Kozai Mechanism Research Articles

Orbital stability of two circumbinary planets around misaligned eccentric binaries

ABSTRACT With n-body simulations, we investigate the stability of tilted circumbinary planetary systems consisting of two non-zero mass planets. The planets are initially in circular orbits that are coplanar to each other, as would be expected if they form in a flat but tilted circumbinary gas disc and decouple from the disc within a time difference that is much less than the disc nodal precession period. We constrain the parameters of stable multiple planet circumbinary systems. Both planet–planet and planet–binary interactions can cause complex planet tilt oscillations that can destabilize the orbits of one or both planets. The system is considerably more unstable than the effects of these individual interactions would suggest, due to the interplay between these two interactions. The stability of the system is sensitive to the binary eccentricity, the orbital tilt, and the semimajor axes of the two circumbinary planets. With an inner planet semimajor axis of $5\, a_{\rm b}$, where ab is the semimajor axis of the binary, the system is generally stable if the outer planet is located at $\gtrsim 8\, a_{\rm b}$, beyond the 2:1 mean motion resonance with the inner planet. For larger inner planet semimajor axis, the system is less stable because the von-Zeipel–Kozai–Lidov mechanism plays a significant role, particularly for low binary-eccentricity cases. For the unstable cases, the most likely outcome is that one planet is ejected and the other remains bound on a highly eccentric orbit. Therefore, we suggest that this instability is an efficient mechanism for producing free-floating planets.

Semi-analytical orbital model around an oblate body with an inclined eccentric perturber

The semi-analytical model (based on the averaging technique) for long-term orbital evolution has proven to be useful in both astrophysical and astrodynamical contexts. In this secular approximation, orbits exhibit rich evolutionary behaviors under the effects of various perturbations. For example, in the hierarchical three-body systems, the Lidov-Kozai mechanism based on the quadrupole-level third-body perturbation model can produce large-amplitude oscillations of the eccentricity and inclination. In recent years, the octupole order has been found to induce dramatically new features when the perturbing body’s orbit is eccentric, including extremely high eccentricities and orbit flips between prograde and retrograde. Motivated by the striking effects of the octupole-order terms, we intend to derive a more general dynamical model by incorporating J2 of the central body and the inclined eccentric third-body perturbation to the hexadecapole order with its non-spherical gravity also included. This issue can be relevant for astrophysical and astrodynamical systems such as planets in stellar binaries, irregular satellites in planetary systems, and artificial probes about binary asteroid systems. Applications to the binary asteroid system 4951 Iwamoto and a fictitious exoplanetary system are illustrated as examples to validate our secular model. From these numerical results, we show the high accuracy of our secular model. Also, we show the important role of these high-order terms and the effects of the third-body’s inclination and eccentricity. Besides, we have found several different secular effects that could drive large eccentricities.

Precession and polar alignment of accretion discs in triple (or multiple) stellar systems

ABSTRACTWe investigate the mechanism of polar alignment for accretion discs in hierarchical systems (HSs) with more than two stars. In eccentric binary systems, low-mass discs that are sufficiently tilted to the binary orbit align in a polar configuration with respect to the binary plane by aligning their angular momentum to the binary eccentricity vector. In HSs, secular evolution of the orbital parameters makes the eccentricity vector of the system precess with time. This precession undermines the stability of the polar orbit for accretion discs hosted in HSs. We analytically show that the binary criteria for polar alignment derived in the literature are necessary but not sufficient conditions for polar alignment in HSs. Then, we derive an analytical criterion for polar alignment in HSs. In general, we find that discs orbiting the innermost level of an HS can go polar. Conversely, radially extended discs orbiting the outer levels of an HS cannot polarly align and evolve as orbiting around a circular binary. We confirm our findings through detailed numerical simulations. Also, our results are compatible with the observed distribution of disc–orbit mutual inclination. Finally, we compare the observed distribution of disc inclinations in the binary and in the HS populations. Binaries host mainly coplanar discs, while HSs show a wide range of disc inclinations. We suggest that the wider range of inclinations in HSs results from the secular oscillation of their orbital parameters (such as Kozai–Lidov oscillations), rather than from a different initial condition or evolution between HSs and binaries.

