Eccentricity Oscillations Research Articles

Circumbinary Disk Evolution in the Presence of an Outer Companion Star

Abstract We consider a hierarchical triple system consisting of an inner eccentric binary with an outer companion. A highly misaligned circumbinary disk around the inner binary is subject to two competing effects: (i) nodal precession about the inner binary eccentricity vector that leads to an increase in misalignment (polar alignment) and (ii) Kozai–Lidov (KL) oscillations of eccentricity and inclination driven by the outer companion that leads to a reduction in the misalignment. The outcome depends upon the ratio of the timescales of these effects. If the inner binary torque dominates, then the disk aligns to a polar orientation. If the outer companion torque dominates, then the disk undergoes KL oscillations. In that case, the highly eccentric and misaligned disk is disrupted and accreted by the inner binary, while some mass is transferred to the outer companion. However, when the torques are similar, the outer parts of the circumbinary disk can undergo large eccentricity oscillations while the inclination remains close to polar orientation. The range of initial disk inclinations that evolve to a polar orientation is smaller in the presence of the outer companion. Disk breaking is also more likely, at least temporarily, during the polar alignment process. The stellar orbits in HD 98800 have parameters such that polar alignment of the circumbinary disk is expected. In the absence of gas, solid particles are unstable at much smaller radii than the gas-disk inner tidal truncation radius because KL-driven eccentricity leads to close encounters with the binary.

Stability of planetary systems within the S-star cluster: the Solar system analogues

ABSTRACT A dynamically relaxed dense cluster comprised of about 40 stars (the so-called S-stars) inhabits the central region of our Galaxy. Their stars revolve around the Sgr A* massive object. To understand the dynamical evolution of planetary systems in a particular environment like that around Sgr A*, we carry out direct N-body simulations of planetary systems embedded in the S-star cluster. In this work, we investigate the short-term stability of the planets orbiting around S-stars after their close interactions with the central massive black hole of our Galaxy. We find that planetary systems go through encounters with the supermassive black hole (SMBH) and the nearby stars. We determine the frequency and the strength of planetary systems’ encounters with the nearby stars as these encounters remarkably increase for systems assigned to S-stars closer to the SMBH. The SMBH severely destabilizes the planetary systems, though we noted that the small oscillations in the mutual eccentricity and inclination of the planetary system could be caused by the planet–planet coupling and the near-resonance effect between the two planets. We obtain estimates of the fraction of survivor planets ($\sim 51{{\ \rm per\ cent}}$), and find that planets stripped from their hosting star are generally captured on close orbits around Sgr A*. We notify while gas giants are tidally disrupted, terrestrial planets do not. We estimate that Sgr A* flares can be due to the tidal disruption events of starless giant planets.

The Chaotic History of the Retrograde Multi-planet System in K2-290A Driven by Distant Stars

Abstract The equator of star K2-290A was recently found to be inclined by 124° ± 6° relative to the orbits of both its known transiting planets. The presence of a companion star B at ∼100 au suggested that the birth protoplanetary disk could have tilted, thus providing an explanation for the peculiar retrograde state of this multi-planet system. In this work, we show that a primordial misalignment is not required and that the observed retrograde state is a natural consequence of the chaotic stellar obliquity evolution driven by a wider-orbit companion C at ≳2000 au long after the disk disperses. The star C drives eccentricity and/or inclination oscillations on the inner binary orbit, leading to widespread chaos from the periodic resonance passages between the stellar spin and planetary secular modes. Based on a population synthesis study, we find that the observed stellar obliquity is reached in ∼40%–70% of the systems, making this mechanism a robust outcome of the secular dynamics, regardless of the spin-down history of the central star. This work highlights the unusual role that very distant companions can have on the orbits of close-in planets and the host star’s spin evolution, connecting four orders of magnitude in distance scale over billions of orbits. We finally comment on the application to other exoplanet systems, including multi-planet systems in wide binaries.

Eccentric Neutron Star Disk Driven Type II Outburst Pairs in Be/X-ray Binaries

Abstract Be star X-ray binaries are transient systems that show two different types of outbursts. Type I outbursts occur each orbital period while type II outbursts have a period and duration that are not related to any periodicity of the binary system. Type II outbursts may be caused by mass transfer to the neutron star from a highly eccentric Be star disk. A sufficiently misaligned Be star decretion disk undergoes secular Von Zeipel–Lidov–Kozai (ZLK) oscillations of eccentricity and inclination. Observations show that in some systems the type II outbursts come in pairs with the second being of lower luminosity. We use numerical hydrodynamical simulations to explore the dynamics of the highly misaligned disk that forms around the neutron star as a consequence of mass transfer from the Be star disk. We show that the neutron star disk may also be ZLK unstable and that the eccentricity growth leads to an enhancement in the accretion rate onto the neutron star that lasts for several orbital periods, resembling a type II outburst. We suggest that in a type II outburst pair, the first outburst is caused by mass transfer from the eccentric Be star disk while the second and smaller outburst is caused by the eccentric neutron star disk. We find that the timescale between outbursts in a pair may be compatible with the observed estimates.

