Lidov Kozai Research Articles

The OATMEAL Survey. I. Low Stellar Obliquity in the Transiting Brown Dwarf System GPX-1

We introduce the OATMEAL survey, an effort to measure the obliquities of stars with transiting brown dwarf companions. We observed a transit of the close-in (P orb = 1.74 days) brown dwarf GPX-1 b using the Keck Planet Finder spectrograph to measure the sky-projected angle between its orbital axis and the spin axis of its early F-type host star (λ). We measured λ = 6.°9 ± 10.°0, suggesting an orbit that is prograde and well aligned with the stellar equator. Hot Jupiters around early F stars are frequently found to have highly misaligned orbits, with polar and retrograde orbits being commonplace. It has been theorized that these misalignments stem from dynamical interactions, such as von Zeipel–Kozai–Lidov cycles, and are retained over long timescales due to weak tidal dissipation in stars with radiative envelopes. By comparing GPX-1 to similar systems under the frameworks of different tidal evolution theories, we argued that the rate of tidal dissipation is too slow to have re-aligned the system. This suggests that GPX-1 may have arrived at its close-in orbit via coplanar high-eccentricity migration or migration through an aligned protoplanetary disk. Our result for GPX-1 is one of few measurements of the obliquity of a star with a transiting brown dwarf. By enlarging the number of such measurements and comparing them with hot-Jupiter systems, we will more clearly discern the differences between the mechanisms that dictate the formation and evolution of both classes of objects.

KPF Confirms a Polar Orbit for KELT-18 b

We present the first spectroscopic transit results from the newly commissioned Keck Planet Finder on the Keck-I telescope at W. M. Keck Observatory. We observed a transit of KELT-18 b, an inflated ultrahot Jupiter orbiting a hot star (T eff = 6670 K) with a binary stellar companion. By modeling the perturbation to the measured cross-correlation functions using the Reloaded Rossiter–McLaughlin technique, we derived a sky-projected obliquity of λ = − 94.°8 ± 0.°7 ( ψ=93.8−1.8+1.6° for isotropic i ⋆). The data are consistent with an extreme stellar differential rotation (α = 0.9), though a more likely explanation is moderate center-to-limb variations of the emergent stellar spectrum. We see additional evidence for the latter from line widths increasing toward the limb. Using loose constraints on the stellar rotation period from observed variability in the available TESS photometry, we were able to constrain the stellar inclination and thus the true 3D stellar obliquity to ψ=91.7−1.8+2.2° . KELT-18 b could have obtained its polar orbit through high-eccentricity migration initiated by Kozai–Lidov oscillations induced by the binary stellar companion KELT-18 B, as the two likely have a large mutual inclination as evidenced by Gaia astrometry. KELT-18 b adds another data point to the growing population of close-in polar planets, particularly around hot stars.

Detailed Abundances of the Planet-hosting TOI-1173 A/B System: Possible Evidence of Planet Engulfment in a Very Wide Binary

Over the last decade, studies of large samples of binary systems have identified chemical anomalies and shown that they might be attributed to planet formation or planet engulfment. However, both scenarios have primarily been tested in pairs without known exoplanets. In this work, we explore these scenarios in the newly detected planet-hosting wide binary TOI-1173 A/B (projected separation ∼11,400 au), using high-resolution MAROON-X and ARCES spectra. We determined photospheric stellar parameters both by fitting stellar models and via the spectroscopic equilibrium approach. Both analyses agree and suggest that they are cool main-sequence stars located in the thin disk. A line-by-line differential analysis between the components (B−A) displays an abundance pattern in the condensation temperature plane, where the planet-hosting star TOI-1173 A is enhanced in refractory elements such as iron by more than 0.05 dex. This suggests the engulfment of ∼18 M ⊕ of rocky material in star A. Our hypothesis is supported by the dynamics of the system (detailed in our companion paper), which suggest that the super-Neptune TOI-1173 A b might have been delivered to its current short period (∼7 days) through circularization and von Zeipel–Lidov–Kozai mechanisms, thereby triggering the engulfment of inner rocky exoplanets.

