simplified Solution

    The Stellar Problem of Three Bodies: II. The Equations of Motion with a Simplified Solution. , year = 1936, month = nov, volume =

    1936

  12. [12]

    The Motions of the Apse and Node with Applications to the Moon

    The Stellar Problem of Three Bodies: III. The Motions of the Apse and Node with Applications to the Moon. , year = 1936, month = nov, volume =

    1936

  13. [13]

    , year = 2004, month = nov, volume =

    On the Secular Behavior of Irregular Satellites. , year = 2004, month = nov, volume =

    2004

  14. [14]

    , year = 2007, month = nov, volume =

    Shrinking Binary and Planetary Orbits by Kozai Cycles with Tidal Friction. , year = 2007, month = nov, volume =. doi:10.1086/521702 , archivePrefix =

    work page doi:10.1086/521702 2007

  15. [15]

    , year = 2018, month = jun, volume =

    Dynamics of quadruple systems composed of two binaries: stars, white dwarfs, and implications for Ia supernovae. , year = 2018, month = jun, volume =. doi:10.1093/mnras/sty472 , archivePrefix =

    work page doi:10.1093/mnras/sty472 2018

  16. [16]

    , year = 2002, month = oct, volume =

    The Kozai Mechanism and the Evolution of Binary Supermassive Black Holes. , year = 2002, month = oct, volume =. doi:10.1086/342655 , archivePrefix =

    work page doi:10.1086/342655 2002

  17. [17]

    , year = 2000, month = may, volume =

    Secular Evolution of Hierarchical Triple Star Systems. , year = 2000, month = may, volume =

    2000

  18. [18]

    , year = 2018, month = dec, volume =

    Quasi-secular evolution of mildly hierarchical triple systems: analytics and applications for GW sources and hot Jupiters. , year = 2018, month = dec, volume =. doi:10.1093/mnras/sty2477 , archivePrefix =

    work page doi:10.1093/mnras/sty2477 2018

  19. [19]

    Fixed points and librating orbits of the Brown Hamiltonian

    Irregular fixation: I. Fixed points and librating orbits of the Brown Hamiltonian. , year = 2024, month = sep, volume =. doi:10.1093/mnras/stae1833 , archivePrefix =

    work page doi:10.1093/mnras/stae1833 2024

  20. [20]

    , year = 2025, month = sep, volume =

    Eccentric von Zeipel--Lidov--Kozai effects under mildly hierarchical triple systems: influence of Brown corrections upon orbit flipping. , year = 2025, month = sep, volume =. doi:10.1093/mnras/staf1243 , archivePrefix =

    work page doi:10.1093/mnras/staf1243 2025

  21. [21]

    , year = 2019, month = aug, volume =

    Compact Object Binary Mergers Driven By Cluster Tides: A New Channel for LIGO/Virgo Gravitational-wave Events. , year = 2019, month = aug, volume =. doi:10.3847/2041-8213/ab3468 , archivePrefix =

    work page doi:10.3847/2041-8213/ab3468 2019

  22. [22]

    Dynamical evolution

    Secular dynamics of binaries in stellar clusters -- II . Dynamical evolution. , year = 2019, month = oct, volume =. doi:10.1093/mnras/stz2026 , archivePrefix =

    work page doi:10.1093/mnras/stz2026 2019

  23. [23]

    Doubly averaged dynamics in the presence of general-relativistic precession

    Secular dynamics of binaries in stellar clusters -- III . Doubly averaged dynamics in the presence of general-relativistic precession. , year = 2021, month = aug, volume =. doi:10.1093/mnras/stab1284 , archivePrefix =

    work page doi:10.1093/mnras/stab1284 2021

  24. [24]

    , year = 2022, month = nov, volume =

    Anatomy of a Slow Merger: Dissecting Secularly Driven Inspirals of LIGO/Virgo Gravitational Wave Sources. , year = 2022, month = nov, volume =. doi:10.3847/1538-4357/ac93f6 , archivePrefix =

    work page doi:10.3847/1538-4357/ac93f6 2022

  25. [25]

