Oscillations In Inclination Research Articles

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

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

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.

Quasi-secular evolution of mildly hierarchical triple systems: analytics and applications for GW sources and hot Jupiters

In hierarchical triple systems, the inner binary is perturbed by a distant companion. For large mutual inclinations, the Lidov–Kozai mechanism secularly excites large eccentricity and inclination oscillations of the inner binary. The maximal eccentricity attained, emax, is simply derived and widely used. However, for mildly hierarchical systems (i.e. the companion is relatively close and massive), non-secular perturbations affect the evolution. Here we account for fast non-secular variations and find a new analytic formula for emax, in terms of the system’s hierarchy level, correcting previous work and reproducing the orbital flip criteria. We find that emax is generally enhanced, allowing closer encounters between the inner binary components, thus significantly changing their interaction and its final outcome. We then extend our approach to include additional relativistic and tidal forces. Using our results, we show that the merger time of gravitational-wave (GW) sources orbiting massive black holes in galactic nuclei is enhanced compared with previous analysis accounting only for the secular regime. Consequently, this affects the distribution and rates of such GW sources in the relevant mild-hierarchy regime. We test and confirm our predictions with direct N-body and 2.5-level post-Newtonian codes. Finally, we calculate the formation and disruption rates of hot Jupiters (HJs) in planetary systems using a statistical approach, which incorporates our novel results for emax. We find that more HJs migrate from further out, but they are also tidally disrupted more frequently. Remarkably, the overall formation rate of HJs remains similar to that found in previous studies. Nevertheless, the different rates could manifest in different underlying distribution of observed warm Jupiters.

THE ROLE OF THE KOZAI–LIDOV MECHANISM IN BLACK HOLE BINARY MERGERS IN GALACTIC CENTERS

ABSTRACT 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 mechanism trades relative inclination of the BHB to the SMBH for eccentricity of the BHB, and for some orientations can bring the BHB to an eccentricity near unity. At very high eccentricities, GW emission from the BHB can become efficient, causing the members of the BHB to coalesce. We use a novel combination of two N-body codes to follow this evolution. We are required to simulate small systems to follow the behavior accurately. We have completed 400 simulations that range from ∼300 stars around a 103 black hole to ∼4500 stars around a 104 black hole. These simulations are the first to follow the internal orbit of a binary near an SMBH while also following the changes to its external orbit self-consistently. We find that this mechanism could produce mergers at a maximum rate per volume of ∼100 Gpc−3 yr−1 or considerably less if the inclination oscillations of the binary remain constant as the BHB inclination to the SMBH changes, or if the binary black hole fraction is small.

Merging binaries in the Galactic Center: the eccentric Kozai–Lidov mechanism with stellar evolution

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 drive the inner stellar binaries to merge. We study the mergers of stellar binaries in the inner 0.1 pc of the GC caused by gravitational perturbations due to the MBH. We run a large set of Monte Carlo simulations that include the secular evolution of the orbits, general relativistic precession, tides, and post-main-sequence stellar evolution. We find that about 13 % of the initial binary population will have merged after a few million years and about 29 % after a few billion years. These expected merged systems represent a new class of objects at the GC and we speculate that they are connected to G2-like objects and the young stellar population.

Oscillations of relative inclination angles in compact extrasolar planetary systems

The Kepler Mission has detected dozens of compact planetary systems with more than four transiting planets. This sample provides a collection of close-packed planetary systems with relatively little spread in the inclination angles of the inferred orbits. A large fraction of the observational sample contains limited multiplicity, begging the question whether there is a true diversity of multi transiting systems, or if some systems merely possess high mutual inclinations, allowing them to appear as single-transiting systems in a transit-based survey. This paper begins an exploration of the effectiveness of dynamical mechanisms in exciting orbital inclination within exoplanetary systems of this class. For these tightly packed systems, we determine that the orbital inclination angles are not spread out appreciably through self-excitation. In contrast, the two Kepler multi-planet systems with additional non-transiting planets are susceptible to oscillations of their inclination angles, which means their currently observed configurations could be due to planet-planet interactions alone. We also provide constraints and predictions for the expected transit duration variations (TDVs) for each planet. In these multi-planet compact Kepler systems, oscillations of their inclination angles are remarkably hard to excite; as a result, they tend to remain continually mutually transiting (CMT-stable). We study this issue further by augmenting the planet masses and determining the enhancement factor required for oscillations to move the systems out of transit. The oscillations of inclination found here inform the recently suggested dichotomy in the sample of solar systems observed by Kepler.

