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.
In this paper we answer a simple question: can a misaligned circumbinary planet induce Kozai-Lidov cycles on an inner stellar binary? We use known analytic equations to analyse the behaviour of the Kozai-Lidov effect as the outer mass is made small. We demonstrate a significant departure from the traditional symmetry, critical angles and amplitude of the effect. Aside from massive planets on near-polar orbits, circumbinary planetary systems are devoid of Kozai-Lidov cycles. This has positive implications for the existence of highly misaligned circumbinary planets: an observationally unexplored…
Exploring weakly perturbed Keplerian motion within the restricted three-body problem, Lidov (1962) and, independently, Kozai (1962) discovered coupled oscillations of eccentricity and inclination (the KL-cycles). Their classical studies were based on an integrable model of the secular evolution, obtained by double averaging of the disturbing function approximated with its first non-trivial term. This was the quadrupole term in the series expansion with respect to the ratio of the semimajor axis of the disturbed body to that of the disturbing body. If the next (octupole) term is kept in the exp…
A test particle in a noncoplanar 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 nonlinear 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…
The hierarchical triple body approximation has useful applications to a variety of systems from planetary and stellar scales to supermassive black holes. In this approximation, the energy of each orbit is separately conserved and therefore the two semi-major axes are constants. On timescales much larger than the orbital periods, the orbits exchange angular momentum which leads to eccentricity and orientation (i.e., inclination) oscillations. The orbits' eccentricity can reach extreme values leading to a nearly radial motion, which can further evolve into short orbit periods and merging binarie…
The Kozai-Lidov mechanism can be applied to a vast variety of astrophysical systems involving hierarchical three-body systems. Here, we study the Kozai-Lidov mechanism systematically in the test particle limit at the octupole level of approximation. We investigate the chaotic and quasiperiodic orbital evolution by studying surfaces of section and the Lyapunov exponents. We find that the resonances introduced by the octupole level of approximation cause orbits to flip from prograde to retrograde and back as well as cause significant eccentricity excitation, and the chaotic behaviors occur when …
As the discoveries of more minor bodies in retrograde resonances with giant planets, such as 2015 BZ509 and 2006 RJ2, our curiosity about the Kozai-Lidov dynamics inside the retrograde resonance has been sparked. In this study, we focus on the 3D retrograde resonance problem and investigate how the resonant dynamics of a minor body impacts on its own Kozai-Lidov cycle. Firstly we deduce the action-angle variables and canonical transformations that deal with the retrograde orbit specifically. After obtaining the dominant Hamiltonian of this problem, we then carry out the numerical averaging pro…
The recent discoveries of WD J091405.30+191412.25 (WD J0914 hereafter), a white dwarf likely accreting material from an ice giant planet, and WD 1856+534 b (WD 1856 b hereafter), a Jupiter-sized planet transiting a white dwarf, are the first direct evidence of giant planets orbiting white dwarfs. However, for both systems the observations indicate that the planets' current orbital distances would have put them inside the stellar envelope during the red giant phase, implying that the planets must have migrated to their current orbits after their host stars became white dwarfs. Furthermore, WD J…
Kozai-Lidov oscillations of Jupiter-mass planets, excited by comparable planetary or brown dwarf mass perturbers were recently shown in numerical experiments to be slowly modulated and to exhibit striking features, including extremely high eccentricities and the generation of retrograde orbits with respect to the perturber. Here we solve this problem analytically for the case of a test particle orbiting a host star and perturbed by a distant companion whose orbit is eccentric and highly inclined. We give analytic expressions for the conditions that produce retrograde orbits and high eccentrici…
Kozai-Lidov (KL) oscillations in hierarchical triple systems have found application to many astrophysical contexts, including planet formation, type Ia supernovae, and supermassive black hole dynamics. The period of these oscillations is known at the order-of-magnitude level, but dependences on the initial mutual inclination or inner eccentricity are not typically included. In this work I calculate the period of KL oscillations ($t_{\textrm{KL}}$) exactly in the test particle limit at quadrupole order (TPQ). I explore the parameter space of all hierarchical triples at TPQ and show that except …
Many triple-star systems have an inner pair with an orbital period of a few days only. A common mechanism to explain the short-period pile-up present in the observations is the migration through Lidov-Kozai cycles combined with tidal friction. Here, we revisit this mechanism and aim to determine the initial orbital configurations leading to this process. We show that the mutual inclination of the triple-star system is not the only critical parameter, since the eccentricity as well as the argument of the pericenter of the inner orbit also play an important role in the establishment of the Lidov…
Both General Relativistic (GR) precession and the Lidov-Kozai mechanism, separately, are known to play an important role in the orbital evolution of solar system bodies. Previous works have studied these two mechanisms independently in great detail. However, both these phenomena occurring at the same time in real solar system bodies have rarely been explored. In this work, we find a continuum connecting the GR precession dominant and Lidov-Kozai like mechanism dominant regimes, i.e. an intermediate regime where the competing effects of GR precession and Lidov-Kozai like oscillations co-exist s…
We use three dimensional hydrodynamical simulations to show that a highly misaligned accretion disk around one component of a binary system can exhibit global Kozai-Lidov cycles, where the inclination and eccentricity of the disk are interchanged periodically. This has important implications for accreting systems on all scales, for example, the formation of planets and satellites in circumstellar and circumplanetary disks, outbursts in X-ray binary systems and accretion on to supermassive black holes.
