Exterior Resonances Research Articles

Non-perturbative investigation of low-eccentricity exterior mean motion resonances

ABSTRACT Mean motion resonances are important in the analysis and understanding of the dynamics of planetary systems. While perturbative approaches have been dominant in many previous studies, recent non-perturbative approaches have revealed novel properties in the low-eccentricity regime for interior mean motion resonances of Jupiter in the fundamental model of the circular planar restricted three-body model. Here, we extend the non-perturbative investigation to exterior mean motion resonances in the low-eccentricity regime (up to about 0.1) and for perturber mass in the range of ∼5 × 10−5 to 1 × 10−3 (in units of the central mass). Our results demonstrate that first-order exterior resonances have two branches at low eccentricity as well as low-eccentricity bridges connecting neighbouring first-order resonances. With increasing perturber mass, higher order resonances dissolve into chaos, whereas low-order resonances persist with larger widths in their radial extent but smaller azimuthal widths. For low-order resonances, we also detect secondary resonances arising from small-integer commensurabilities between resonant librations and the synodic frequency. These secondary resonances contribute significantly to generating the chaotic sea that typically occurs near mean motion resonances of higher mass perturbers.

Dynamical Stability of a 3-DOF Auto-Parametric Vibrating System

PurposeThis article concentrates on the oscillating movement of an auto-parametric dynamical system comprising of a damped Duffing oscillator and an associated simple pendulum in addition to a rigid body as main and secondary systems, respectively.MethodsAccording to the system generalized coordinates, the controlling equations of motion are derived utilizing Lagrange's approach. These equations are solved applying the perturbation methodology of multiple scales up to higher orders of approximation to achieve further precise unique outcomes. The fourth-order Runge–Kutta algorithm is employed to obtain numerical outcomes of the governing system.ResultsThe comparison between both solutions demonstrates their high level of consistency and highlights the great accuracy of the adopted analytical strategy. Despite the conventional nature of the applied methodology, the obtained results for the studied dynamical system are considered new.ConclusionsIn light of the solvability criteria, all resonance scenarios are classified, in which two of the fundamental exterior resonances are examined simultaneously with one of the interior resonances. Therefore, the modulation equations are achieved. The conditions of Routh–Hurwitz are employed to inspect the stability/instability regions and to analyze them in accordance with the solutions in the steady-state case. For various factors of the examined structure, the temporary history solutions, the curves of resonance in terms of the adjusted amplitudes and phases, and the stability zones are graphically presented and discussed.ApplicationsThe results of the current study will be of interest to wide range experts in the fields of mechanical and aerospace technology, as well as those working to reduce rotors dynamical vibrations and attenuate vibration caused by swinging structures.

Analysis of Resonance Transition Periodic Orbits in the Circular Restricted Three-Body Problem

Resonance transition periodic orbits exist in the chaotic regions where the 1:1 resonance overlaps with nearby interior or exterior resonances in the circular restricted three-body problem (CRTBP). The resonance transition periodic orbits have important applications for tour missions between the interior and the exterior regions of the system. In this work, following the increase of the mass parameter μ in the CRTBP model, we investigate the breakup of the first-order resonant periodic families and their recombination with the resonance transition periodic families. In this process, we can describe in detail how the 1:1 resonance gradually overlaps with nearby first-order resonances with increasing strength of the secondary’s perturbation. Utilizing the continuation method, features of the resonance transition periodic families are discussed and characterized. Finally, an efficient approach to finding these orbits is proposed and some example resonance transition periodic orbits in the Sun–Jupiter system are presented.

A new expansion of planetary disturbing function and applications to interior, co-orbital and exterior resonances with planets

In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination, and the harmonic arguments are linear combinations of the mean longitude, longitude of pericenter and longitude of ascending node of each mass. The resulting new expansion is valid for arbitrary inclinations and semimajor axis ratios. In the case of mean motion resonant configuration, the disturbing function can be easily averaged to produce the analytical expansion of resonant disturbing function. Based on the analytical expansion, the Hamiltonian model of mean motion resonances is formulated, and the resulting analytical developments are applied to Jupiter’s inner and co-orbital resonances and Neptune’s exterior resonances. Analytical expansion is validated by comparing the analytical results with the associated numerical outcomes.

