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. Whereas 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 timescales, all as a function of $γ\equiv (1/2) L_1/G_2$, 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 $\cos(i_\mathrm{rel,0}) = -γ$, where $i_\mathrm{rel,0}$ is the initial relative inclination. We provide a Python script which can be used to quickly compute these properties.
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…
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…
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…
Martin et al. (2014b) showed that a substantially misaligned accretion disk around one component of a binary system can undergo global damped Kozai-Lidov oscillations. During these oscillations, the inclination and eccentricity of the disk are periodically exchanged. However, the robustness of this mechanism and its dependence on the system parameters were unexplored. In this paper, we use three-dimensional hydrodynamical simulations to analyze how various binary and disk parameters affect the Kozai-Lidov mechanism in hydrodynamical disks. The simulations include the effect of gas pressure and…
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…
We study a hierarchical triple system with the Kozai-Lidov mechanism, and analyse the cumulative shift of periastron time of a binary pulsar by the emission of gravitational waves. Time evolution of the osculating orbital elements of the triple system is calculated by directly integrating the first-order post-Newtonian equations of motion. The Kozai-Lidov mechanism will bend the evolution curve of the cumulative shift when the eccentricity becomes large. We also investigate the parameter range of mass and semi-major axis of the third companion with which the bending of the cumulative-shift cur…
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 …
We consider the secular dynamics of a binary and a planet in terms of non-restricted, hierarchical three-body problem, including the general relativity corrections to the Newtonian gravity. We determine regions in the parameter space where the relativistic corrections may be important for the long-term dynamics. We try to constrain the inclinations of putative Jovian planets in recently announced binary systems of HD 4113 and HD 156846.
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…
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…
$γ$ 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…
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 …
Previously we showed that a substantially misaligned viscous accretion disk with pressure that orbits around one component of a binary system can undergo global damped Kozai-Lidov (KL) oscillations. These oscillations produce periodic exchanges of the disk eccentricity with inclination. The disk KL mechanism is quite robust and operates over a wide range of binary and disk parameters. However, the effects of self-gravity, which are expected to suppress the KL oscillations for sufficiently massive disks, were ignored. Here, we analyze the effects of disk self-gravity by means of hydrodynamic si…
Breaking the Mass Inclination Degeneracy of Radial Velocity Measurements via Monitoring von Zeipel-Lidov-Kozai Cycles: Implications in the HD 41004 System
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…
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 …
Dynamical scattering of binaries and triple systems of stars, planets, and compact objects may produce highly inclined triple systems subject to Kozai-Lidov (KL) oscillations, potentially leading to collisions, mergers, Type Ia supernovae, and other phenomena. We present the results of more than 400 million gravitational scattering experiments of binary-binary, triple-single, and triple-binary scattering. We compute the cross sections for all possible outcomes and explore their dependencies on incoming velocity, mass, semi-major axis, and eccentricity, including analytic fits and discussion of…
Most, if not all, stars in the field are born in binary configurations or higher multiplicity systems. In dense stellar environment such as the Galactic Center (GC), many stars are expected to be in binary configurations as well. These binaries form hierarchical triple body systems, with the massive black hole (MBH) as the third, distant object. The stellar binaries are expected to undergo large amplitude eccentricity and inclination oscillations via the so-called "eccentric Kozai-Lidov" (EKL) mechanism. These eccentricity excitations, combined with post main sequence stellar evolution, can dr…
In order to understand the rate of merger of stellar-mass black hole binaries (BHBs) by gravitational wave (GW) emission it is important to determine the major pathways to merger. We use numerical simulations to explore the evolution of BHBs inside the radius of influence of supermassive black holes (SMBHs) in galactic centers. In this region the evolution of binaries is dominated by perturbations from the central SMBH. In particular, as first pointed out by Antonini and Perets, the Kozai-Lidov (KL) mechanism trades relative inclination of the BHB to the SMBH for eccentricity of the BHB, and f…
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 …
White dwarf-white dwarf (WD-WD) mergers may lead to type Ia supernovae events. Thompson (2011) suggested that many such binaries are produced in hierarchical triple systems. The tertiary induces eccentricity oscillations in the inner binary via the Kozai-Lidov mechanism, driving the binary to high eccentricities, and significantly reducing the gravitational wave merger timescale (T_GW) over a broad range of parameter space. Here, we investigate the role of tidal forces in these systems. We show that tidal effects are important in the regime of moderately high initial relative inclination betwe…
Compact stellar binaries are expected to survive in the dense environment of the Galactic Center. The stable binaries may undergo Kozai-Lidov oscillations due to perturbations from the central supermassive black hole (Sgr A*), yet the General Relativistic precession can suppress the Kozai-Lidov oscillations and keep the stellar binaries from merging. However, it is challenging to resolve the binary sources and distinguish them from single stars. The close separation of the stable binaries allow higher eclipse probabilities. Here, we consider the massive star SO2 as an example and calculate the…
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 explicit analytical formulae for the maximal eccentricity, $e_{\rm max}$, of the inner binary undergoing ZLK oscillations, where both the test particle limit (parametrised by the inner-to-outer angular momentum ratio $η$) and the…
Over a broad range of initial inclinations and eccentricities an appreciable fraction of hierarchical triple star systems with similar masses are essentially unaffected by the Kozai-Lidov mechanism (KM) until the primary in the central binary evolves into a compact object. Once it does, it may be much less massive than the other components in the ternary, enabling the "eccentric Kozai mechanism (EKM):" the mutual inclination between the inner and outer binary can flip signs driving the inner binary to very high eccentricity, leading to a close binary or collision. We demonstrate this "Mass-los…
The recently discovered system Gaia 0007-1605 consisting of a white dwarf with a close brown dwarf companion and a distant white dwarf tertiary very much resembles the triple system containing the first transiting planet candidate around a white dwarf 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 analyze different formation scenarios for Gaia 0007-1605. We reconstructed the potential common envelope evolution of the system and f…
Zeta Aquarii is a bright and nearby (28 pc) triple star with a 26-year astrometric subsystem. Almost a half of the outer 540-year visual orbit has been covered in 238 years of its observations. Both inner and outer orbits are revised here taking into account recent direct resolution of the inner pair Aa,Ab. The inner orbit has a high eccentricity of 0.87 and is inclined to the outer orbit by 140+-10 degrees, suggesting that Kozai-Lidov cycles take place. The masses of the stars Aa, B, and Ab are 1.4, 1.4, and 0.6 solar. The age of the system is about 3 Gyr, and the two main components have jus…
Many close stellar binaries are accompanied by a far-away star. The "eccentric Kozai-Lidov" (EKL) mechanism can cause dramatic inclination and eccentricity fluctuations, resulting in tidal tightening of inner binaries of triple stars. We run a large set of Monte-Carlo simulations including the secular evolution of the orbits, general relativistic precession and tides, and we determine the semimajor axis, eccentricity, inclination and spin-orbit angle distributions of the final configurations. We find that the efficiency of forming tight binaries (<~16 d) when taking the EKL mechanism into acco…