Lidov-Kozai oscillations of planets in stellar binaries, combined with tidal dissipation, can lead to the formation of hot Jupiters (HJs) or tidal disruption of planets. Recent population synthesis studies have found that the fraction of systems resulting in HJs (F_HJ) depends strongly on the planet mass, host stellar type and tidal dissipation strength, while the total migration fraction F_mig = F_ HJ + F_dis (including both HJ formation and tidal disruption) exhibits much weaker dependence. We present an analytical method for calculating F_HJ and F_mig in the Lidov-Kozai migration scenario. …
Recent results by Martin et al. (2014) showed in 3D SPH simulations that tilted discs in binary systems can be unstable to the development of global, damped Kozai-Lidov (KL) oscillations in which the discs exchange tilt for eccentricity. We investigate the linear stability of KL modes for tilted inviscid discs under the approximations that the disc eccentricity is small and the disc remains flat. By using 1D equations, we are able to probe regimes of large ratios of outer to inner disc edge radii that are realistic for binary systems of hundreds of AU separations and are not easily probed by m…
The stability of planets in the alpha-Centauri AB stellar system has been studied extensively. However, most studies either focus on the orbital plane of the binary or consider inclined circular orbits. Here, we numerically investigate the stability of a possible planet in the alpha-Centauri AB binary system for S-type orbits in an arbitrary spatial configuration. In particular, we focus on inclined orbits and explore the stability for different eccentricities and orientation angles. We show that large stable and regular regions are present for very eccentric and inclined orbits, corresponding…
Circumplanetary discs can be linearly unstable to the growth of disc tilt in the tidal potential of the star-planet system. We use three-dimensional 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 $H/r\gtrsim 0.05$, 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 insta…
We investigate the possible origin of the transiting giant planet WD1856+534b, 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 an 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 ($e \gtrsim 0.999$), allowing the giant planet to migrate for a wide range of…
We analyze the gravitational instability (GI) of a locally isothermal inclined disk around one component of a binary system. Such a disk can undergo global Kozai-Lidov (KL) cycles if the initial disk tilt is above the critical KL angle (of about 40 degrees). During these cycles, an initially circular disk exchanges its inclination for eccentricity, and vice versa. Self-gravity may suppress the cycles under some circumstances. However, with hydrodynamic simulations including self-gravity we show that for a sufficiently high initial disk tilts and for certain disk masses, disks can undergo KL os…
Recent SPH simulations by Martin et al. (2014) suggest a circumstellar gaseous disk may exhibit coherent eccentricity-inclination oscillations due to the tidal forcing of an inclined binary companion, in a manner that resembles Lidov-Kozai oscillations in hierarchical triple systems. We carry out linear stability analysis for the eccentricity growth of circumstellar disks in binaries, including the effects of gas pressure and viscosity and secular (orbital-averaged) tidal force from the inclined companion. We find that the growth of disk eccentricity depends on the dimensionless ratio ($S$) be…
$γ$ Cep Ab is a typical S-type planet, which occupies a nearly perpendicular planetary orbit relative to the binary. Here we use the Markov Chain Monte Carlo (MCMC) sampler to conduct full N-body fitting and derive self-consistent orbital solutions for this hierarchical system. Then we employ the Eccentric Kozai-Lidov (EKL) mechanism to explain the extremely inclined orbit of S-type planet $γ$ Cep Ab. The EKL mechanism plays an essential role in exploring significant oscillations of the mutual inclination $i_{\mathrm{mut}}$ between the planet and the secondary star. We perform qualitative anal…
A highly misaligned gas disk around one component of a binary star system can undergo global Kozai-Lidov (KL) oscillations for which the disk inclination and eccentricity are exchanged. With hydrodynamical simulations of a gas and dust disk we explore the effects of these oscillations on the dust density distribution. For dust that is marginally coupled to the gas (${\rm St} \approx 1$), we find that the dust undergoes similar dynamical behaviour to the gas disk but the radial distribution of dust may be very different to the gas. The inward radial drift of the dust is faster in an eccentric d…
Atira-class asteroids have aphelia greater than 0.718 au and smaller than 0.983 au. It has been found that the orbits of many of them remain relatively stable for at least 1 Myr, but it is unclear what can make them so stable when they cross the orbit of Venus and some may reach, in heliocentric terms, as far as Earth's perihelion and as close as inside Mercury's perihelion. The analysis of their orbits shows that many of them have values of the argument of perihelion close to 0 or 180 degrees. Such an orbital arrangement means that the nodes are located at perihelion and at aphelion, i.e. for…