ECCENTRICITY GROWTH AND ORBIT FLIP IN NEAR-COPLANAR HIERARCHICAL THREE-BODY SYSTEMS

The dynamical evolution of a hierarchical three body system is well characterized by the eccentric Kozai-Lidov mechanism, where the inner orbit can undergo large eccentricity and inclination oscillations. It was shown before that starting with a circular inner orbit, large mutual inclination (40-140 degree) can produce long timescale modulations that drives the eccentricity to extremely large value and can flip the orbit. Here, we demonstrate that starting with an almost coplanar configuration, for eccentric inner and outer orbits, the eccentricity of the inner orbit can still be excited to high values, and the orbit can flip by ~180 degree, rolling over its major axis. The ~180 degree flip criterion and the flip timescale are described by simple analytic expressions that depend on the initial orbital parameters. With tidal dissipation, this mechanism can produce counter-orbiting exo-planetary systems. In addition, we also show that this mechanism has the potential to enhance the tidal disruption or collision rates for different systems.

Numerical simulation of capsule deformation in simple shear flow

The transient deformation of two-dimensional non-circular and three-dimensional non-spherical capsules in simple shear flow is studied numerically, using the hybrid immersed boundary and multi-block lattice Boltzmann method recently proposed by the present authors. The capsules are modeled as Newtonian liquid drops enclosed by elastic membranes; the liquids inside and outside the capsule have the same physical properties. The present results show important different behaviors between two-dimensional and three-dimensional capsules in shear flow. For two-dimensional non-circular capsules without considering the membrane bending rigidity, or considering the bending with the minimum bending-energy configurations (shapes at which the bending-energy has a global minimum) being uniform-curvature shapes, the capsules will always achieve the steady tank treading motion (a capsule deforms to a steady shape with a steady inclination and the membrane rotates around the liquid inside). However, for three-dimensional non-spherical capsules without membrane bending rigidity, such a steady mode does not exist; with the shear rate decreasing, the three-dimensional capsules’ motion changes from swinging mode (a capsule undergoes periodic shape deformation and inclination oscillation while its membrane is rotating around the liquid inside) to flipping mode. The deformation of two-dimensional capsules, with their initial non-circular shapes taken as the minimum bending-energy configurations, is also considered. It is quite interesting to find such two-dimensional capsules behave qualitatively similar to three-dimensional capsules: there is a swinging-to-flipping transition induced by lowering the shear rate.

Dynamic motion of red blood cells in simple shear flow

A three-dimensional numerical model is proposed to simulate the dynamic motion of red blood cells (RBCs) in simple shear flow. The RBCs are approximated by ghost cells consisting of Newtonian liquid drops enclosed by Skalak membranes which take into account the membrane shear elasticity and the membrane area incompressibility. The RBCs have an initially biconcave discoid resting shape, and the internal liquid is assumed to have the same physical properties as the matrix fluid. The simulation is based on a hybrid method, in which the immersed boundary concept is introduced into the framework of the lattice Boltzmann method, and a finite element model is incorporated to obtain the forces acting on the nodes of the cell membrane which is discretized into flat triangular elements. The dynamic motion of RBCs is investigated in simple shear flow under a broad range of shear rates. At large shear rates, the cells are found to carry out a swinging motion, in which periodic inclination oscillation and shape deformation superimpose on the membrane tank treading motion. With the shear rate decreasing, the swinging amplitude of the cell increases, and finally triggers a transition to tumbling motion. This is the first direct numerical simulation that predicts both the swinging motion of the RBCs and the shear rate induced transition, which have been observed in a recent experiment. It is also found that as the mode changes from swinging to tumbling, the apparent viscosity of the suspension increases monotonically.