On the Dynamical Evolution of the Asteroid Belt in a Massive Star–Neutron Star Binary

Some fast radio bursts (FRBs) exhibit repetitive behaviors, and their origins remain enigmatic. It has been argued that repeating FRBs could be produced by the interaction between a neutron star and an asteroid belt. Here, we consider the systems in which an asteroid belt dwells around a massive star, while a neutron star, as a companion of the massive star, interacts with the belt through gravitational force. Various orbital configurations are assumed for the system. Direct N-body simulations are performed to investigate the dynamical evolution of the asteroids' belt. It is found that a larger orbital eccentricity of the neutron star will destroy the belt more quickly, with a large number of asteroids being scattered out of the system. A low inclination not only suppresses the collisions but also inhibits the ejection rate at early stages. However, highly inclined systems may undergo strong oscillations, resulting in the Kozai–Lidov instabilities. Among the various configurations, a clear periodicity is observed in the collision events for the case with an orbital eccentricity of 0.7 and mutual inclination of 0◦. It is found that such a periodicity can be sustained for at least eight neutron star orbital periods, supporting this mechanism as a possible explanation for periodically repeating FRBs. Our studies also suggest that the active stage of these kinds of FRB sources should be limited, since the asteroid belt would finally be destroyed by the neutron star after multiple passages.

Analytical Models for Secular Descents in Hierarchical Triple Systems

Three-body systems are prevalent in nature, from planetary to stellar to supermassive black hole scales. In a hierarchical triple system, oscillations of the inner orbit’s eccentricity and inclination can be induced on secular timescales. Over many cycles, the octupole-level terms in the secular equations of motion can drive the system to extremely high eccentricities via the eccentric Kozai–Lidov (EKL) mechanism. The overall decrease in the inner orbit’s pericenter distance has potentially dramatic effects for realistic systems, such as tidal disruption events. We present an analytical approximation in the test-particle limit to describe individual stepwise increases in eccentricity of the inner orbit. A second approximation, also in the test-particle limit, is obtained by integrating the equations of motion and calibrating to numerical simulations to estimate the overall octupole-level time evolution of the eccentricity. The latter approach is then extended beyond the test particle to the general case. The three novel analytical approximations are compared to numerical solutions to show that the models accurately describe the form and timescale of the secular descent from large distances to a close-encounter distance (e.g., the Roche limit). By circumventing the need for numerical simulations to obtain the long-term behavior, these approximations can be used to readily estimate properties of close encounters and descent timescales for populations of systems. We demonstrate this by calculating rates of EKL-driven migration for Hot Jupiters in stellar binaries.

Dynamical and Secular Stability of Mutually Inclined Planetary Systems

Multiple analytical, semi-analytical, and empirical stability criteria have been derived in the literature for two-planet systems. But, the dependence of the stability limit on the initial mutual inclination between the inner and outer orbits is not well modeled by previous stability criteria. Here, we derive a semi-analytical stability criteria for two-planet systems, at arbitrary inclinations, in which the inner planet is a test particle. Using perturbation theory we calculate the characteristic fractional change in the semimajor axis of the inner binary β = δ a 1/a 1 caused by perturbations from the companion. Stability criteria can be derived by setting a threshold on β. Focusing initially on circular orbits, we derive an analytical expression for β for coplanar prograde and retrograde orbits. For noncoplanar configurations, we evaluate a semi-analytical expression. We then generalize to orbits with arbitrary eccentricities and account for the secular effects. Our analytical and semi-analytical results are in excellent agreement with direct N-body simulations. In addition, we show that contours of β ∼ 0.01 can serve as criteria for stability. More specifically, we show that (1) retrograde orbits are generally more stable than prograde ones; (2) systems with intermediate mutual inclination are less stable due to von Ziepel–Lidov–Kozai (vZLK) dynamics; and (3) mean motion resonances (MMRs) can stabilize intermediate inclination secularly unstable regions in phase space, by quenching vZLK secular processes (4) MMRs can destabilize some of the dynamically stable regions. We also point out that these stability criteria can be used to constrain the orbital properties of observed systems and their age.