Formation of polar circumstellar discs in binary star systems

ABSTRACT We investigate the flow of material from highly misaligned and polar circumbinary discs that feed the formation of circumstellar discs around each binary component. With 3D hydrodynamic simulations we consider equal mass binaries with low eccentricity. We also simulate inclined test particles and highly misaligned circumstellar discs around one binary component for comparison. During Kozai–Lidov (KL) cycles, the circumstellar disc structure is altered through exchanges of disc eccentricity with disc tilt. Highly inclined circumstellar discs and test particles around individual binary components can experience very strong KL oscillations. The continuous accretion of highly misaligned material from the circumbinary disc allows the KL oscillations of circumstellar discs to be long-lived. In this process, the circumbinary material is continuously delivered with a high inclination to the lower inclination circumstellar discs. We find that the simulation resolution is important for modelling the longevity of the KL oscillations. An initially polar circumbinary disc forms nearly polar, circumstellar discs that undergo KL cycles. The gas steams accreting onto the polar circumstellar discs vary in tilt during each binary orbital period, which determines how much material is accreted onto the discs. The long-lived KL cycles in polar circumstellar discs may lead to the formation of polar S-type planets in binary star systems.

Lagrange versus Lyapunov Stability of Hierarchical Triple Systems: Dependence on the Mutual Inclination between Inner and Outer Orbits

While there have been many studies examining the stability of hierarchical triple systems, the meaning of “stability” is somewhat vague and has been interpreted differently in previous literatures. The present paper focuses on “Lagrange stability,” which roughly refers to the stability against the escape of a body from the system, or “disruption” of the triple system, in contrast to “Lyapunov-like stability,” which is related to the chaotic nature of the system dynamics. We compute the evolution of triple systems using direct N-body simulations up to 107 P out, which is significantly longer than previous studies (with P out being the initial orbital period of the outer body). We obtain the resulting disruption timescale T d as a function of the triple orbital parameters with particular attention to the dependence on the mutual inclination between the inner and outer orbits, i mut. By doing so, we have clarified explicitly the difference between Lagrange and Lyapunov stabilities in astronomical triples. Furthermore, we find that the von Zeipel–Kozai–Lidov oscillations significantly destabilize inclined triples (roughly with 60° < i mut < 150°) relative to those with i mut = 0°. On the other hand, retrograde triples with i mut > 160° become strongly stabilized with much longer disruption timescales. We show the sensitivity of the normalized disruption timescale T d/P out to the orbital parameters of triple system. The resulting T d/P out distribution is practically more useful in a broad range of astronomical applications than the stability criterion based on the Lyapunov divergence.

Common envelope evolution and triple dynamics as potential pathways to form the inner white dwarf + brown dwarf binary of the triple star system Gaia 0007−1605

ABSTRACT The recently discovered system Gaia 0007−1605 consisting of a white dwarf (WD) with a close brown dwarf companion and a distant WD tertiary very much resembles the triple system containing the first transiting planet candidate around a WD ever discovered: WD 1856+534. We have previously argued that the inner binary in WD 1856+534 most likely formed through common envelope evolution but triple star dynamics represent an alternative scenario. Here, we analyse different formation scenarios for Gaia 0007−1605. We reconstructed the potential common envelope evolution of the system and found that assuming standard parameters for the energy budget provides a reasonable solution. In agreement with other close white dwarf + brown dwarf binaries, and in contrast to WD 1856+534, no energy sources other than orbital energy during common envelope evolution are required to understand the current configuration of the system. In addition, using analytical prescriptions for triple dynamics, we show that Von Zeipel–Lidov–Kozai oscillations might have triggered tidal migration due to high-eccentricity incursions (e ≳ 0.997). We conclude that the inner binary in Gaia 0007−1605, as its sibling WD 1856+534, formed either through common envelope evolution, triple dynamics, or a combination of both mechanisms.