Many exoplanetary systems containing hot Jupiters (HJs) exhibit significant misalignment between the spin axes of the host stars and the orbital angular momentum axes of the planets ("spin-orbit misalignment"). High-eccentricity migration involving Lidov-Kozai oscillations of the planet's orbit induced by a distant perturber is a possible channel for producing such misaligned HJ systems. Previous works have shown that the dynamical evolution of the stellar spin axis during the high-$e$ migration plays a dominant role in generating the observed spin-orbit misalignment. Numerical studies have al…
The problem of orbit flips caused by eccentric von Zeipel-Lidov-Kozai effects is systematically investigated by means of three approaches, including Poincaré sections, dynamical system theory (periodic orbits and invariant manifolds) and perturbation treatments. Poincaré sections show that orbit flips are due to the existence of islands of libration centered at inclination of $90^{\circ}$, dynamical system theory shows that orbit flips are due to the existence of polar periodic orbits and invariant manifolds, and perturbative treatments indicate that orbit flips are due to the libration of a c…
We investigate the formation and evolution of misaligned accretion discs around the secondary component of a binary through mass transfer driven by Kozai-Lidov oscillations of the circumprimary disc's eccentricity and inclination. We perform SPH simulations to study the amount of mass transferred to the secondary star as a function of both the disc and binary parameters. For the range of parameters we explore, we find that increasing the disc aspect ratio, viscosity parameter and initial inclination as well as decreasing the binary mass ratio leads to larger amount of mass transfer, up to a ma…
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 compa…
Many exoplanetary systems containing hot Jupiters are found to possess significant misalignment between the spin axis of the host star and the planet's orbital angular momentum axis. A possible channel for producing such misaligned hot Jupiters involves Lidov-Kozai oscillations of the planet's orbital eccentricity and inclination driven by a distant binary companion. In a recent work (Storch, Anderson & Lai 2014), we have shown that a proto-hot Jupiter undergoing Lidov-Kozai oscillations can induce complex, and often chaotic, evolution of the spin axis of its host star. Here we explore the ori…
Observed hot Jupiter (HJ) systems exhibit a wide range of stellar spin-orbit misalignment angles. The origin of these HJs remains unclear. This paper investigates the inward migration of giant planets due to Lidov-Kozai (LK) oscillations induced by a distant (100-1000 AU) stellar companion. We conduct a large population synthesis study, including the octupole gravitational potential from the stellar companion, mutual precession of the host stellar spin axis and planet orbital axis, tidal dissipation in the planet, and stellar spin-down in the host star due to magnetic braking. We consider a ra…
Luna 3 (or Lunik 3 in Russian sources) was the first spacecraft to perform a flyby of the Moon. Launched in October 1959 on a translunar trajectory with large semi-major axis and eccentricity, it collided with the Earth in late March 1960. The short, 6-month dynamical lifetime has often been explained through an increase in eccentricity due to the Lidov-Kozai effect. However, the classical Lidov-Kozai solution is only valid in the limit of small semi-major axis ratio, a condition that is satisfied only for solar (but not for lunar) perturbations. We undertook a study of the dynamics of Luna 3 …
The secular approximation of the hierarchical three body systems has been proven to be very useful in addressing many astrophysical systems, from planets, stars to black holes. In such a system two objects are on a tight orbit, and the tertiary is on a much wider orbit. Here we study the dynamics of a system by taking the tertiary mass to zero and solve the hierarchical three body system up to the octupole level of approximation. We find a rich dynamics that the outer orbit undergoes due to gravitational perturbations from the inner binary. The nominal result of the precession of the nodes is …
Secular dynamics inside 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, semi-analytical developments are made to explore the von Zeipel-Lidov-Kozai (ZLK) resonance inside mean motion resonances (MMRs). To this end, a semi-secular model is formulated by 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…
We study the steady-state orbital distributions of giant planets migrating through the combination of the Kozai-Lidov (KL) mechanism due to a stellar companion and friction due to tides raised on the planet by the host star. We run a large set of Monte Carlo simulations that describe the secular evolution of a star-planet-star triple system including the effects from general relativistic precession, stellar and planetary spin evolution, and tides. Our simulations show that KL migration produces Hot Jupiters (HJs) with semi-major axes that are generally smaller than in the observations and they…
With hydrodynamical simulations we determine the conditions under which an initially coplanar planet-disc system that orbits a member of a misaligned binary star evolves to form a planet that undergoes Kozai-Lidov (KL) oscillations once the disc disperses. These oscillations may explain the large orbital eccentricities, as well as the large misalignments with respect to the spin of the central star, observed for some exoplanets. The planet is assumed to be massive enough to open a gap in the disc. The planet's tilt relative to the binary orbital plane is subject to two types of oscillations. T…