Hierarchical three-body problem at high eccentricities = simple pendulum I: octupole

ABSTRACT The gradual evolution of the restricted hierarchical three body problem is analysed analytically, focusing on conditions of Kozai–Lidov cycles that may lead to orbital flips from prograde to retrograde motion due to the octupole (third order) term which are associated with extremely high eccentricities. We revisit the approach described by Katz, Dong and Malhotra (2011) and show that for most initial conditions, to an excellent approximation, the analytical derivation can be greatly simplified and reduces to a simple pendulum model allowing an explicit flip criterion. The resulting flip criterion is much simpler than the previous one but the latter is still needed in a small fraction of phase space. We identify a logical error in the earlier derivation but clarify why it does not affect the final results.

Merging hierarchical triple black hole systems with intermediate-mass black holes in population III star clusters

ABSTRACT Theoretical predictions suggest that very massive stars have the potential to form through multiple collisions and eventually evolve into intermediate-mass black holes (IMBHs) within Population III star clusters embedded in mini dark matter haloes. In this study, we investigate the long-term evolution of Population III star clusters, including models with a primordial binary fraction of $f_{\rm b}=0$ and 1, using the N-body simulation code petar. We comprehensively examine the phenomenon of hierarchical triple black holes in the clusters, specifically focusing on their merging inner binary black holes (BBHs), with post-Newtonian correction, by using the tsunami code. Our findings suggest a high likelihood of the inner BBHs containing IMBHs with masses on the order of $\mathcal {O}(100)\,{\rm M}_{\odot }$, and as a result, their merger rate could be up to $0.1{\rm Gpc}^{-3}{\rm yr}^{-3}$. The orbital eccentricities of some merging inner BBHs oscillate over time periodically, known as the Kozai–Lidov oscillation, due to dynamical perturbations. Detectable merging inner BBHs for mHz GW detectors LISA/TianQin/Taiji concentrate within $z\lt 3$. More distant sources would be detectable for CE/ET/LIGO/KAGRA/DECIGO, which are sensitive from $\mathcal {O}(0.1)$Hz to $\mathcal {O}(100)$ Hz. Furthermore, compared with merging isolated BBHs, merging inner BBHs affected by dynamical perturbations from tertiary BHs tend to have higher eccentricities, with a significant fraction of sources with eccentricities closing to 1 at mHz bands. GW observations would help constrain the formation channels of merging BBHs, whether through isolated evolution or dynamical interaction, by examining eccentricities.

Irregular fixation: I. Fixed points and librating orbits of the Brown Hamiltonian

ABSTRACT In hierarchical triple systems, the inner binary is slowly perturbed by a distant companion, giving rise to large-scale oscillations in eccentricity and inclination, known as von-Zeipel–Lidov–Kozai oscillations. Stable systems with a mild hierarchy, where the period ratio is not too small, require an additional corrective term, known as the Brown Hamiltonian, to adequately account for their long-term evolution. Although the Brown Hamiltonian has been used to accurately describe the highly eccentric systems on circulating orbits where the periapse completes a complete revolution, the analysis near its elliptical fixed points had been overlooked. We derive analytically the modified fixed points including the Brown Hamiltonian and analyse its librating orbits (where the periapse motion is limited in range). We compare our result to the direct three-body integrations of millions of orbits and discuss the regimes of validity. We numerically discover the regions of orbital instability, allowed and forbidden librating zones with a complex, fractal, structure. The retrograde orbits, where the mutual inclination is $\iota \ \gt\ 90\ \rm deg$, are more stable and allowed to librate for larger areas of the parameter space. We find numerical fits for the librating-circulating boundary. Finally, we discuss the astrophysical implications for systems of satellites, stars, and compact objects. In a companion paper (Paper II), we apply our formalism to the orbits of irregular satellites around giant planets.