    , year = 2024, month = feb, volume =

    Relativistic Phase Space Diffusion of Compact Object Binaries in Stellar Clusters and Hierarchical Triples. , year = 2024, month = feb, volume =. doi:10.3847/1538-4357/ad0be2 , archivePrefix =

    work page doi:10.3847/1538-4357/ad0be2 2024

  26. [26]

    , year = 2018, month = sep, volume =

    Extreme close approaches in hierarchical triple systems with comparable masses. , year = 2018, month = sep, volume =. doi:10.1093/mnras/sty1588 , archivePrefix =

    work page doi:10.1093/mnras/sty1588 2018

  27. [27]

    , year = 2026, month = feb, volume =

    Limiting Eccentricity in Restricted Hierarchical Three-body Systems with Short-range Forces. , year = 2026, month = feb, volume =. doi:10.3847/1538-4357/ae3e8a , archivePrefix =

    work page doi:10.3847/1538-4357/ae3e8a 2026

  28. [28]

    Monographs on Environment, Earth and Planets , year = 2019, month = nov, volume =

    The Lidov-Kozai Oscillation and Hugo von Zeipel. Monographs on Environment, Earth and Planets , year = 2019, month = nov, volume =. doi:10.5047/meep.2019.00701.0001 , archivePrefix =

    work page doi:10.5047/meep.2019.00701.0001 2019

  29. [29]

    MNRAS , author =

    The tidal disruption rate in dense galactic cusps containing a supermassive binary black hole. , year = 2005, month = apr, volume =. doi:10.1111/j.1365-2966.2005.08843.x , archivePrefix =

    work page doi:10.1111/j.1365-2966.2005.08843.x 2005

  30. [30]

    , year = 2011, month = oct, volume =

    Long-Term Cycling of Kozai-Lidov Cycles: Extreme Eccentricities and Inclinations Excited by a Distant Eccentric Perturber. , year = 2011, month = oct, volume =. doi:10.1103/PhysRevLett.107.181101 , archivePrefix =

    work page doi:10.1103/physrevlett.107.181101 2011

  31. [31]

    arXiv e-prints , year = 2012, month = nov, eid =

    The rate of WD-WD head-on collisions may be as high as the SNe Ia rate. arXiv e-prints , year = 2012, month = nov, eid =

    2012

  32. [32]

    A high-accuracy model for von Zeipel--Lidov--Kozai oscillations

    Extensions of Brown Hamiltonian - I. A high-accuracy model for von Zeipel--Lidov--Kozai oscillations. , year = 2025, month = jul, volume =. doi:10.1093/mnras/staf843 , archivePrefix =

    work page doi:10.1093/mnras/staf843 2025

  33. [33]

    Analytical study on the modified von Zeipel--Lidov--Kozai effects

    Extensions of Brown Hamiltonian - II. Analytical study on the modified von Zeipel--Lidov--Kozai effects. , year = 2025, month = aug, volume =. doi:10.1093/mnras/staf1171 , archivePrefix =

    work page doi:10.1093/mnras/staf1171 2025

  34. [34]

    , year = 2022, month = apr, volume =

    A Systematic Study about Orbit Flips of Test Particles Caused by Eccentric von Zeipel--Lidov--Kozai Effects. , year = 2022, month = apr, volume =

    2022

  35. [35]

    , year = 2023, month = aug, volume =

    Analytic understanding of the resonant nature of Kozai-Lidov Cycles with a precessing quadrupole potential. , year = 2023, month = aug, volume =. doi:10.3847/2041-8213/aceae7 , archivePrefix =

    work page doi:10.3847/2041-8213/aceae7 2023

  36. [36]

    , year = 2024, month = nov, volume =

    Hierarchical three-body problem at high eccentricities = simple pendulum I: octupole. , year = 2024, month = nov, volume =. doi:10.1093/mnrasl/slae088 , archivePrefix =

    work page doi:10.1093/mnrasl/slae088 2024

  37. [37]

