Kozai Mechanism Research Articles

Planetary migration in precessing discs for S-type wide binaries

ABSTRACT The discovery of numerous circumprimary planets in the last few years has brought to the fore the question of planet formation in binary systems. The significant dynamical influence, during the protoplanetary disc phase, of a binary companion on a giant planet has previously been highlighted for wide binary stars. In particular, highly inclined binary companion can induce perturbations on the disc and the planets through the Lidov–Kozai resonance, which could inhibit the formation process. In this work, we aim to study how the disc gravitational potential acting on the planet and the nodal precession induced by the wide binary companion with separation of 1000 au on the disc act to suppress the Lidov–Kozai perturbations on a migrating giant planet. We derive new approximate formulas for the evolution of the disc’s inclination and longitude of the ascending node in case of a rigidly precessing disc with a decreasing mass and perturbed by a wide binary companion, which are suitable for N-body simulations. We carry out 3200 simulations with several eccentricity and inclination values for the binary companion. The gravitational and damping forces exerted by the disc on the planet tend to keep the latter in the mid-plane of the former, and suppress the effect of the binary companion by preventing the planet from getting locked in the Lidov–Kozai resonance during the disc phase. We also confirm that because of nodal precession induced by the binary, a primordial spin–orbit misalignment could be generated for circumprimary planets with an inclined binary companion.

An analytic solution to the Kozai–Lidov evolution equations

ABSTRACT A test particle in a non-coplanar orbit about a member of a binary system can undergo Kozai–Lidov oscillations in which tilt and eccentricity are exchanged. An initially circular highly inclined particle orbit can reach high eccentricity. We consider the non-linear secular evolution equations previously obtained in the quadrupole approximation. For the important case that the initial eccentricity of the particle orbit is zero, we derive an analytic solution for the particle orbital elements as a function of time that is exact within the quadrupole approximation. The solution involves only simple trigonometric and hyperbolic functions. It simplifies in the case that the initial particle orbit is close to being perpendicular to the binary orbital plane. The solution also provides an accurate description of particle orbits with non-zero but sufficiently small initial eccentricity. It is accurate over a range of initial eccentricity that broadens at higher initial inclinations. In the case of an initial inclination of π/3, an error of 1 per cent at maximum eccentricity occurs for initial eccentricities of about 0.1.

Station-keeping in high near-circular polar orbits around the Moon

High circular polar orbits around the Moon are considered as one of the options for placing a prospective lunar station. Due to gravitational perturbations from the Earth, such orbits are subject to the Lidov–Kozai effect. This effect results in the increase in the average value of the eccentricity, which eventually leads to the collision of the spacecraft with the lunar surface. Therefore, when used in lunar missions, these orbits require periodic corrections. The work proposes a station-keeping technique for high near-circular polar lunar orbits based on the dynamical structures of the double-averaged equations of motion that include the central gravitational fields of the Earth and the Moon. The annual station-keeping delta-v for a 10,000 km polar orbit is presented considering navigation errors and maneuver execution errors. The nominal parameters of the orbit are selected to increase its lifetime.

Effective two-body approach to the hierarchical three-body problem

The motion of three bodies can be solved perturbatively when a tightly bound inner binary is orbited by a distant perturber, giving rise for example to the well-known Kozai-Lidov oscillations. We propose to study the relativistic hierarchical three-body orbits by adapting the Effective Field Theory techniques used in the two-body problem. This allows us to conveniently treat the inner binary as an effective point-particle, thus reducing the complexity of the three-body problem to a simpler spinning two-body motion. We present in details the mapping between the inner binary osculating elements and the resulting spin of the effective point-particle. Our study builds towards a derivation of three-body analytic waveforms.