Enhanced Kozai–Lidov Eccentricity Oscillations in Nuclear Star Clusters

Abstract Stellar motions in the innermost parts of galactic nuclei, where the gravity of a supermassive black hole dominates, follow Keplerian ellipses to the first order of approximation. These orbits may be subject to periodic (Kozai–Lidov) oscillations of their orbital elements if some nonspherically distributed matter (e.g., a secondary massive black hole, coherent stellar subsystem, or large-scale gaseous structure) perturbs the gravity of the central supermassive black hole. These oscillations are, however, affected by the overall potential of the host nuclear star cluster. In this paper, we show that its influence strongly depends on the properties of the particular system, as well as the considered timescale. We demonstrate that for systems with astrophysically relevant parameters, the Kozai–Lidov oscillations of eccentricity can be enhanced by the extended potential of the cluster in terms of reaching significantly higher maximal values. In a more general statistical sense, the oscillations of eccentricity are typically damped. The efficiency of the damping, however, may be small to negligible for the suitable parameters of the system. This applies, in particular, in the case when the perturbing body is on an eccentric orbit.

Analysis of Contact Characteristics of Carbon Brush/Slip Ring under Eccentric Oscillation of Hydrogenerator Rotor

In this paper, the contact characteristics of the carbon brush/slip ring of the hydrogenerator under the eccentric vibration of the rotor are studied. The influence of spring parameters on the contact loss of the carbon brush/slip ring was studied by establishing the motion model of the carbon brush/slip ring for theoretical calculation and fitting the motion trajectory curve with MATLAB software. Studies have shown that the loss of contact is mainly related to the spring damping coefficient and the precompression. By optimizing these two parameters, the contact stability of the carbon brush on the ring surface of the slip ring can be improved. It provides a theoretical reference for the design and optimization of the parameters of the carbon brush/slip ring spring.

Formation of Polar Terrestrial Circumbinary Planets

Abstract All circumbinary planets (CBPs) currently detected are in orbits that are almost coplanar to the binary orbit. While misaligned CBPs are more difficult to detect, observations of polar-aligned circumbinary gas and debris disks around eccentric binaries suggest that polar planet formation may be possible. A polar-aligned planet has a stable orbit that is inclined by 90° to the orbital plane of the binary with an angular momentum vector that is aligned to the binary eccentricity vector. With n-body simulations we model polar terrestrial planet formation using hydrodynamic gas disk simulations to motivate the initial particle distribution. Terrestrial planet formation around an eccentric binary is more likely in a polar alignment than in a coplanar alignment. Similar planetary systems form in a polar alignment around an eccentric binary and a coplanar alignment around a circular binary. The polar planetary systems are stable even with the effects of general relativity. Planetary orbits around an eccentric binary exhibit tilt and eccentricity oscillations at all inclinations; however, the oscillations are larger in the coplanar case than the polar case. We suggest that polar-aligned terrestrial planets will be found in the future.

Secular dynamics of a test particle perturbed by a Miyamoto–Nagai disc

ABSTRACT We study the secular behaviour of a test particle orbiting a dominant central body and perturbed by a Miyamoto–Nagai (MN) disc. We derive a quadrupole-level secular Hamiltonian of this system, which involves a dimensionless parameter η that is used to characterize the flattening of MN disc. (The smaller η, the flatter disc; and η = 0 gives the infinitely thin Kuzmin disc.) We find that, in the quadrupole approximation, the perturbation of the MN disc can give rise to the von Zeipel–Lidov–Kozai (ZLK)-like dynamics and depending on the value of η the dynamics has two different manifestations: (i) when η < 1/3, the test particle’s behaviour is similar to that described in the classical ZLK problem. In particular, as η increases from 0 to 1/3, the critical inclination for the large eccentricity oscillations decreases from $26{_{.}^{\circ}}56$ to 0°, ; (ii) when η > 1/3, the orbital evolution of the test particle and the phase-space morphology are opposite to the classical ZLK case. This leads to a striking result that the test particle cannot remain on a near-coplanar orbit if its eccentricity is sufficiently large. However, as η increases further the ZLK-like dynamics would be gradually suppressed by the spherical term in the Hamiltonian. We also survey the global secular dynamics numerically in which the quadrupole approximation is no longer valid.