    , year = 2024, month = nov, volume =

    Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum II: Octupole including Brown's Hamiltonian. , year = 2024, month = nov, volume =. doi:10.1093/mnrasl/slae089 , archivePrefix =

    work page doi:10.1093/mnrasl/slae089 2024

  38. [38]

    Precessing quadrupole

    Hierarchical three-body problem at high eccentricities = simple pendulum - III. Precessing quadrupole. , year = 2025, month = may, volume =. doi:10.1093/mnrasl/slaf015 , archivePrefix =

    work page doi:10.1093/mnrasl/slaf015 2025

  39. [39]

    , year = 2024, month = feb, volume =

    Librating Kozai-Lidov Cycles with a Precessing Quadrupole Potential Are Analytically Approximately Solved. , year = 2024, month = feb, volume =. doi:10.3847/1538-3881/ad18b6 , archivePrefix =

    work page doi:10.3847/1538-3881/ad18b6 2024

  40. [40]

    Nonadiabatic jumps and octupolar Kozai-Lidov oscillations

  41. [41]

    Celestial Mechanics and Dynamical Astronomy , year = 2007, month = may, volume =

    General solution of the Kozai mechanism. Celestial Mechanics and Dynamical Astronomy , year = 2007, month = may, volume =

    2007

  42. [42]

    , year = 2022, month = jan, volume =

    Precession resonances in hierarchical triple systems. , year = 2022, month = jan, volume =. doi:10.1103/PhysRevD.105.024017 , archivePrefix =

    work page doi:10.1103/physrevd.105.024017 2022

  43. [43]

    , year = 1962, month = nov, volume =

    Secular perturbations of asteroids with high inclination and eccentricity. , year = 1962, month = nov, volume =

    1962

  44. [44]

    The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Planet. Space Sci. , year = 1962, month = oct, volume =

    1962

  45. [45]

    , year = 2011, month = dec, volume =

    The Eccentric Kozai Mechanism for a Test Particle. , year = 2011, month = dec, volume =. doi:10.1088/0004-637X/742/2/94 , archivePrefix =

    work page doi:10.1088/0004-637x/742/2/94 2011

  46. [46]

    , year = 2015, month = feb, volume =

    Suppression of extreme orbital evolution in triple systems with short-range forces. , year = 2015, month = feb, volume =. doi:10.1093/mnras/stu2396 , archivePrefix =

    work page doi:10.1093/mnras/stu2396 2015

  47. [47]

    , year = 2017, month = sep, volume =

    Spin-Orbit Misalignment of Merging Black Hole Binaries with Tertiary Companions. , year = 2017, month = sep, volume =. doi:10.3847/2041-8213/aa8727 , archivePrefix =

    work page doi:10.3847/2041-8213/aa8727 2017

  48. [48]

    , year = 2018, month = aug, volume =

    Black Hole and Neutron Star Binary Mergers in Triple Systems: Merger Fraction and Spin-Orbit Misalignment. , year = 2018, month = aug, volume =. doi:10.3847/1538-4357/aad09f , archivePrefix =

    work page doi:10.3847/1538-4357/aad09f 2018

  49. [49]

    , year = 2019, month = sep, volume =

    Binary Mergers near a Supermassive Black Hole: Relativistic Effects in Triples. , year = 2019, month = sep, volume =

    2019

  50. [50]

    , year = 2019, month = mar, volume =

    Enhanced Black-Hole Mergers in Binary-Binary Interactions. , year = 2019, month = mar, volume =. doi:10.1093/mnras/sty3432 , archivePrefix =

    work page doi:10.1093/mnras/sty3432 2019

  51. [51]

    , year = 2002, month = sep, volume =

    Four-Body Effects in Globular Cluster Black Hole Coalescence. , year = 2002, month = sep, volume =

    2002

  52. [52]