A hybrid method to study flow-induced deformation of three-dimensional capsules

A hybrid method is proposed to study the transient deformation of liquid filled capsules with elastic membranes under flow. In this method, the immersed boundary concept is introduced into the framework of lattice Boltzmann method, and the multi-block strategy is employed to refine the mesh near the capsule to increase the accuracy and efficiency of computation. A finite element model is incorporated to obtain the forces acting on the membrane nodes of the three-dimensional capsule which is discretized into flat triangular elements. The present method was validated by studying the transient deformation of initially spherical and oblate-spheroidal capsules with various membrane constitutive laws under shear flow; and there were good agreements with previous theory or numerical results. The versatility of the present method was demonstrated by studying the effects of inertia on the deformation of capsules in shear flow; and the inertia effects were found to be significant. The transient deformation of capsules with initially biconcave discoid shape in shear flow was also studied. The unsteady tank-treading motion was observed, in which the capsule undergoes periodic shape deformation and inclination oscillation while its membrane is rotating around the liquid inside. To our knowledge, this motion of three-dimensional biconcave discoid capsules has not been fully recovered by numerical simulation so far.

Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow

The dynamic motion of three-dimensional (3D) capsules in a shear flow is investigated by direct numerical simulation. The capsules are modeled as Newtonian liquid droplets enclosed by elastic membranes, with or without considering the membrane-area incompressibility. The internal liquid of the capsules is the same as that outside. The dynamic motion of capsules with initially spherical and oblate spheroidal unstressed shapes is studied under various shear rates. The results show that spherical capsules deform to stationary configurations and then the membranes rotate around the liquid inside (steady tank-treading motion). Such a steady mode is not observed for oblate spheroidal capsules. It is shown that with the shear rate decreasing, the motion of oblate spheroidal capsules changes from the swinging mode (a capsule undergoes periodic shape deformation and inclination oscillation while its membrane is rotating around the liquid inside) to tumbling mode.

Planet dispersal in binary systems during transient multiple star phases

Aims. If a significant fraction of binary star systems spent some time as inclined triple systems, either during their formation process or as the outcome of several close dynamical encounters in a crowded stellar environemnt, then the number of planets in binaries would be significantly lower than around single stars. The stellar chaotic phase preceding the instability of the triple system and the wide oscillations in eccentricity and inclination of the companion star due to the high mutual inclination between the companion and the singleton would quickly eject planets orbiting the binary in S-type orbits. Methods. We perform numerical simulations of the dynamical evolution of hierarchical triple star systems with planets hosted around the primary star of the inner binary. Different values of mutual inclination, binary separation and singleton initial semimajor axis are explored in a statistical way. Results. We find that a significant mutual inclination i m between the singleton and the binary is a key factor for instability of the planetary system. When i m is larger than ~$40^{\circ}$ the fraction of planets in the binary surviving the chaotic phase of the triple declines dramatically. The combination of eccentricity and inclination oscillations of the binary companion induced by the secular perturbations of the singleton and the sequence of close encounters preceding the ejection of one star fully destabilize a planetary system extending beyond 1 AU from the star. For i m around $90^{\circ}$ the percentage of surviving planets is lower than 20% for all binaries with a semimajor axis smaller than 200 AU. Conclusions. The frequency of planets in binaries with low separation may be strongly reduced by the residence of the pair in the past in a temporary inclined hierarchical triple.

Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

It was believed until very recently that a near-equatorial satellite would always keep up with the planet's equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich (2004), this opinion originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet's obliquity change. Some of these terms will be nontrivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet's obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite's orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in the subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics.