On the observed excess of large Mars-crossers in high-inclination orbits

Mars-crossing asteroids (MCAs) constitute a population of transient bodies coming from the asteroid belt that end up in the near-Earth region or are ejected hyperbolically. We analyze the dynamical evolution of a sample of 1066 MCAs with absolute magnitudes H<16 for 800 Myr. We find that the sample can be split in two groups: the high-inclination group (i≳15°) and the low-inclination group (i≲15°). Asteroids of the first group generally have much longer dynamical lifetimes than those of the latter group. We focus our attention on the largest members of the MCAs in high-i orbits which seem to be in excess relative to those of smaller sizes. We followed in particular the dynamical evolution of (132) Aethra – the largest MCA – and 9 clones for an extended period of 2 Gyr. We found that Aethra and all its clones remain in a stable orbit for the whole studied period. Despite Aethra partially crosses Mars’s orbit, no close encounters with this planet are registered (within a Hill radius) which is due to the Kozai mechanism by which the oscillations of Aethra’s perihelion distance are coupled with the precession of the argument of perihelion. Given its size, the Yarkovsky mechanism, that causes a size-dependent drift in the semimajor axis, is probably too small to drive Aethra’s semimajor axis to an unstable zone in the asteroid belt on a Gyr timescale. Long dynamical lifetimes (at least several hundreds Myr) are also found for other large MCAs (D≳10 km) in high-i orbits, which also share Aethra-like dynamics. Their long dynamical lifetimes may explain their observed excess in comparison with what would be expected from the extrapolation of the size distribution of smaller members of the population, whose dynamics may be affected by the Yarkovsky mechanism that destabilizes their orbits and shortens their lifetimes. By contrast, there are very few large MCAs in the low-i population, which suggests an insufficient replenishment rate of big bodies in low-i orbits. The time scale to shift big fragments produced in catastrophic collisions to unstable zones of the main asteroid belt by the Yarkovsky mechanism may very likely be too long in comparison with the dynamical lifetimes of low-i MCAs. The frequency of collisions of the Earth with any of the large MCAs is estimated to be of one every several hundreds Myr and it will most likely move on a low-i orbit.

On the masses, age, and architecture of the VHS J1256−1257AB b system

ABSTRACT VHS J1256−1257 AB is an ultracool dwarf binary that hosts a wide-separation planetary-mass companion that is a key target of the JWST Exoplanet Early Release Science programme. Using Keck adaptive optics imaging and aperture masking interferometry, we have determined the host binary’s orbit (a = 1.96 ± 0.03 au, P = 7.31 ± 0.02 yr, e = 0.883 ± 0.003) and measured its dynamical total mass (0.141 ± 0.008 M⊙). This total mass is consistent with VHS J1256−1257 AB being a brown dwarf binary or pair of very low-mass stars. In addition, we measured the orbital motion of VHS J1256−1257 b with respect to the barycentre of VHS J1256−1257 AB, finding that the wide companion’s orbit is also eccentric ($e=0.68^{+0.11}_{-0.10}$), with a mutual inclination of 115○ ± 14○ with respect to the central binary. This orbital architecture is consistent with VHS J1256−1257 b attaining a significant mutual inclination through dynamical scattering and thereafter driving Kozai–Lidov cycles to pump the eccentricity of VHS J1256−1257 AB. We derive a cooling age of 140 ± 20 Myr for VHS J1256−1257 AB from low-mass stellar/substellar evolutionary models. At this age, the luminosity of VHS J1256−1257 b is consistent with both deuterium-inert and deuterium-fusing evolutionary tracks. We thus find a bimodal probability distribution for the mass of VHS J1256−1257 b, either 12.0 ± 0.1 MJup or 16 ± 1 MJup, from these models. Future spectroscopic data to measure isotopologues such as HDO and CH3D could break this degeneracy and provide a strong test of substellar models at the deuterium-fusion mass boundary.