ν Gem: A Hierarchical Triple System with an Outer Be Star

Abstract Time series of spectroscopic, speckle-interferometric, and optical long-baseline-interferometric observations confirm that ν Gem is a hierarchical triple system. It consists of an inner binary composed of two B-type stars and an outer classical Be star. Several photospheric spectral lines of the inner components were disentangled, revealing two stars with very different rotational broadening (∼260 and ∼140 km s−1, respectively), while the photospheric lines of the Be star remain undetected. From the combined spectroscopic and astrometric orbital solution it is not possible to unambiguously cross identify the inner astrometric components with the spectroscopic components. In the preferred solution based on modeling of the disentangled line profiles, the inner binary is composed of two stars with nearly identical masses of 3.3 M ⊙ and the more rapidly rotating star is the fainter one. These two stars are in a marginally elliptical orbit (e = 0.06) about each other with a period of 53.8 days. The third star also has a mass of 3.3 M ⊙ and follows a more eccentric (e = 0.24) orbit with a period of 19.1 yr. The two orbits are codirectional, and at inclinations of 79° and 76° of the inner and the outer orbit, respectively, about coplanar. No astrometric or spectroscopic evidence could be found that the Be star itself is double. The system appears dynamically stable and not subject to eccentric Lidov–Kozai oscillations. After disentangling, the spectra of the components of the inner binary do not exhibit peculiarities that would be indicative of past interactions. Motivations for a wide range of follow-up studies are suggested.

The Rarity of Very Red Trans-Neptunian Objects in the Scattered Disk

Abstract We investigate the origins of the photometrically very red (VR) and less red (LR) trans-Neptunian objects (TNOs). We first reanalyze the data set of Marsset et al. and find that in addition to the known color–inclination correlation in hot TNOs, a similar trend exists for color–eccentricity. We show that VR TNOs are sharply constrained to eccentricities <0.42 and inclinations <21°, leading to a paucity of VR scattered disk and distant mean motion resonance objects. We then interpret these findings using N-body simulations accounting for Neptune’s outward migration into a massless particles disk and find that these observations are best reproduced with an LR-to-VR color transition line between ∼38 and 42 au in the primordial disk, separating the objects’ formation locations. For an initial surface density profile (Σ ∝ 1/r 2), a color transition around 38 au is needed to explain the high abundance of VR plutinos, but it creates too many VR scattered disk objects, while a transition line around 42 au seems to better reproduce the scattered disk colors but creates virtually no VR plutinos. Our simulations furthermore show that the rarity of VR particles at high eccentricity is possibly due to the absence of sweeping higher-order MMRs, and secular resonances, beyond 42 au. Inspecting individual populations, we show that the majority of VR SDOs originate as objects trapped in Neptune’s second- and third-order MMRs. These then evolve due to diffusion, scattering, Kozai–Lidov cycles, and secular resonances into their current orbits. Future unbiased color surveys are crucial to better constrain the TNOs dynamical origins.

The mass-ratio distribution of tertiary-induced binary black hole mergers

ABSTRACT Many proposed scenarios for black hole (BH) mergers involve a tertiary companion that induces von Zeipel–Lidov–Kozai (ZLK) eccentricity cycles in the inner binary. An attractive feature of such mechanisms is the enhanced merger probability when the octupole-order effects, also known as the eccentric Kozai mechanism, are important. This can be the case when the tertiary is of comparable mass to the binary components. Since the octupole strength [∝(1 − q)/(1 + q)] increases with decreasing binary mass ratio q, such ZLK-induced mergers favour binaries with smaller mass ratios. We use a combination of numerical and analytical approaches to fully characterize the octupole-enhanced binary BH mergers and provide semi-analytical criteria for efficiently calculating the strength of this enhancement. We show that for hierarchical triples with semimajor axial ratio a/aout ≳ 0.01–0.02, the binary merger fraction can increase by a large factor (up to ∼20) as q decreases from unity to 0.2. The resulting mass-ratio distribution for merging binary BHs produced in this scenario is in tension with the observed distribution obtained by the LIGO/VIRGO collaboration, although significant uncertainties remain about the initial distribution of binary BH masses and mass ratios.