Determination of scattering frequencies for two-dimensional acoustic problems using boundary element method

This paper presents a numerical approach for the determination of scattering frequencies of two-dimensional acoustic exterior resonance problems using boundary element method combined with a contour integral method. Since the solving range is surrounded by a Jordan curve in the complex plane, the complex eigenvalues can be acquired through evaluating the contour integral along the Jordan curve using the contour integral method. Complex scattering frequencies are directly extracted by the proposed approach, and their corresponding eigenmodes are also presented by evaluating the vibration amplitudes of internal points. The behaviors of the fictitious eigenfrequencies for unbounded domain are also studied. Numerical examples demonstrate the effectiveness of the approach for the resonance problems of unbounded domains.

Long-term Hydrodynamic Simulations on the Planetesimals Trapped in the First-order Mean Motion Resonances

Abstract The resonant perturbations from planets are able to halt the drag-induced migration, and capture the inwardly drifting planetesimals into mean motion resonances. The equilibrium eccentricity of planetesimals in resonances, and the minimum size of planetesimal that can trigger resonance trapping, have been analyzed and formulated. However, the analytical works based on the assumption that the disk is axisymmetric, which is violated by the asymmetric structures developed by planets. We perform long-term 2D hydrodynamic simulations to study the dynamics of planetesimals in the j: (j + 1) first-order exterior resonances, and reexamine the theoretical expressions. We find the expression of equilibrium eccentricity underestimates the values for resonances with j < 5, in particular the 1:2 resonance that the underestimation can be 30%–40%. Within the parameter space we explored, we find the equilibrium eccentricity and the minimum size are reduced in an asymmetric disk. The amount of discrepancy in eccentricity depends on the degree of asymmetric structures. For cases of Earth-sized planets, where the disk is less disturbed, the planetesimal’s eccentricity can reach to the values predicted by our modified expression. For gaseous planets, however, the eccentricity can be 0.01–0.02 smaller in value. We find the minimum size is 10 times smaller, and the factor seems to be independent of the planet’s mass. The influences of asymmetric profiles on the eccentricity and the minimum size could affect the outcome of collisions between resonant and nonresonant planetesimals, and the amount of planetesimals migrated into the planet’s feeding zone.

The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits

In the framework of the perturbed photo-gravitational restricted three-body problem, the first order exterior resonant orbits and the first, third and fifth order interior resonant periodic orbits are analyzed. The location, eccentricity and period of the first order exterior and interior resonant orbits are investigated in the unperturbed and perturbed cases for a specified value of Jacobi constant C.It is observed that as the number of loops increases successively from one loop to five loops, the period of infinitesimal body increases in such a way that the successive difference of periods is either 6 or 7 units. It is further observed that for the exterior resonance, as the number of loops increases, the location of the periodic orbit moves towards the Sun whereas for the interior resonance as the number of loops increases, location of the periodic orbit moves away from the Sun. Thereby we demonstrate that the location of resonant orbits of the given order moves away from the Sun when perturbation is included.The evolution of interior first order resonant orbit with three loops is studied for different values of Jacobi constant C. It is observed that when the value of C increases, the size of the loop decreases and degenerates finally into a circle, the eccentricity of periodic orbit decreases and location of the periodic orbit moves towards the second primary body.

Strength, stability and three dimensional structure of mean motion resonances in the solar system

In the framework of the circular restricted three body problem we show that the numerically computed strength SR(e, i, ω) is a good indicator of the strength and width of the mean-motion resonances in the full space (e, i, ω). We present a survey of strengths in the space (e, i) for typical interior and exterior resonances. The resonance strength is highly dependent on (e, i, ω) except for exterior resonances of the type 1:k for which the dependence with (i, ω) is softer. Such resonances are thus strong even for retrograde orbits. All other resonances are weaker at very-high eccentricities for ω ∼ 90° or 270° and 60° ≲ i ≲ 120°. We explore the resonance structure in the space (a, i) by means of dynamical maps and we find structures similar to those of space (a, e).