Making hot Jupiters in stellar clusters: The importance of binary exchange

ABSTRACT It has been suggested that the occurrence rate of hot Jupiters (HJs) in open clusters might reach several per cent, significantly higher than that of the field (∼a per cent). In a stellar cluster, when a planetary system scatters with a stellar binary, it may acquire a companion star, which may excite large-amplitude von Zeipel–Lidov–Kozai oscillations in the planet’s orbital eccentricity, triggering high-eccentricity migration, and the formation of an HJ. We quantify the efficiency of this mechanism by modelling the evolution of a gas giant around a solar mass star under the influence of successive scatterings with binary and single stars. We show that the chance that a planet ∈ (1, 10) au becomes an HJ in a Gyr in a cluster of stellar density n* = 50 pc−3, and binary fraction fbin = 0.5 is about 2 per cent and an additional 4 per cent are forced by the companion star into collision with or tidal disruption by the central host. An empirical fit shows that the total percentage of those outcomes asymptotically reaches an upper limit determined solely by fbin (e.g. 10 per cent at fbin = 0.3 and 18 per cent at fbin = 1) on a time-scale inversely proportional to n* (∼Gyr for n* ∼ 100 pc−3). The ratio of collisions to tidal disruptions is roughly a few, and depends on the tidal model. Therefore, if the giant planet occurrence rate is 10 per cent, our mechanism implies an HJ occurrence rate of a few times 0.1 per cent in a Gyr and can thus explain a substantial fraction of the observed rate.

Ephemerides of the Irregular Saturnian Satellites from Earth-based Astrometry and Cassini Imaging* *©2022. All rights reserved.

We report a new determination of the orbits of the irregular Saturnian satellites. We fit their numerically integrated orbits to a data set containing Earth-based observations and imaging data from the Cassini spacecraft. We include the statistics of the observation residuals, the satellites’ orbital elements, and projected accuracies of the satellites’ positions. We also provide astrometric positions derived from the Cassini imaging. Two of the satellites are considered lost because they have not been observed for more than one epoch and have indeterminate uncertainties in their positions. Three of the satellites appear to be in a Kozai resonance, with one being the first irregular satellite of any planet found to be in a 270° rather than 90° resonance.

Orbital Flips Caused by the Eccentric Von Zeipel–Lidov–Kozai Effect in Nonrestricted Hierarchical Planetary Systems

The eccentric von Zeipel–Lidov–Kozai effect is widely applied to diverse astrophysical settings. In the restricted hierarchical three-body system, the topic of orbital flips has been extensively studied. However, it is far from being understood for nonrestricted circumstances. In this work, the dynamics of orbital flips are investigated under the Hamiltonian model at the octupole-level approximation for nonrestricted hierarchical planetary systems, where the outer planet is more massive than the inner one. Numerical distribution of flipping orbits shows that there are three major flipping regions, distributed in low-, intermediate-, and high-eccentricity spaces. Poincaré sections indicate that those islands of libration centered near i tot = 90° can lead to orbital flips. Thus, we refer to the behavior of orbital flips as a resonance phenomenon. From this viewpoint, dynamical models of orbital flips can be described by a separable Hamiltonian, which can be treated by a means of perturbation theory. The resonant model for orbital flips is formulated based on the adiabatic invariant approximation and then phase portraits are generated by plotting level curves of adiabatic invariants with the given Hamiltonian. By analyzing phase portraits, analytical boundaries of libration and circulation zones causing orbital flips are obtained. As expected, the numerical and analytical conditions that allow orbits to flip agree well with each other. The phenomenon of orbital flips in nonrestricted hierarchical problems can be well understood with the help of dynamical structures of secular resonance.