Resolving the Complex Evolution of a Supermassive Black Hole Triplet in a Cosmological Simulation

Abstract We present here a self-consistent cosmological zoom-in simulation of a triple supermassive black hole (SMBH) system forming in a complex multiple galaxy merger. The simulation is run with an updated version of our code KETJU, which is able to follow the motion of SMBHs down to separations of tens of Schwarzschild radii while simultaneously modeling the large-scale astrophysical processes in the surrounding galaxies, such as gas cooling, star formation, and stellar and AGN feedback. Our simulation produces initially an SMBH binary system for which the hardening process is interrupted by the late arrival of a third SMBH. The KETJU code is able to accurately model the complex behavior occurring in such a triple SMBH system, including the ejection of one SMBH to a kiloparsec-scale orbit in the galaxy due to strong three-body interactions as well as Lidov–Kozai oscillations suppressed by relativistic precession when the SMBHs are in a hierarchical configuration. One pair of SMBHs merges ∼3 Gyr after the initial galaxy merger, while the remaining binary is at a parsec-scale separation when the simulation ends at redshift z = 0. We also show that KETJU can capture the effects of the SMBH binaries and triplets on the surrounding stellar population, which can affect the binary merger timescales as the stellar density in the system evolves. Our results demonstrate the importance of dynamically resolving the complex behavior of multiple SMBHs in galactic mergers, as such systems cannot be readily modeled using simple orbit-averaged semianalytic models.

The Kozai-Lidov Mechanism in Binary Star Systems and the Stabilizing Effect of Binary Stars

This study used Python simulations to compare the orbits of planets in single-planet binary systems and two-planet binary systems with equivalent single star systems to examine the differences of the Kozai-Lidov mechanism in three-body single star systems and four-body binary star systems. The Python program simulated the forces between the stars and the planets, and produced inclination graphs and orbital diagrams. In the single-planet scenario, no Kozai-Lidov effect is observed as the additional star did not affect the planet’s inclination, but the orbit is considerably rotated. In the two-planet scenario, the Kozai-Lidov oscillation period decreases from 650 years to 370 years, and the maximum inclination decreases from 0.690 radians to 0.280 radians, indicating that the binary star system acts against the Kozai-Lidov Mechanism. Additional trials with modified initial configurations of the two-planet binary star system confirm the universality of the suppressive effect in varying initial conditions in binary star systems.

The TESS–Keck Survey. IV. A Retrograde, Polar Orbit for the Ultra-low-density, Hot Super-Neptune WASP-107b

Abstract We measured the Rossiter–McLaughlin effect of WASP-107b during a single transit with Keck/HIRES. We found the sky-projected inclination of WASP-107b’s orbit, relative to its host star’s rotation axis, to be degrees. This confirms the misaligned/polar orbit that was previously suggested from spot-crossing events and adds WASP-107b to the growing population of hot Neptunes in polar orbits around cool stars. WASP-107b is also the fourth such planet to have a known distant planetary companion. We examined several dynamical pathways by which this companion could have induced such an obliquity in WASP-107b. We find that nodal precession and disk dispersal-driven tilting can both explain the current orbital geometry while Kozai–Lidov cycles are suppressed by general relativity. While each hypothesis requires a mutual inclination between the two planets, nodal precession requires a much larger angle, which for WASP-107 is on the threshold of detectability with future Gaia astrometric data. As nodal precession has no stellar type dependence, but disk dispersal-driven tilting does, distinguishing between these two models is best done on the population level. Finding and characterizing more extrasolar systems like WASP-107 will additionally help distinguish whether the distribution of hot-Neptune obliquities is a dichotomy of aligned and polar orbits or if we are uniformly sampling obliquities during nodal precession cycles.