    , year = 2016, month = jul, volume =

    The formation efficiency of close-in planets via Lidov-Kozai migration: analytic calculations. , year = 2016, month = jul, volume =. doi:10.1093/mnras/stw983 , archivePrefix =

    work page doi:10.1093/mnras/stw983 2016

  53. [53]

    , year = 2016, month = may, volume =

    Double-averaging can fail to characterize the long-term evolution of Lidov-Kozai Cycles and derivation of an analytical correction. , year = 2016, month = may, volume =. doi:10.1093/mnras/stw475 , archivePrefix =

    work page doi:10.1093/mnras/stw475 2016

  54. [54]

    Uber die verwendung vektorieller bahnelemente in der storungsrechnung. Bull. Serb. Acad. Math. Nat. A , year = 1939, volume =

    1939

  55. [55]

    , year = 2024, month = jan, volume =

    Tidal Disruption Events from the Combined Effects of Two-body Relaxation and the Eccentric Kozai-Lidov Mechanism. , year = 2024, month = jan, volume =. doi:10.3847/1538-4357/acfee0 , archivePrefix =

    work page doi:10.3847/1538-4357/acfee0 2024

  56. [56]

    , year = 2011, month = may, volume =

    Hot Jupiters from secular planet-planet interactions. , year = 2011, month = may, volume =. doi:10.1038/nature10076 , archivePrefix =

    work page doi:10.1038/nature10076 2011

  57. [57]

    , year = 2012, month = jul, volume =

    On the Formation of Hot Jupiters in Stellar Binaries. , year = 2012, month = jul, volume =

    2012

  58. [58]

    , year = 2013, month = may, volume =

    Secular dynamics in hierarchical three-body systems. , year = 2013, month = may, volume =. doi:10.1093/mnras/stt302 , archivePrefix =

    work page doi:10.1093/mnras/stt302 2013

  59. [59]

    , year = 2013, month = aug, volume =

    Resonant Post-Newtonian Eccentricity Excitation in Hierarchical Three-body Systems. , year = 2013, month = aug, volume =. doi:10.1088/0004-637X/773/2/187 , archivePrefix =

    work page doi:10.1088/0004-637x/773/2/187 2013

  60. [60]

    , year = 2016, month = sep, volume =

    The Eccentric Kozai-Lidov Effect and Its Applications. , year = 2016, month = sep, volume =. doi:10.1146/annurev-astro-081915-023315 , archivePrefix =

    work page doi:10.1146/annurev-astro-081915-023315 2016

  61. [61]

    , year = 2021, month = feb, volume =

    Enhanced Lidov-Kozai migration and the formation of the transiting giant planet WD 1856+534 b. , year = 2021, month = feb, volume =. doi:10.1093/mnras/staa3723 , archivePrefix =

    work page doi:10.1093/mnras/staa3723 2021

  62. [62]

    , year = 2018, month = oct, volume =

    Photoevaporation and high-eccentricity migration created the sub-Jovian desert. , year = 2018, month = oct, volume =. doi:10.1093/mnras/sty1760 , archivePrefix =

    work page doi:10.1093/mnras/sty1760 2018

  63. [63]

    , year = 2015, month = jan, volume =

    Steady-state Planet Migration by the Kozai-Lidov Mechanism in Stellar Binaries. , year = 2015, month = jan, volume =. doi:10.1088/0004-637X/799/1/27 , archivePrefix =

    work page doi:10.1088/0004-637x/799/1/27 2015

  64. [64]

    , year = 2017, month = sep, volume =

    Greatly Enhanced Merger Rates of Compact-object Binaries in Non-spherical Nuclear Star Clusters. , year = 2017, month = sep, volume =. doi:10.3847/1538-4357/aa8628 , archivePrefix =

    work page doi:10.3847/1538-4357/aa8628 2017

  65. [65]