Revisiting the Tertiary-induced Binary Black Hole Mergers: The Role of Superthermal Wide Tertiary Eccentricity Distributions

Recent studies show that the eccentricity distribution of wide binaries (semimajor axis ≳103 au) observed by Gaia tends to favor large eccentricities more strongly than the canonical thermal distribution (P(e) ∝ e)—such distributions are termed “superthermal.” Motivated by this observation, we revisit the formation channel of black hole (BH) binary mergers in triple stellar systems and study the impact of superthermal eccentricity distributions in the outer binaries. We explore the persistence of the highly eccentric outer orbits after each component in a stellar triple has undergone mass loss due to supernova explosions. We find that the outer eccentricity distribution can remain significantly superthermal for modestly hierarchical BH triples satisfying a in/a out ≳ 0.005 (where a in and a out are the semimajor axes of the inner and outer orbits), and are otherwise shaped by mass-loss induced kicks and dynamical instability. We then study the impact of these different outer eccentricity distributions of the remaining BH triples on mergers via the tertiary-induced channel. Of interest, we find that mergers can sometimes be produced even when the initial stellar orbits are near alignment (not subject to the von-Zeipel–Lidov–Kozai effect; ZLK effect) as long as the system is sufficiently hierarchical. On the other hand, although the impact of the octupole-order ZLK effect is much greater when the outer binary is more eccentric, we find that the merger fraction only changes modestly for extreme outer eccentricity distributions because the largest eccentricities tend to lead to dynamical instability.

Polar Orbits around the Newly Formed Earth–Moon Binary System

We examine the dynamics and stability of circumbinary particles orbiting around the Earth–Moon binary system. The moon formed close to the Earth (semimajor axis a EM ≈ 3 R ⊕) and expanded through tides to its current day semimajor axis (a EM = 60 R ⊕). Circumbinary orbits that are polar or highly inclined to the Earth–Moon orbit are subject to two competing effects: (i) nodal precession about the Earth–Moon eccentricity vector and (ii) Kozai–Lidov oscillations of eccentricity and inclination driven by the Sun. While we find that there are no stable polar orbits around the Earth–Moon orbit with the current day semimajor axis, polar orbits were stable immediately after the formation of the Moon, at the time when there was a lot of debris around the system, up to when the semimajor axis reached about a EM ≈ 10 R ⊕. We discuss implications of polar orbits on the evolution of the Earth–Moon system and the possibility of polar orbiting moons around exoplanet–moon binaries.

Detecting Gravitational-wave Bursts from Black Hole Binaries in the Galactic Center with LISA

Stellar-mass black hole binaries (BHBs) in galactic nuclei are gravitationally perturbed by the central supermassive black hole (SMBH) of the host galaxy, potentially inducing strong eccentricity oscillations through the eccentric Kozai–Lidov mechanism. These highly eccentric binaries emit a train of gravitational-wave (GW) bursts detectable by the Laser Interferometer Space Antenna (LISA)—a planned space-based GW detector—with signal-to-noise ratios up to ∼100 per burst. In this work, we study the GW signature of BHBs orbiting our galaxy’s SMBH, Sgr A*, which are consequently driven to very high eccentricities. We demonstrate that an unmodeled approach using a wavelet decomposition of the data effectively yields the time-frequency properties of each burst, provided that the GW frequency peaks between 10−3 and 10−1 Hz. The wavelet parameters may be used to infer the eccentricity of the binary, measuring log10(1−e) within an error of 20%. Our proposed search method can thus constrain the parameter space to be sampled by complementary Bayesian inference methods, which use waveform templates or orthogonal wavelets to reconstruct and subtract the signal from LISA data.