Inclined asymmetric librations in exterior resonances

Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $\pi$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.

Did Jupiter's core form in the innermost parts of the Sun's protoplanetary disc?

Jupiter's core is generally assumed to have formed beyond the snow line. Here we consider an alternative scenario, that Jupiter's core may have accumulated in the innermost parts of the protoplanetary disk. A growing body of research suggests that small particles ("pebbles") continually drift inward through the disk. If a fraction of drifting pebbles is trapped at the inner edge of the disk a several Earth-mass core can quickly grow. Subsequently, the core may migrate outward beyond the snow line via planet-disk interactions. Of course, to reach the outer Solar System Jupiter's core must traverse the terrestrial planet-forming region. We use N-body simulations including synthetic forces from an underlying gaseous disk to study how the outward migration of Jupiter's core sculpts the terrestrial zone. If the outward migration is fast (Tmig~10^4 years), the core simply migrates past resident planetesimals and planetary embryos. However, if its migration is slower (Tmig~10^5 years) the core removes solids from the inner disk by shepherding objects in mean motion resonances. In many cases the disk interior to 0.5-1 AU is cleared of embryos and most planetesimals. By generating a mass deficit close to the Sun, the outward migration of Jupiter's core may thus explain the absence of terrestrial planets closer than Mercury. Jupiter's migrating core often stimulates the growth of another large (~Earth-mass) core -- that may provide a seed for Saturn's core -- trapped in exterior resonance. The migrating core also may transport a fraction of terrestrial planetesimals, such as the putative parent bodies of iron meteorites, to the asteroid belt.

Robust and Efficient Solution of the Drum Problem via Nyström Approximation of the Fredholm Determinant

The “drum problem''---finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition---has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce the following two ideas to remedy this: (1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant, and we show that this is many times faster than the usual iterative minimization of a singular value. (2) We fix the problem of spurious exterior resonances via a combined field representation. This also provides the first robust boundary integral eigenvalue method for non--simply connected domains. We implement the new method in two dimensions using spectrally accurate Nyström product quadrature. We prove exponential convergence of the determinant at roots for domains with analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency, in a variety of domain shapes including a nonconvex cavity shape with strong exterior resonances.

Origin and sustainability of the population of asteroids captured in the exterior resonance 1:2 with Mars

At present, approximately 1500 asteroids are known to evolve inside or sticked to the exterior 1:2 resonance with Mars at a ≃ 2.418 AU, being (142) Polana the largest member of this group. The effect of the forced secular modes superposed to the resonance gives rise to a complex dynamical evolution. Chaotic diffusion, collisions, close encounters with massive asteroids and mainly orbital migration due to the Yarkovsky effect generate continuous captures to and losses from the resonance, with a fraction of asteroids remaining captured over long time scales and generating a concentration in the semimajor axis distribution that exceeds by 20% the population of background asteroids. The Yarkovsky effect induces different dynamics according to the asteroid size, producing an excess of small asteroids inside the resonance. The evolution in the resonance generates a signature on the orbits, mainly in eccentricity, that depends on the time the asteroid remains captured inside the resonance and on the magnitude of the Yarkovsky effect. The greater the asteroids, the larger the time they remain captured in the resonance, allowing greater diffusion in eccentricity and inclination. The resonance generates a discontinuity and mixing in the space of proper elements producing misidentification of dynamical family members, mainly for Vesta and Nysa-Polana families. The half-life of resonant asteroids large enough for not being affected by the Yarkovsky effect is about 1 Gyr. From the point of view of taxonomic classes, the resonant population does not differ from the background population and the excess of small asteroids is confirmed.