The mystery in Gaia DR3 triples: occurrence rates, orientations, and eccentricities of wide tertiaries around close binaries

ABSTRACT The formation of close binaries has been an open question for decades. A large fraction of close binaries are in triple systems, suggesting that their formation may be associated with the Kozai–Lidov mechanism. However, this picture remains under debate because the configurations of many observed triples are unlikely to trigger the Kozai–Lidov mechanism. In this paper, we use the close binary samples, including eclipsing, spectroscopic, and astrometric binaries, from Gaia Data Release 3 to investigate the mysterious connection between inner binaries and their wide tertiaries. We show that the wide tertiary (at 103–104 au) fraction increases with decreasing orbital periods of the inner binaries. The wide tertiary fraction of eclipsing binaries (a median orbital period of 0.41 d) is 2.33 ± 0.11 times higher than the field wide binary fraction. Furthermore, there is a tentative excess at ∼104 au for tertiaries of eclipsing binaries. Using the v − r angle distributions, we show that wide tertiaries have isotropic orientations with respect to the inner binaries, and the co-planar orbits can be ruled out. The inferred eccentricity distribution of wide tertiaries is consistent (&amp;lt;1σ) with being thermal (f(e) ∝ e), similar to wide binaries at similar separations. The dynamical unfolding scenario is disfavoured because it predicts highly eccentric wide tertiaries, which is inconsistent with our findings. For the Kozai–Lidov mechanism to be effective for wide tertiaries at &amp;gt;103 au, the initial separations of the inner binaries need to be &amp;gt;3 au. Future theoretical investigations are needed to explore the parameter space at these large initial separations and large tertiary separations.

Production of hot Jupiter candidates from high-eccentricity mechanisms for different initial planetary mass configurations

ABSTRACT Hot Jupiters (HJs) are giant planets with orbital periods of the order of a few days with semimajor axis within ∼0.1 au. Several theories have been invoked in order to explain the origin of this type of planets, one of them being the high-eccentricity migration. This migration can occur through different high-eccentricity mechanisms. Our investigation focused on six different kinds of high-eccentricity mechanisms, namely, direct dispersion, coplanar, Kozai–Lidov, secular chaos, E1 and E2 mechanisms. We investigated the efficiency of these mechanisms for the production of HJ candidates in multiplanet systems initially tightly-packed in the semimajor axis, considering a large set of numerical simulations of the exact equations of motion in the context of the N-body problem. In particular, we analyzed the sensitivity of our results to the initial number of planets, the initial semimajor axis of the innermost planetary orbit, the initial configuration of planetary masses, and to the inclusion of general relativity (GR) effects. We found that the E1 mechanism is the most efficient in producing HJ candidates both in simulations with and without the contribution of GR, followed by the Kozai–Lidov and E2 mechanisms. Our results also revealed that, except for the initial equal planetary mass configuration, the E1 mechanism was notably efficient in the other initial planetary mass configurations considered in this work. Finally, we investigated the production of HJ candidates with prograde, retrograde, and alternating orbits. According to our statistical analysis, the Kozai–Lidov mechanism has the highest probability of significantly exciting the orbital inclinations of the HJ candidates.

Extremely Inclined Orbit of the S-type Planet γ Cep Ab Induced by the Eccentric Kozai–Lidov Mechanism