Kozai–Lidov oscillations triggered by a tilt instability of detached circumplanetary discs

ABSTRACT Circumplanetary discs can be linearly unstable to the growth of disc tilt in the tidal potential of the star–planet system. We use 3D hydrodynamical simulations to characterize the disc conditions needed for instability, together with its long-term evolution. Tilt growth occurs for disc aspect ratios, evaluated near the disc outer edge, of H/r ≳ 0.05, with a weak dependence on viscosity in the wave-like regime of warp propagation. Lower mass giant planets are more likely to have circumplanetary discs that satisfy the conditions for instability. We show that the tilt instability can excite the inclination to above the threshold where the circumplanetary disc becomes unstable to Kozai–Lidov (KL) oscillations. Dissipation in the KL unstable regime caps further tilt growth, but the disc experiences large oscillations in both inclination and eccentricity. Planetary accretion occurs in episodic accretion events. We discuss implications of the joint tilt–KL instability for the detectability of circumplanetary discs, for the obliquity evolution of forming giant planets, and for the formation of satellite systems.

Hierarchical black hole mergers in multiple systems: constrain the formation of GW190412-, GW190814-, and GW190521-like events

ABSTRACT The merging black hole (BH) binaries GW190412, GW190814, and GW190521 from the third LIGO/VIRGO observing run exhibit some extraordinary properties, including highly asymmetric masses, significant spin, and component mass in the ‘mass gap’. These features can be explained if one or both components of the binary are the remnants of previous mergers. In this paper, we explore hierarchical mergers in multiple stellar systems, taking into account the natal kick and mass-loss due to the supernova explosion (SN) on each component, as well as the merger kick received by the merger remnant. The binaries that have survived the SNe and kicks generally have too wide orbital separations to merge by themselves, but can merge with the aid of an external companion that gives rise to Lidov–Kozai oscillations. The BH binaries that consist of second-generation BHs can also be assembled in dense star clusters through binary interactions. We characterize the parameter space of these BH binaries by merger fractions in an analytical approach. Combining the distributions of the survived binaries, we further constrain the parameters of the external companion, using the analytically formulated tertiary perturbation strength. We find that to produce the three LIGO/VIRGO O3 events, the external companions must be at least a few hundreds M⊙, and fall in the intermediate-mass BH and supermassive BH range. We suggest that GW190412, GW190814, and GW190521 could all be produced via hierarchical mergers in multiples, likely in a nuclear star cluster, with the final merger induced by a massive BH.

Sustained Kozai–Lidov Oscillations in Misaligned Circumstellar Gas Disks

Abstract A disk around one component of a binary star system with sufficiently high inclination can undergo Kozai–Lidov (KL) oscillations during which the disk inclination and disk eccentricity are exchanged. Previous studies show that without a source of accretion, KL unstable disks exhibit damped oscillations, due to viscous dissipation, that leave the disk stable near or below the critical inclination for KL oscillations. With three-dimensional hydrodynamical simulations we show that a highly misaligned circumbinary disk that flows onto the binary components forms highly inclined circumstellar disks around each component. We show that a continuous infall of highly inclined material allows the KL oscillations to continue. The KL disk oscillations produce shocks and eccentricity growth in the circumstellar disks that affect the conditions for planet formation.

Evolution of a migrating giant planet in the presence of an inclined binary companion

Aims. There are a growing number of giant planets discovered moving around one stellar component of a binary star, most of which have very diverse eccentricity. These discoveries raise the question of their formation and long-term evolution because the stellar companion can strongly affect the planet formation process. We aim to study the dynamical influence of a wide binary companion on the evolution of a single giant planet migrating in a protoplanetary disk. Methods. Using a symplectic N-body integrator adapted for binary star systems and modeling the dissipation due to the disk by appropriate formulae emerging from hydrodynamical simulations, we carried out 3200 simulations with different orbital parameters for the planet and different eccentricity and inclination values for the binary companion. The long-term evolution of the planets was followed for 100 Myr and the different dynamical behaviors were unveiled using a quadrupolar Hamiltonian approach. Results. We show that a capture in a Lidov-Kozai resonant state is far from automatic when the binary companion star is highly inclined, since only 36% of the systems end up locked in the resonance at the end of the simulations. Nevertheless, in the presence of a highly inclined binary companion, all the planetary evolutions are strongly influenced by the Lidov-Kozai resonance and the nonresonant evolutions present high eccentricity and inclination variations associated with circulation around the Lidov-Kozai islands.