Exomoons in Systems with a Strong Perturber: Applications to α Cen AB

Abstract The presence of a stellar companion can place constraints on occurrence and orbital evolution of satellites orbiting exoplanets, i.e., exomoons. In this work we revise earlier orbital stability limits for retrograde orbits in the case of a three-body system consisting of a star, planet, and satellite. The revised limit reads a sat crit ≈ 0.668 ( 1 − 1.236 e p ) for e p ≤ 0.8 in units of the Hill Radius and represents the lower critical orbit as a function of the planetary eccentricity e p. A similar formula is determined for exomoons hosted by planets in binary star systems, where e p is replaced with the components of free and forced eccentricity from secular orbit evolution theory. By exploring the dynamics of putative exomoons in α Centauri AB we find that the outer stability limit can be much less than half the Hill Radius due to oscillations in the planetary orbital eccentricity caused by the gravitational interaction with the binary star. We show, furthermore, how the resulting truncation of the outer stability limit can affect the outward tidal migration and potential observability of exomoons through transit-timing variations (TTVs). Typical TTV (rms) amplitudes induced by exomoons in binary systems are ≲10 minutes and appear more likely for planets orbiting the less massive stellar component.

Orbital Stability Around the Primary of a Binary Asteroid System

Orbiting the primary of a binary asteroid system is extremely challenging due to the perturbative effects of the primary’s nonspherical gravity and the secondary’s close-proximity third-body gravity. In this work, the stability of perturbed Keplerian orbits around the primary is investigated from the point of view of the long-term eccentricity oscillation. Numerical investigations indicate that the eccentricity undergoes a large-amplitude oscillation, caused by the secular perturbation of the secondary’s gravity and may cause impact. A two-degree-of-freedom dynamic model is established, based on the doubly averaged, semi-analytical orbital dynamics incorporating effects of the primary’s oblateness and the secondary’s nonspherical third-body gravity. The oscillation of eccentricity, including its phase and amplitude, and its dependence on initial orbital geometry, is investigated through the phase space structure. The results can reveal the origin of the instability, predict stable and unstable regions in the space of orbital elements, and determine the initial orbital geometry that can ensure the secular stability. The binary asteroid system 2003 YT1 is used as an example to present our verifications and analyses, and the results can also be applied to other binary asteroid systems and even planetary systems with the central body’s oblateness and the third-body gravity dominating the perturbative environments.

On the Correlation between Hot Jupiters and Stellar Clustering: High-eccentricity Migration Induced by Stellar Flybys

Abstract A recent observational study suggests that the occurrence of hot Jupiters (HJs) around solar-type stars is correlated with stellar clustering. We study a new scenario for HJ formation, called “Flyby Induced High-e Migration,” that may help explain this correlation. In this scenario, stellar flybys excite the eccentricity and inclination of an outer companion (giant planet, brown dwarf, or low-mass star) at large distance (10–300 au), which then triggers high-e migration of an inner cold Jupiter (at a few astronomical units) through the combined effects of von Zeipel–Lidov–Kozai (ZLK) eccentricity oscillation and tidal dissipation. Using semianalytical calculations of the effective ZLK inclination window, together with numerical simulations of stellar flybys, we obtain the analytic estimate for the HJ occurrence rate in this formation scenario. We find that this “flyby induced high-e migration” could account for a significant fraction of the observed HJ population, although the result depends on several uncertain parameters, including the density and lifetime of birth stellar clusters, and the occurrence rate of the “cold Jupiter + outer companion” systems.