TOI-1173 A b: The First Inflated Super-Neptune in a Wide Binary System

Among Neptunian mass exoplanets (20−50 M ⊕), puffy hot Neptunes are extremely rare, and their unique combination of low mass and extended radii implies very low density (ρ < 0.3 g cm−3). Over the last decade, only a few puffy planets have been detected and precisely characterized with both transit and radial velocity observations, most notably including WASP-107 b, TOI-1420 b, and WASP-193 b. In this paper, we report the discovery of TOI-1173 A b, a low-density ( ρ=0.195−0.017+0.018 g cm−3) super-Neptune with P = 7.06 days in a nearly circular orbit around the primary G-dwarf star in the wide binary system TOI-1173 A/B. Using radial velocity observations with the MAROON-X and HIRES spectrographs and transit photometry from TESS, we determine a planet mass of M p = 27.4 ± 1.7 M ⊕ and radius of R p = 9.19 ± 0.18 R ⊕. TOI-1173 A b is the first puffy super-Neptune planet detected in a wide binary system (projected separation ∼11,400 au). We explore several mechanisms to understand the puffy nature of TOI-1173 A b and show that tidal heating is the most promising explanation. Furthermore, we demonstrate that TOI-1173 A b likely has maintained its orbital stability over time and may have undergone von-Zeipel–Lidov–Kozai migration followed by tidal circularization, given its present-day architecture, with important implications for planet migration theory and induced engulfment into the host star. Further investigation of the atmosphere of TOI-1173 A b will shed light on the origin of close-in low-density Neptunian planets in field and binary systems, while spin–orbit analyses may elucidate the dynamical evolution of the system.

Runaway Eccentricity Growth: A Pathway for Binary Black Hole Mergers in AGN Disks

Binary black holes (BBHs) embedded within the accretion disks that fuel active galactic nuclei (AGN) are promising progenitors for the source of gravitational wave (GW) events detected by LIGO/VIRGO. Several recent studies have shown that when these binaries form, they are likely to be highly eccentric and retrograde. However, many uncertainties remain concerning the orbital evolution of these binaries as they either inspiral toward merger or disassociate. Previous hydrodynamical simulations exploring their orbital evolution have been predominantly two-dimensional or have been restricted to binaries on nearly circular orbits. We present the first high-resolution, three-dimensional local shearing-box simulations of both prograde and retrograde eccentric BBHs embedded in AGN disks. We find that retrograde binaries shrink several times faster than their prograde counterparts and exhibit significant orbital eccentricity growth, the rate of which monotonically increases with binary eccentricity. Our results suggest that retrograde binaries may experience runaway orbital eccentricity growth, which may bring them close enough together at pericenter for GW emission to drive them to coalescence. Although their eccentricity is damped, prograde binaries shrink much faster than their orbital eccentricity decays, suggesting they should remain modestly eccentric as they contract toward merger. Finally, binary precession driven by the AGN disk may dominate over precession induced by the supermassive black hole depending on the binary accretion rate and its location in the AGN disk, which can subdue the evection resonance and von Ziepel–Lidov–Kozai cycles.

Eccentricity and inclination of massive planets inside low-density cavities: results of 3D simulations

ABSTRACT We study the evolution of eccentricity and inclination of massive planets in low-density cavities of protoplanetary discs using three-dimensional (3D) simulations. When the planet’s orbit is aligned with the equatorial plane of the disc, the eccentricity increases to high values of 0.7–0.9 due to the resonant interaction with the inner parts of the disc. For planets on inclined orbits, the eccentricity increases due to the Kozai–Lidov mechanism, where the disc acts as an external massive body, which perturbs the planet’s orbit. At small inclination angles, ${\lesssim}30^\circ$, the resonant interaction with the inner disc strongly contributes to the eccentricity growth, while at larger angles, eccentricity growth is mainly due to the Kozai–Lidov mechanism. We conclude that planets inside low-density cavities tend to acquire high eccentricity if favourable conditions give sufficient time for growth. The final value of the planet’s eccentricity after the disc dispersal depends on the planet’s mass and the properties of the cavity and protoplanetary disc.

Formation of the inner Oort cloud in the presence of an early solar binary

ABSTRACT Consistent with the notion that most Sun-like stars form in multistellar systems, this study explores the impact of a temporarily bound stellar binary companion on the early dynamical evolution of the Solar system. Using N-body simulations, we illustrate how such a companion markedly enhances the trapping of scattered bodies on inner Oort cloud-like orbits, with perihelion distances exceeding $q \gt 40$ au. We further find that the orbital geometry of the Sun-binary system plays a central role in regulating the efficiency of small-body implantation on to high-perihelion orbits, and demonstrate that this process is driven by the von Zeipel–Kozai–Lidov mechanism. Incorporating the transiency of stellar clusters and the eventual Sun-binary pair dissociation due to passing stars, we show how the binary can be stripped away by an approximately solar-mass ejector star, with only a modest impact on the generated inner Oort cloud population. Collectively, our results highlight a previously underappreciated process that could have contributed to the formation of the inner Oort cloud.