A SEARCH FOR ADDITIONAL PLANETS IN FIVE OF THE EXOPLANETARY SYSTEMS STUDIED BY THE NASAEPOXIMISSION

We present time series photometry and constraints on additional planets in five of the exoplanetary systems studied by the EPOCh (Extrasolar Planet Observation and Characterization) component of the NASA EPOXI mission: HAT-P-4, TrES-3, TrES-2, WASP-3, and HAT-P-7. We conduct a search of the high-precision time series for photometric transits of additional planets. We find no candidate transits with significance higher than our detection limit. From Monte Carlo tests of the time series using putative periods from 0.5 days to 7 days, we demonstrate the sensitivity to detect Neptune-sized companions around TrES-2, sub-Saturn-sized companions in the HAT-P-4, TrES-3, and WASP-3 systems, and Saturn-sized companions around HAT-P-7. We investigate in particular our sensitivity to additional transits in the dynamically favorable 3:2 and 2:1 exterior resonances with the known exoplanets: if we assume coplanar orbits with the known planets, then companions in these resonances with HAT-P-4b, WASP-3b, and HAT-P-7b would be expected to transit, and we can set lower limits on the radii of companions in these systems. In the nearly grazing exoplanetary systems TrES-3 and TrES-2, additional coplanar planets in these resonances are not expected to transit. However, we place lower limits on the radii of companions that would transit if the orbits were misaligned by 2.0 degrees and 1.4 degrees for TrES-3 and TrES-2, respectively.

Inner Solar System dynamical analogs of plutinos

By studying orbits of asteroids potentially in 3:2 exterior mean motion resonance with Earth, Venus, and Mars, we have found plutino analogs. We identify at least 27 objects in the inner Solar System dynamically protected from encounter through this resonance. These are four objects associated with Venus, six with Earth, and seventeen with Mars. Bodies in the 3:2 exterior resonance (including those in the plutino resonance associated with Neptune) orbit the Sun twice for every three orbits of the associated planet, in such a way that with sufficiently low libration amplitude close approaches to the planet are impossible. As many as 15% of Kuiper Belt objects share the 3:2 resonance, but are poorly observed. One of several resonance sweeping mechanisms during planetary migration is likely needed to explain the origin and properties of 3:2 resonant Kuiper Belt objects. Such a mechanism likely did not operate in the inner Solar System. We suggest that scattering by the next planet out allows entry to, and exit from, 3:2 resonance for objects associated with Venus or Earth. 3:2 resonators of Mars, on the other hand, do not cross the paths of other planets, and have a long lifetime. There may exist some objects trapped in the 3:2 Mars resonance which are primordial, with our tests on the most promising objects known to date indicating lifetimes of at least tens of millions of years. Identifying 3:2 resonant systems in the inner Solar System permits this resonance to be studied on shorter timescales and with better determined orbits than has been possible to date, and introduces new mechanisms for entry into the resonant configuration.

The Mars 1:2 resonant population

An excess of around 400 asteroids in the distribution of the semimajor axes of the asteroids is identified by means of numerical integrations as generated by a population of approximately 1000 asteroids evolving inside the exterior resonance 1:2 with Mars. Approximately 200 asteroids are librating around the asymmetric libration centers and their evolution in a time-scale of 1 million years appears stable but with a strong influence of Mars' eccentricity. The biggest Mars 1:2 resonant asteroid is (142) Polana.