Figures of Pluto and Charon and their relative motion

The comparative effect of two factors on the translatory motion of the centres of mass of the Pluto-Charon system is investigated. The first important factor is the non-sphericity of the shape and gravitational field of the bodies in the system. The second is the gravitation of the Sun. As a measure of the influence of both factors we use the ratio of the corresponding perturbing acceleration to the main one. The main acceleration is caused by the mutual Newtonian attraction of Pluto and Charon. It has been established that for the first factor this measure is of the order of 10^−6, while for the second factor it is two orders of magnitude smaller. This explains why the Lidov-Kozai effect (despite a large mutual slope of 96 between the planes of the satellite’s orbit around the planet, and the barycentre of the system around the Sun) does not appear. The situation is similar to the case with the satellites of Uranus. As a result, the Pluto-Charon system remains stable at least on a timescale of millions of years. The tidal effect of the Sun on the surface shape of the bodies under study is also estimated. The ratio of the tidal potential of the Sun at a point on the surface of the body to the gravitational potential of the body itself at this point is taken as a measure of impact. It turned out to be of the order of 3 · 10^−12, which is more than six orders less than the influence of rotation and mutual attraction of Pluto and Charon. In fact, the Sun does not affect the figures of the bodies of the system.

Mildly Hierarchical Triple Dynamics and Applications to the Outer Solar System

Abstract Three-body interactions are ubiquitous in astrophysics. For instance, Kozai–Lidov oscillations in hierarchical triple systems have been studied extensively and applied to a wide range of astrophysical systems. However, mildly hierarchical triples also play an important role, but they are less explored. In this work, we consider the secular dynamics of a test particle in a mildly hierarchical configuration. We find the limit within which the secular approximation is reliable when the outer perturber is in a circular orbit. In addition, we present resonances and chaotic regions using surface-of-section plots, and characterize regions of phase space that allow large eccentricity and inclination variations. Finally, we apply the secular results to the outer Solar System. We focus on the distribution of extreme trans-Neptunian objects (eTNOs) under the perturbation of a possible outer planet (Planet 9), and find that in addition to a low-inclination Planet 9, a polar or a counter-orbiting one could also produce pericenter clustering of eTNOs, while the polar one leads to a wider spread of eTNO inclinations.

Tidal evolution of the Pluto–Charon binary

A giant collision is believed to be at the origin of the Pluto–Charon system. As a result, the initial orbit and spins after impact may have substantially differed from those observed today. More precisely, the distance at periapse may have been shorter, subsequently expanding to its current separation by tides raised simultaneously on the two bodies. Here we provide a general 3D model to study the tidal evolution of a binary composed of two triaxial bodies orbiting a central star. We apply this model to the Pluto–Charon binary, and notice some interesting constraints on the initial system. We observe that when the eccentricity evolves to high values, the presence of the Sun prevents Charon from escaping because of Lidov-Kozai cycles. However, for a high initial obliquity for Pluto or a spin-orbit capture of Charon’s rotation, the binary eccentricity is damped very efficiently. As a result, the system can maintain a moderate eccentricity throughout its evolution, even for strong tidal dissipation on Pluto.