Orbital Behavior of Captured Satellites: The Effect of Solar Gravity on Triton's Post-Capture Orbit

The effects of solar perturbations on the postcapture orbital behavior of satellites are investigated in the context of the restricted, circular three-body problem as applied to Neptune, Triton, and the Sun. Highly eccentric and inclined satellite orbits are considered; thus a numerical, phenomenological approach is taken to describe variations of the satellite's orbital elements. We focus on harmonic variations in specific orbital angular momentum h, and thus pericenter distance q, eccentricity e , semimajor axis a, and inclination to Neptune's orbital plane i . From prograde and retrograde simulations over a range of eccentricities and semimajor axes, a momentum oscillation is found with a period of half a Neptune year and an amplitude proportional to a2e2 cos i. Inclined orbits also experience a longer period, secular-torque-driven variation in h associated with orbital precession and nutation, upon which the semiannual oscillation is superimposed. The amplitude of the longer period variation can exceed and dominate the semiannual variation, and the two can combine to produce much larger variations in the elements q, e, and i than is possible for noninclined orbits, leading in some circumstances to "Neptune,grazing." Consequently, if Triton was temporarily gravitationally captured, solar perturbations could have increased e and reduced h sufficiently to drive the pericenter close to Neptune. There, interactions with a gaseous protoplanetary nebula or a collision with an existing satellite could have dissipated enough orbital energy to make capture permanent. It is more likely, though, that Triton was promptly captured by collision or gas drag into a lower q state to begin with. In either situation, capture at lower q ensures that further orbital variation does not bring Triton dangerously close to Neptune. Repeated close flybys following permanent capture are likely (and could also occur in the less likely event of an extended temporary capture). Multiple close flybys increase the probability of further satellite collisions and multiply the effects of gas drag. For gas-drag capture, possibly recurring larger pericenters associated with the longer period inclination oscillation imply longer gas-drag evolution time scales, because Triton's passages through the nebula are reduced in frequency. Similarly, for tidal evolution, pericenter oscillations could have delayed the onset of large-scale melting within Triton's interior and extended Triton's tidally driven postcapture orbital contraction if large inclination variations were present. The behavior described here may have influenced other satellite captures and can be identified in integrations of the present motion of the outer jovian satellites.

Internal Resonance of a Cracked Rotor. Major Critical Speed and Critical Speeds in Precritical Range.

Internal resonance of cracked rotors is investigated using 2-degree-of-freedom rotor systems. Phenomena at the major critical speed and other subcritical speeds in the precritical speed range are discussed theoretically and experimentally. Phenomena in a lateral oscillation model with internal resonance and in an inclination oscillation model with no internal resonance are compared. In the former model, the two critical speeds of forward and backward whirling modes coincide. The following is clarified. (a) The vibration of the synchronous backward whirling mode changes from a steady-state oscillation to an unstable oscillation due to internal resornance. (b) The superharmonic oscillations of 2nd and 3rd order of the backward whirling mode and the super-subharmonic oscillation of 3/2-order of the backward whirling mode occur due to internal resornance.

The connection between comet P/Machholz and the Quadrantid meteor stream

The parent comet of the Quadrantid meteors was not observed until, perhaps, quite recently. Comet P/Maccholz (1986 VIII) currently has orbital elements that differ drastically from those of the meteor stream, but its secular perturbations involve large-scale oscillations in inclination and perihelion distance similar to those earlier found for the Quadrantids. Whether there may have been an epoch in the past, when the Quadrantids were shed by comet P/Machholz depends upon whether or not there has been enough time since that epoch for differential Jovian perturbations to cause the 180° discrepancy in nodal longitudes presently observed.