Effect of radiation on dust particles in orbital resonances

The effect of electromagnetic radiation on the dynamics of arbitrarily shaped cosmic dust particles is investigated. The paper concentrates on the motion of dust grains near commensurability resonances with a planet—mean-motion resonances—and possible capture of the grains in the resonances. A particle is in resonance with a planet when the ratio of the mean motions of the two objects is a ratio of two small integers. The most fundamental properties of the orbital evolution of spherical dust particles in the mean-motion resonances are shortly rederived: the solar wind effect is also included and the existing result is improved. The results for spherical particles are compared with the detailed numerical calculations for nonspherical particles. It is shown that the fundamental results valid for spherical grains do not hold, in general, for nonspherical particles. While spherical particles are always characterized by the secular decrease of the semi-major axes near mean-motion resonances, this may not be true for nonspherical particles. Nonspherical grains may exhibit an increase of the semi-major axes before capturing in the mean-motion resonances. This is caused by the effect of electromagnetic radiation on nonspherical dust grains. The eccentricities of spherical particles in the exterior resonances approach a limiting value, but nonspherical grains may not follow this behaviour. The interior resonances are characterized by a systematic decrease of eccentricity for spheres, but various behaviours exist in the case of irregularly shaped particles. The motion of a nonspherical dust particle under the action of electromagnetic radiation may be characterized by a small change of the semi-major axis during a long-time interval, but the particle is not captured in any mean-motion resonance. This kind of motion does not exist for spherical grains.

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem.

Planar periodic orbits in exterior resonances with Neptune

In the framework of the planar restricted three-body problem we study a considerable number of resonances associated to the basic dynamical features of Kuiper belt and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is an indication of the existence of asymmetric ones only in a few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies, the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem, the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals.

A Signature of Planetary Migration: The Origin of Asymmetric Capture in the 2 : 1 Resonance

The spatial distribution of Kuiper Belt objects (KBOs) in 2 : 1 exterior resonance with Neptune constrains that planet's migration history. Numerical simulations demonstrate that fast planetary migration generates a larger population of KBOs trailing rather than leading Neptune in orbital longitude. This asymmetry corresponds to a greater proportion of objects caught into asymmetric resonance such that their resonance angles ϕ librate about values greater than π (trailing) as opposed to less than π (leading). We provide, for the first time, an explanation of this phenomenon, using physical, analytic, and semianalytic arguments. Central to our understanding is how planetary migration shifts the equilibrium points of the superposed direct and indirect potentials. Symmetric libration, in which ϕ librates about ~π, precedes capture into asymmetric resonance. As a particle transitions from symmetric to asymmetric libration, if ϕ exceeds the value ψ at the unstable point of asymmetric resonance, then the particle is caught into trailing resonance, while if ϕ < ψ, the particle is caught into leading resonance. The probability that the KBO is caught into trailing resonance is determined by the fraction of time it spends with ϕ > ψ while in symmetric libration. This fractional time increases with faster migration because migration not only shifts ψ to values less than π but also shifts the stable point of symmetric libration to values greater than π. Smaller eccentricities prior to capture strengthen the effect of these shifts. Large capture asymmetries appear for exponential timescales of migration τ shorter than ~107 yr. The observed distribution of 2 : 1 KBOs (two trailing and seven leading) excludes τ ≤ 106 yr with 99.65% confidence.

Motion of dust near exterior resonances with planet

Motion of non-spherical cosmic dust particle under the action of electromagnetic radiation and gravitational forces of central star and a planet orbiting the star is investigated. The contribution concentrates on the problem if the particle can be trapped in commensurability resonances with the planet (a particle is in resonance with a planet when the ratio of their orbital periods is the ratio of two small integers). Trapping in the circular restricted three-body problem (planetary orbit is circle) with electromagnetic radiation coming from the star is considered. The effect of radiation plays an important role as for the possible capture in exterior resonances (particle's orbital period is larger than that for the planet): particle's optical properties not only define positions of the resonances, but also the capture times and evolution of orbital elements. Results for spherical and non-spherical particles are compared: physical difference between possibilities for resonant captures for spherical and non-spherical dust particles is pointed out.We concentrate on possible trapping of the grain in commensurability resonances – mean motion exterior resonances – with planet Neptune. The orbital evolution of non-spherical dust grains of radius ≈ 2 micrometers is numerically calculated.Capture of spherical grain in a resonance is characterized by decrease of semimajor axis outside the resonance. Non-spherical grain can exhibit also an increase of semimajor axis outside the resonance. This is the first case when a physically justified force – force given by interaction between electromagnetic radiation and non-spherical dust particle – can generate such a behaviour.