γ Cep Ab is a typical S-type planet, which occupies a nearly perpendicular planetary orbit relative to the binary. Here, we use a Markov Chain Monte Carlo sampler to conduct a full N-body fitting and derive self-consistent orbital solutions for this hierarchical system. We then employ the eccentric Kozai–Lidov (EKL) mechanism to explain the extremely inclined orbit of the S-type planet γ Cep Ab. The EKL mechanism plays an essential part in our exploration of the significant oscillations of the mutual inclination i mut between the planet and the secondary star. We perform a qualitative analysis and use extensive numerical integrations to investigate the flip conditions and timescales of γ Cep Ab’s orbit. When the planetary mass is 15 M Jup, the planet can reach i mut ∼ 113° given the critical initial conditions of i mut < 60° and e 1 < 0.7. The timescale for the first orbital flip decreases with the increase of the perturbation Hamiltonian. The flipping orbits of γ Cep Ab are confirmed to have a large possibility of remaining stable, based on surfaces of section and the secular stability criterion. Furthermore, we extend the application of EKL to general S-type planetary systems with a 1/a 2 ≤ 0.1, where the most intense excitation of i mut occurs when a 1/a 2 = 0.1 and e 2 ∼ 0.8, and the variation in planetary mass mainly affects the flip possibility where e 1 ≤ 0.3.

On the Mass Ratio Distribution of Black Hole Mergers in Triple Systems

Observations have shown that the majority of massive stars, the progenitors of black holes (BHs), have on average more than one stellar companion. In triple systems, wide inner binaries can be driven to a merger by a third body due to long-term secular interactions, most notably by the eccentric Lidov–Kozai effect. In this study, we explore the properties of BH mergers in triple systems and compare their population properties to those of binaries produced in isolation and assembled in dense star clusters. Using the same stellar physics and identical assumptions for the initial populations of binaries and triples, we show that stellar triples yield a significantly flatter mass ratio distribution from q = 1 down to q ∼ 0.3 than either binary stars or dense stellar clusters, similar to the population properties inferred from the most recent catalog of gravitational-wave events, though we do not claim that all the observed events can be accounted for with triples. While hierarchical mergers in clusters can also produce asymmetric mass ratios, the unique spins of such mergers can be used to distinguish them from those produced from stellar triples. All three channels occupy distinct regions in the total mass–mass ratio space, which may allow them to be disentangled as more BH mergers are detected by LIGO, Virgo, and KAGRA.

Non-gravitational parameters and orbital stability of asteroids in retrograde orbits

ABSTRACT We studied the dynamical evolution of all numbered and selected unnumbered asteroids in a retrograde orbit. We computed the absolute magnitude, size range, and non-gravitational parameters A2 and $Lyapunov\, times$ during the 2 Myr forward and backward orbital evolution of all 21 studied retrograde asteroids. We also studied non-gravitation parameters of 10 unnumbered retrograde asteroids brighter than 18.1 mag. It turns out that the values of A2 are of a similar order as for comets, i.e. on average, a few orders more than in the case of asteroids that are moving in prograde orbits. We investigated how the change of the non-gravitational effect influences the Lyapunov time and the evolution of the semimajor-axis-selected retrograde orbits during the forward and backward integration. We also indicated the roles of Lidov–Kozai resonance in the orbital evolution of the retrograde asteroid.

Dynamical essence of the eccentric von Zeipel-Lidov-Kozai effect in restricted hierarchical planetary systems

Aims. The eccentric von Zeipel–Lidov–Kozai (ZLK) effect is widely used to explain dynamical phenomena in a variety of astrophysical systems. The purpose of this work is to clarify the dynamical essence of the eccentric ZLK effect by constructing an inherent connection between this effect and the dynamics of secular resonance in restricted hierarchical planetary systems. Methods. Dynamical structures of apsidal resonance were studied analytically by means of perturbative treatments. The resonant model was formulated by averaging the Hamiltonian (up to octupole order) over rotating ZLK cycles, producing an additional motion integral. The phase portraits under the resonant model can be used to analyse dynamical structures, including resonant centres, dynamical separatrices, and islands of libration. Results. By analysing phase portraits, five branches of libration centres and eight libration zones are found in eccentricity-inclination space. The analytical results of the libration zone and the numerical distributions of the resonant orbit agree very well, indicating that the resonant model for apsidal resonances is valid and applicable. Additionally, we found that in the test-particle limit, the distributions of flipping orbits are dominated by the apsidal resonances that are centred at an inclination of i = 90°. Conclusions. The eccentric ZLK effect is dynamically equivalent to the effect of apsidal resonance in restricted hierarchical planetary systems. The dynamical response of the eccentric ZLK effect (or of the effect of apsidal resonance) is to significantly excite the eccentricities and/or inclinations of test particles in the very long-term evolution.