Geomagnetic secular variations (inclination) of high latitude fiord cores: eastern Canadian Arctic

Paleomagnetic measurements are reported from 11 piston cores, from the fiords and shelf of eastern Baffin Island, N.W.T., between latitudes 66 and 72 degrees north. The majority of the measurements arc from bioturbated, massive, or laminated mud, with some drop-stones and graded sand beds. Corrected radiocarbon dates on the acid-insoluble organic matter fraction, supplemented by AMS dates on in situbivalves, indicate that all cores extend into the early Holocene, and three extend into the latest Pleistocene Sedimentation rates averaged between 0.2m/ka and 1.4m/ka. Because of varying sedimentation rates, the depth scales are converted to 100 or 200 yr/sample time series. The results indicate a scries of geomagnetic secular oscillations with amplitudes in inclination of c. 10 degrees. A stacked record from four piston cores suggests seven major oscillations in inclination. Times when inclinations consistently exceeded 80° occurred c. 1,400, 4,500, and 8,000 B.P. The most characteristic oscillation occurred c.1,400 ± B.P., when inclinations were nearly vertical. Inclination errors are associated with gravity flows and/or with an increase in sand content, or changes in physical properties. In one core an interval of reverse polarity is attributed to a slump.

Geomagnetic secular variations (inclination) of high latitude fiord cores: eastern Canadian Arctic

Paleomagnetic measurements are reported from 11 piston cores, from the fiords and shelf of eastern Baffin Island, N.W.T., between latitudes 66 and 72 degrees north. The majority of the measurements arc from bioturbated, massive, or laminated mud, with some drop-stones and graded sand beds. Corrected radiocarbon dates on the acid-insoluble organic matter fraction, supplemented by AMS dates on in situbivalves, indicate that all cores extend into the early Holocene, and three extend into the latest Pleistocene Sedimentation rates averaged between 0.2m/ka and 1.4m/ka. Because of varying sedimentation rates, the depth scales are converted to 100 or 200 yr/sample time series. The results indicate a scries of geomagnetic secular oscillations with amplitudes in inclination of c. 10 degrees. A stacked record from four piston cores suggests seven major oscillations in inclination. Times when inclinations consistently exceeded 80° occurred c. 1,400, 4,500, and 8,000 B.P. The most characteristic oscillation occurred c.1,400 ± B.P., when inclinations were nearly vertical. Inclination errors are associated with gravity flows and/or with an increase in sand content, or changes in physical properties. In one core an interval of reverse polarity is attributed to a slump.

Palaeomagnetic study of the Holocene and Upper Pleistocene sediments from Lake Barombi Mbo, Cameroun: first results

The analyses of the stable (deposition and post-depositional) remanent magnetization carried by the lacustrine sediments contained in a 24 m sequence collected from the bottom of the maar lake, Barombi Mbo (Cameroun) yield well-defined declination and inclination oscillations which can be attributed to palaeosecular variations of the local geomagnetic field. The transformation to a time-scale, based on the interpolation and extrapolation of 14C ages, allows comparison of this record with contemporaneous records obtained in other parts of the world. The successive inclination peaks can be individually identified with those found in western Europe, in spite of small chronological discrepancies which are in the range of the standard deviation of the ages. Normalization of the NRMs by anhysteretic magnetization provides a profile of ‘relative palaeointensities’ which shows similarities with that reconstructed from the same time interval in a sequence from western Europe although an important phase shift appears.

A 10 000 yr geomagnetic secular variation record from three Australian maars

Summary. Detailed within-lake magnetic stratigraphies enable composite declination and inclination versus depth curves to be derived for sets of 6m cores from each of three volcanic crater lakes in western Victoria. Comprehensive radiocarbon chronologies for each sequence enable reliable timescales to be assigned to the variations in direction of the remanence. The resultant time-scaled curves are in satisfactory agreement and are therefore thought to provide an estimate of the geomagnetic secular variation in SE Australia during the last 10 000 yr. Amplitude attenuation occurs as an increasing function of age, and probably results from long time-scale remagnetization processes. Mean directions coincide with the geocentric axial dipole field directions at the three sites. Inclination oscillations are stronger and better resolved than those in declination, whereas the opposite tends to be true for the European records. Reduction of records to a common ‘virtual’ site suggests that the difference is largely of geometrical origin. Comparisons of VGP paths do not reveal a common dipole wobble component. The sense of looping of the magnetic vector favours westward rather than eastward drift of features of the non-dipole field. Relative field strength estimates based on normalization of NRM with respect to measures of the magnetic mineral content do not agree with Australian and worldwide’ archaeointensity results.