Kepler-78b is one of a growing sample of planets similar, in composition and size, to the Earth. It was first detected with NASA's \emph{Kepler} spacecraft and then characterised in more detail using radial velocity follow-up observations. Not only is its size very similar to that of the Earth ($1.2 R_\oplus$), it also has a very similar density ($5.6$ g cm$^{-2}$). What makes this planet particularly interesting is that it orbits its host star every $8.5$ hours, giving it an orbital distance of only $0.0089$ au. What we investigate here is whether or not such a planet could have been perturbe…
Stellar companions can influence the formation and evolution of planetary systems, but there are currently few observational constraints on the properties of planet-hosting binary star systems. We search for stellar companions around 77 transiting hot Jupiter systems to explore the statistical properties of this population of companions as compared to field stars of similar spectral type. After correcting for survey incompleteness, we find that $47\%\pm7\%$ of hot Jupiter systems have stellar companions with semi-major axes between 50-2000 AU. This is 2.9 times larger than the field star compa…
Aim. The eccentric von Zeipel-Lidov-Kozai (ZLK) effect is widely used to explain dynamical phenomena in varieties of astrophysical systems. The purpose of this work is to make clear the dynamical essence of the eccentric ZLK effect by constructing an inherent connection between such an effect and dynamics of secular resonance in restricted hierarchical planetary systems. Methods. Dynamical structures of apsidal resonance are analytically studied by means of perturbative treatments. The resonant model is formulated by averaging the Hamiltonian (up to octupole order) over rotating ZLK cycles, pr…
We analyze the behavior of the argument of pericenter $ω_2$ of an outer particle in the elliptical restricted three-body problem, focusing on the inverse Lidov-Kozai resonance. First, we calculate the contribution of the terms of quadrupole, octupole, and hexadecapolar order of the secular approximation of the potential to the outer particle's $ω_2$ precession rate $(dω_2/dτ)$. Then, we derive analytical criteria that determine the vanishing of the $ω_2$ quadrupole precession rate $(dω_2/dτ)_{\text{quad}}$ for different values of the inner perturber's eccentricity $e_1$. Finally, we use such a…
The double-averaging (DA) approximation is widely employed as the standard technique in studying the secular evolution of the hierarchical three-body system. We show that effects stemmed from the short-timescale oscillations ignored by DA can accumulate over long timescales and lead to significant errors in the long-term evolution of the Lidov-Kozai cycles. In particular, the conditions for having an orbital flip, where the inner orbit switches between prograde and retrograde with respect to the outer orbit and the associated extremely high eccentricities during the switch, can be modified sig…
We report the results of the photodynamical analyses of four compact, tight triple stellar systems, KICs 6964043, 5653126, 5731312, 8023317, based largely on $Kepler$ and $TESS$ data. All systems display remarkable eclipse timing and eclipse depth variations, the latter implying a non-aligned outer orbit. Moreover, KIC 6964043 is also a triply eclipsing system. We combined photometry, ETV curves, and archival spectral energy distribution data to obtain the astrophysical parameters of the constituent stars and the orbital elements with substantial precision. KICs 6964043 and 5653126 were found …
The New Horizons flyby of the cold classical Kuiper Belt object MU69 showed it to be a contact binary. The existence of other contact binaries in the 1-10km range raises the question of how common these bodies are and how they evolved into contact. Here we consider that the lobes of MU69 formed as a binary in the Solar nebula, and calculate its orbital evolution in the presence of gas. We find that the sub-Keplerian wind of the disk brings the drag timescales for 10km bodies under 1 Myr for quadratic-velocity drag. In the Kuiper belt, however, the drag is linear with velocity and the effect of…
Shepherding Miorita and its flock: A group of near-Earth asteroids driven by apsidal and von Zeipel-Lidov-Kozai secular resonances. A source of low-perihelion asteroids
Breaking the Mass Inclination Degeneracy of Radial Velocity Measurements via Monitoring von Zeipel-Lidov-Kozai Cycles: Implications in the HD 41004 System
We investigate the effects of gas-disk gravity on the planetesimal dynamics in inclined binary systems, where the circumprimary disk plane is tilted by a significant angle ($i_B$) with respect to the binary disk plane. Our focus is on the Lidov-Kozai mechanism and the evolution of planetesimal eccentricity and inclination. Using both analytical and numerical methods, we find that, on one hand, the disk gravity generally narrows down the Kozai-on region, i.e., the Lidov-Kozai effect can be suppressed in certain parts of (or even the whole of) the disk, depending on various parameters. In the Ko…
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 appropri…