Spin-orbit misalignments in tertiary-induced binary black-hole mergers: Theoretical analysis

Black-hole (BH) binary mergers driven by gravitational perturbations of tertiary companions constitute an important class of dynamical formation channels for compact binaries detected by LIGO/VIRGO. Recent works have examined numerically the combined orbital and spin dynamics of BH binaries that undergo large Lidov-Kozai (LK) eccentricity oscillations induced by a highly inclined companion and merge via gravitational wave radiation. However, the extreme eccentricity variations make such systems difficult to characterize analytically. In this paper, we develop an analytical formalism for understanding the spin dynamics of binary BHs undergoing LK-induced mergers. We show that, under certain conditions, the eccentricity oscillations of the binary can be averaged over to determine the long-term behavior of the BH spin in a smooth way. In particular, we demonstrate that the final spin-orbit misalignment angle $\theta_{\rm sl}$ is often related to the binary's primordial spin orientation through an approximate adiabatic invariant. Our theory explains the "$90^\circ$ attractor" (as found in recent numerical studies) for the evolution of $\theta_{\rm sl}$ when the initial BH spin is aligned with the orbital axis and the octupole LK effects are negligible -- such a "$90^\circ$ attractor" would lead to a small binary effective spin parameter $\chi_{\rm eff}\sim 0$ even for large intrinsic BH spins. We calculate the deviation from adiabaticity in closed form as a function of the initial conditions. We also place accurate constraints on when this adiabatic invariant breaks down due to resonant spin-orbit interactions. We consider both stellar-mass and supermassive BH tertiary companions, and provide simple prescriptions for determining analytically the final spin-orbit misalignment angles of the merging BH binaries.

Observable Predictions from Perturber-coupled High-eccentricity Tidal Migration of Warm Jupiters

Abstract The origin of warm Jupiters (gas giant planets with periods between 10 and 200 days) is an open question in exoplanet formation and evolution. We investigate a particular migration theory in which a warm Jupiter is coupled to a perturbing companion planet that excites secular eccentricity oscillations in the warm Jupiter, leading to periodic close stellar passages that can tidally shrink and circularize its orbit. If such companions exist in warm Jupiter systems, they are likely to be massive and close-in, making them potentially detectable. We generate a set of warm Jupiter-perturber populations capable of engaging in high-eccentricity tidal migration and calculate the detectability of the perturbers through a variety of observational metrics. We show that a small percentage of these perturbers should be detectable in the Kepler light curves, but most should be detectable with precise radial velocity measurements over a 3 month baseline and Gaia astrometry. We find these results to be robust to the assumptions made for the perturber parameter distributions. If a high-precision radial velocity search for companions to warm Jupiters does not find evidence of a significant number of massive companions over a 3 month baseline, it will suggest that perturber-coupled high-eccentricity migration is not the predominant delivery method for warm Jupiters.

Normal forms for the Laplace resonance

We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at second order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms and to the study of its equilibria. Henceforth, resonant normal forms are computed. The main result is the identification of two different classes of equilibria. In the first class, only one kind of stable equilibrium is present: the paradigmatic case is that of the Galilean system. In the second class, three kinds of stable equilibria are possible and at least one of them is characterised by a high value of the forced eccentricity for the ‘first planet’: here, the paradigmatic case is the exo-planetary system GJ-876, in which the combination of libration centres admits triple conjunctions otherwise not possible in the Galilean case. The normal form obtained by averaging with respect to the free eccentricity oscillations describes the libration of the Laplace argument for arbitrary amplitudes and allows us to determine the libration width of the resonance. The agreement of the analytic predictions with the numerical integration of the toy models is very good.

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.

Eccentricity Driven Climate Effects in the Kepler-1649 System

Abstract The discovery of terrestrial exoplanets is uncovering increasingly diverse architectures. Of particular interest are those systems that contain exoplanets at a variety of star–planet separations, allowing direct comparison of exoplanet evolution (comparative planetology). The Kepler-1649 system contains two terrestrial planets similar both in size and insolation flux to Venus and Earth, although their eccentricities remain largely unconstrained. Here we present results of dynamical studies of the system and the potential effects on climate. The eccentricities of the Kepler-1649 system are poorly constrained, and we show that there are dynamically viable regions for further terrestrial planets in between the two known planets for a limited range of eccentricities. We investigate the effect of eccentricity of the outer planet on the dynamics of both planets and show that this results in high-frequency (1000–3000 years) eccentricity oscillations in long-term stable configurations. We calculate the resulting effect of these eccentricity variations on insolation flux and present the results of 3D climate simulations for the habitable zone planet. Our simulations demonstrate that, despite large eccentricity variations, the planet can maintain stable climates with relatively small temperature variations on the substellar hemisphere for a variety of initial climate configurations. Such systems thus provide key opportunities to explore alternative Venus/Earth climate evolution scenarios.