Enhanced Lidov–Kozai migration and the formation of the transiting giant planet WD 1856+534 b

ABSTRACT We investigate the possible origin of the transiting giant planet WD 1856+534 b, the first strong exoplanet candidate orbiting a white dwarf, through high-eccentricity migration (HEM) driven by the Lidov–Kozai (LK) effect. The host system’s overall architecture is a hierarchical quadruple in the ‘2 + 2’ configuration, owing to the presence of a tertiary companion system of two M-dwarfs. We show that a secular inclination resonance in 2 + 2 systems can significantly broaden the LK window for extreme eccentricity excitation (e ≳ 0.999), allowing the giant planet to migrate for a wide range of initial orbital inclinations. Octupole effects can also contribute to the broadening of this ‘extreme’ LK window. By requiring that perturbations from the companion stars be able to overcome short-range forces and excite the planet’s eccentricity to e ≃ 1, we obtain an absolute limit of $a_{1} \gtrsim 8 \, \mathrm{au}\, (a_{3} / 1500 \, \mathrm{au})^{6/7}$ for the planet’s semimajor axis just before migration (where a3 is the semimajor axis of the ‘outer’ orbit). We suggest that, to achieve a wide LK window through the 2 + 2 resonance, WD 1856 b likely migrated from $30 \, \mathrm{au}\lesssim a_{1} \lesssim 60 \, \mathrm{au}$, corresponding to ∼10–$20 \, \mathrm{au}$ during the host’s main-sequence phase. We discuss possible difficulties of all flavours of HEM affecting the occurrence rate of short-period giant planets around white dwarfs.

WD 1856 b: a close giant planet around a white dwarf that could have survived a common envelope phase

ABSTRACT The discovery of a giant planet candidate orbiting the white dwarf WD 1856+534 with an orbital period of 1.4 d poses the questions of how the planet reached its current position. We here reconstruct the evolutionary history of the system assuming common envelope evolution as the main mechanism that brought the planet to its current position. We find that common envelope evolution can explain the present configuration if it was initiated when the host star was on the asymptotic giant branch, the separation of the planet at the onset of mass transfer was in the range 1.69–2.35 au, and if in addition to the orbital energy of the surviving planet either recombination energy stored in the envelope or another source of additional energy contributed to expelling the envelope. We also discuss the evolution of the planet prior to and following common envelope evolution. Finally, we find that if the system formed through common envelope evolution, its total age is in agreement with its membership to the Galactic thin disc. We therefore conclude that common envelope evolution is at least as likely as alternative formation scenarios previously suggested such as planet–planet scattering or Kozai–Lidov oscillations.

Properties of von Zeipel–Lidov–Kozai oscillations in triple systems at the quadrupole order: relaxing the test particle approximation

ABSTRACT Von Zeipel–Lidov–Kozai (ZLK) oscillations in hierarchical triple systems have important astrophysical implications such as triggering strong interactions and producing, e.g. Type Ia supernovae and gravitational wave sources. When considering analytic properties of ZLK oscillations at the lowest (quadrupole) expansion order, as well as complications due to higher order terms, one usually assumes the test particle limit, in which one of the bodies in the inner binary is massless. Although this approximation holds well for, e.g. planetary systems, it is less accurate for systems with more comparable masses such as stellar triples. Although non-test-particle effects are usually taken into account in numerical simulations, a more analytic approach focusing on the differences between the test particle and general case (at quadrupole order) has, to our knowledge, not been presented. Here, we derive several analytic properties of secular oscillations in triples at the quadruple expansion order. The latter applies even to relatively compact triples, as long as the inner bodies are similar in mass such that octupole-order effects are suppressed. We consider general conditions for the character of the oscillations (circular versus librating), minimum and maximum eccentricities, and time-scales, all as a function of $\gamma \equiv (1/2) \, L_1/G_2$, a ratio of inner-to-outer orbital angular momenta variables (γ = 0 in the test particle limit). In particular, eccentricity oscillations are more effective at retrograde orientations for non-zero γ; assuming zero initial inner eccentricity, the maximum eccentricity peaks at $\cos (i_{\mathrm{rel},0}) = -\gamma$, where $i_{\mathrm{rel},0}$ is the initial relative inclination. We provide a python script that can be used to quickly compute these properties.