Dynamical coupling of Keplerian orbits in a hierarchical four-body system: from the Galactic Centre to compact planetary systems

ABSTRACT This study focuses on the long-term evolution of two bodies in nearby initially coplanar orbits around a central dominant body perturbed by a fourth body on a distant Keplerian orbit. Our previous works that considered this setup enforced circular orbits by adding a spherical potential of extended mass, which dampens Kozai–Lidov oscillations; it led to two qualitatively different modes of the evolution of the nearby orbits. In one scenario, their mutual interaction exceeds the effect of differential precession caused by a perturbing body. This results in a long-term coherent evolution, with nearly coplanar orbits experiencing only small oscillations of inclination. We extend the previous work by (i) considering post-Newtonian corrections to the gravity of the central body, either instead of or in addition to the potential of extended mass, (ii) relaxing the requirement of strictly circular orbits, and (iii) removing the strict requirement of complete Kozai–Lidov damping. Thus, we identify the modes of interorbital interaction described for the zero eccentricity case in the more general situation, which allows for its applicability to a much broader range of astrophysical systems than considered initially. In this work, we scale the systems to the orbits of S-stars; we consider the clockwise disc to represent the perturbing body, with post-Newtonian corrections to the gravity of Sagittarius A* playing the role of damping potential. Considering post-Newtonian corrections, even stellar-mass central bodies in compact planetary systems can allow for the coupled evolution of Keplerian orbits.

Lidov–Kozai mechanism in Hildas and Jupiter Trojans

Lidov–Kozai mechanism in Hildas and Jupiter Trojans

The Interaction between Stars and Past AGN Disk: Possible Explanation for the Kinematic Distributions of S-stars in the Galactic Center

The presence of young stars, aged around several million years and situated within the range of ∼0.04–1 pc from our Galactic center raises a question about their origins and dynamical evolutions. Their kinematics provide an opportunity to explore their formation or possible subsequent dynamical evolution. If Sagittarius A* was active in the past as suggested by several observations, the accretion disk may have a significant impact on the dynamics of stars in the Galactic center. The drag force exerted on stars during star–disk interaction could lead some of them to sink into the accretion disk, and these embedded stars will rapidly migrate inward and eventually be disrupted within ∼105 yr. This could roughly explain the absence of stars within 2.5 × 104 R g (∼1000 au). Additionally, Kozai–Lidov oscillations, induced by the gravitational perturbation of the disk, could contribute to the bimodal distribution of S-star inclinations and drive a majority of stars into high-eccentricity orbits.

Inclination instability of circumbinary planets

ABSTRACT We analyse a tilt instability of the orbit of an outer planet in a two-planet circumbinary system that we recently reported. The binary is on an eccentric orbit and the inner circumbinary planet is on a circular polar orbit that causes the binary to undergo apsidal precession. The outer circumbinary planet is initially on a circular or eccentric orbit that is coplanar with respect to the binary. We apply a Hamiltonian in quadrupole order of the binary potential to show that the tilt instability is the result of a secular resonance in which the apsidal precession rate of the binary matches the nodal precession rate of the outer planet. A resonance is possible because the polar inner planet causes the apsidal precession of the binary to be retrograde. The outer planet periodically undergoes large tilt oscillations for which we analytically determine the initial evolution and maximum inclination. Following a typically relatively short adjustment phase, the tilt grows exponentially in time at a characteristic rate that is of order the absolute value of the binary apsidal precession rate. The analytic results agree well with numerical simulations. This instability is analogous to the Kozai–Lidov instability, but applied to a circumbinary object. The instability fails to operate if the binary mass ratio is too extreme. The instability occurs even if the outer planet is instead an object of stellar mass and involves tilt oscillations of the inner binary.