Astronomical Cycles of the Late Permian Lopingian in South China and Their Implications for Third-Order Sea-Level Change

The late Permian (Lopingian) was a crucial climate transition period from the late Paleozoic Ice Age to the early Triassic of exceptionally high temperatures. However, the origins of the third-order sea-level changes during the Lopingian Epoch remain unclear. Here, we presented astronomically calibrated gamma-ray (GR) log and non-U GR (computed gamma ray or CGR) curves from the clastic and carbonate successions of well GFD-1 in the Pingle Depression of South China for studying the sea-level oscillations during the Lopingian. Spectral analyses of the 405 kyr-calibrated GR and CGR time data revealed periodicities close to about 405, about 100, about 44.2, about 35.1, about 21, and about 17.5kyr, supporting the existence of Milankovitch forcing in the sedimentary records. A high-resolution astronomical time scale and high-resolution sedimentation rate curve of the Lopingian from well GFD-1 were constructed by cyclostratigraphic analysis. The eccentricity and obliquity amplitude modulation cycles suggested long periodicities of about 2.4 and about 1.2 myr, respectively. In the Wuchiapingian greenhouse of the Lopingian, the about 2.4 myr eccentricity oscillation controlled’ weak’ glacio-eustasy and/or aquifer eustatic changes related to the global third-order sea-level changes and that a lowstand (W2) was initiated by an eccentricity oscillation minimum. In contrast, during the Changhsingian, which exhibited a cooling event, an about 1.2 myr obliquity cycle was probably strong, with the sea-level records highlighting the link between the ‘icehouse’ sea-level lowering (C2 and C1) and the obliquity nodes. Moreover, dynamic sedimentary noise model as an indicator of sea-level showed local third-order sea-level variations, the coevolution trends in the orbital power, global and local sea-level changes, and sedimentation rate had significant implications for establishing the global nature and synchronicity of these million-year-scale eustatic records and reconstructing the temporal depositional history at a regional scale. In addition, the volcanism and tectonism that continued into the early-middle Wuchiapingian probably led to a series of climate changes that drove the hydrological cycles not paced by the Milankovitch cycles.

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.

Black Hole Mergers from Hierarchical Triples in Dense Star Clusters

Abstract Hierarchical triples are expected to be produced by the frequent binary-mediated interactions in the cores of globular clusters. In some of these triples, the tertiary companion can drive the inner binary to merger following large eccentricity oscillations, as a result of the eccentric Kozai–Lidov mechanism. In this paper, we study the dynamics and merger rates of black hole (BH) hierarchical triples, formed via binary–binary encounters in the CMC Cluster Catalog, a suite of cluster simulations with present-day properties representative of the Milky Way’s globular clusters. We compare the properties of the mergers from triples to the other merger channels in dense star clusters, and show that triple systems do not produce significant differences in terms of mass and effective spin distribution. However, they represent an important pathway for forming eccentric mergers, which could be detected by LIGO–Virgo/KAGRA (LVK), and future missions such as LISA and the DECi-hertz Interferometer Gravitational wave Observatory. We derive a conservative lower limit for the merger rate from this channel of 0.35 Gpc−3 yr−1 in the local universe and up to ∼ 9 % of these events may have a detectable eccentricity at LVK design sensitivity. Additionally, we find that triple systems could play an important role in retaining second-generation BHs, which can later merge again in the core of the host cluster.

Evolution of α Centauri b’s protoplanetary disc

ABSTRACT With hydrodynamical simulations we examine the evolution of a protoplanetary disc around α Centauri B including the effect of the eccentric orbit binary companion α Centauri A. The initially circular orbit disc undergoes two types of eccentricity growth. First, the eccentricity oscillates on the orbital period of the binary, Porb, due to the eccentricity of the binary orbit. Secondly, for a sufficiently small disc aspect ratio, the disc undergoes global forced eccentricity oscillations on a time-scale of around $20\, P_{\rm orb}$. These oscillations damp out through viscous dissipation leaving a quasi-steady eccentricity profile for the disc that oscillates only on the binary orbital period. The time-averaged global eccentricity is in the range 0.05–0.1, with no precession in the steady state. The periastrons of the gas particles are aligned to one another. The higher the disc viscosity, the higher the disc eccentricity. With N-body simulations we examine the evolution of a disc of planetesimals that forms with the orbital properties of the quasi-steady protoplanetary disc. We find that the average magnitude of the eccentricity of particles increases and their periastrons become misaligned to each other once they decouple from the gas disc. The low planetesimal collision velocity required for planet formation suggests that for planet formation to have occurred in a disc of planetesimals formed from a protoplanetary disc around α Centauri B, said disc’s viscosity must be have been small and planet formation must have occurred at orbital radii smaller than about $2.5\, \rm au$. Planet formation may be easier with the presence of gas.