    , year = 2013, month = oct, volume =

    Greatly enhanced eccentricity oscillations in quadruple systems composed of two binaries: implications for stars, planets, and transients. , year = 2013, month = oct, volume =. doi:10.1093/mnras/stt1281 , archivePrefix =

    work page doi:10.1093/mnras/stt1281 2013

  66. [66]

    , year = 1977, month = dec, volume =

    Love numbers of the giant planets. , year = 1977, month = dec, volume =

    1977

  67. [67]

    , year = 2018, month = apr, volume =

    On the Terminal Rotation Rates of Giant Planets. , year = 2018, month = apr, volume =

    2018

  68. [68]

    , year = 2026, month = jun, volume =

    An Enhanced Formation Channel for Galactic Dual-line Gravitational-wave Sources: von Zeipel-Lidov-Kozai Effect in Triples Involving Sgr A*. , year = 2026, month = jun, volume =

    2026

  69. [69]

    , year = 1939, month = mar, volume =

    Apsidal motion in binary stars. , year = 1939, month = mar, volume =

    1939

  70. [70]

    , year = 2016, month = aug, volume =

    Merging binaries in the Galactic Center: the eccentric Kozai-Lidov mechanism with stellar evolution. , year = 2016, month = aug, volume =. doi:10.1093/mnras/stw1220 , archivePrefix =

    work page doi:10.1093/mnras/stw1220 2016

  71. [71]

    , year = 2021, month = nov, volume =

    Giant Planets, Tiny Stars: Producing Short-period Planets around White Dwarfs with the Eccentric Kozai-Lidov Mechanism. , year = 2021, month = nov, volume =. doi:10.3847/1538-4357/ac22a9 , archivePrefix =

    work page doi:10.3847/1538-4357/ac22a9 2021

  72. [72]

    The eccentric Kozai--Lidov effect as a resonance phenomenon. Celest. Mech. Dyn. Astron. , year = 2018, month = dec, volume =

    2018

  73. [73]

    , year = 2025, month = oct, volume =

    Gas meets Kozai: the influence of a gas-rich accretion disc on hierarchical triples undergoing von Zeipel--Lidov--Kozai oscillations. , year = 2025, month = oct, volume =. doi:10.1093/mnras/staf1592 , archivePrefix =

    work page doi:10.1093/mnras/staf1592 2025

  74. [74]

    Application to Algol and lambda Tauri

    The three-body problem and eclipsing binaries. Application to Algol and lambda Tauri. , year = 1975, month = aug, volume =

    1975

  75. [75]

    , year = 2013, month = dec, volume =

    Extreme Orbital Evolution from Hierarchical Secular Coupling of Two Giant Planets. , year = 2013, month = dec, volume =

    2013

  76. [76]

    , year = 2019, month = jun, volume =

    Formation of hot Jupiters through secular chaos and dynamical tides. , year = 2019, month = jun, volume =. doi:10.1093/mnras/stz1011 , archivePrefix =

    work page doi:10.1093/mnras/stz1011 2019

  77. [77]

    , year = 2023, volume =

    The Hamiltonian for von Zeipel--Lidov--Kozai oscillations. , year = 2023, volume =

    2023

  78. [78]

    Astronomy Letters , year = 1999, month = jul, volume =

    Evolution of the orbits of distant satellites of Uranus. Astronomy Letters , year = 1999, month = jul, volume =

    1999

  79. [79]

    , year = 2019, month = apr, volume =

    Chaotic tides in migrating gas giants: forming hot and transient warm Jupiters via Lidov-Kozai migration. , year = 2019, month = apr, volume =. doi:10.1093/mnras/stz354 , archivePrefix =

    work page doi:10.1093/mnras/stz354 2019

  80. [80]

    , year = 2010, month = jun, volume =

    Secular Orbital Dynamics of Hierarchical Two-planet Systems. , year = 2010, month = jun, volume =. doi:10.1088/0004-637X/715/2/803 , archivePrefix =

    work page doi:10.1088/0004-637x/715/2/803 2010

Showing first 80 references.