The Von Zeipel–Lidov–Kozai Effect inside Mean Motion Resonances with Applications to Trans-Neptunian Objects

Secular dynamics inside mean motion resonances (MMRs) plays an essential role in governing the dynamical structure of the trans-Neptunian region and sculpting the orbital distribution of trans-Neptunian objects (TNOs). In this study, semianalytical developments are made to explore the von Zeipel–Lidov–Kozai resonance inside MMRs. To this end, a semi-secular model is formulated from averaging theory and then a single-degree-of-freedom integrable model is achieved based on the adiabatic invariance approximation. In particular, we introduce a modified adiabatic invariant, which is continuous around the separatrices of MMRs. During long-term evolution, both the resonant Hamiltonian and the adiabatic invariant remain unchanged, thus phase portraits can be produced by plotting level curves of the adiabatic invariant with a given Hamiltonian. The phase portraits provide global pictures to predict long-term behaviors of the eccentricity, inclination, and argument of pericenter. Applications to some representative TNOs inside MMRs (2018 VO137, 2005 SD278, 2015 PD312, Pluto, 2004 HA79, 1996 TR66, and 2014 SR373) show good agreements between the numerically propagated trajectories under the full N-body model and the level curves arising in phase portraits. Interestingly, 2018 VO137 and 2005 SD278 exhibit switching behaviors during their long-term evolution and currently they are inside 2:5 MMR with Neptune.

Black Hole Mergers through Evection Resonances

Mechanisms have been proposed to enhance the merger rate of stellar-mass black hole binaries, such as the Von Zeipel–Lidov–Kozai mechanism (vZLK). However, high inclinations are required in order to greatly excite the eccentricity and to reduce the merger time through vZLK. Here, we propose a novel pathway through which compact binaries could merge due to eccentricity increase in general, including in a near coplanar configuration. Specifically, a compact binary migrating in an active galactic nucleus disk could be captured in an evection resonance, when the precession rate of the binary equals the orbital period around the supermassive black hole. In our study we include precession due to first-order post-Newtonian precession as well as that due to disk around one or both components of the binary. Eccentricity is excited when the binary sweeps through the resonance, which happens only when it migrates on a timescale 10–100 times the libration timescale of the resonance. Libration timescale decreases as the mass of the disk increases. The eccentricity excitation of the binary can reduce the merger timescale by up to a factor of ∼103−5.

Extreme Eccentricities of Triple Systems: Analytic Results

Triple stars and compact objects are ubiquitously observed in nature. Their long-term evolution is complex; in particular, the von Zeipel–Lidov–Kozai (ZLK) mechanism can potentially lead to highly eccentric encounters of the inner binary. Such encounters can lead to a plethora of interacting binary phenomena, as well as stellar and compact-object mergers. Here we find implicit analytical formulae for the maximal eccentricity, , of the inner binary undergoing ZLK oscillations, where both the test-particle limit (parameterized by the inner-to-outer angular momentum ratio η) and the double-averaging approximation (parameterized by the period ratio, ϵ SA) are relaxed, for circular outer orbits. We recover known results in both limiting cases (either η or ϵ SA → 0) and verify the validity of our model using numerical simulations. We test our results with two accurate numerical N-body codes, rebound for Newtonian dynamics and tsunami for general-relativistic dynamics, and find excellent correspondence. We discuss the implications of our results for stellar triples and both stellar and supermassive triple black hole mergers.