Co-orbital Motion Research Articles

Secular evolution of co-orbital motion of two exoplanets: semi-analytical investigation

Secular evolution of co-orbital motion of two exoplanets: semi-analytical investigation

Jupiter’s Metastable Companions

Jovian co-orbitals share Jupiter’s orbit and exhibit 1:1 mean-motion resonance with the planet. This includes >10,000 so-called Trojan asteroids surrounding the leading (L4) and trailing (L5) Lagrange points, viewed as stable groups dating back to planet formation. A small number of extremely transient horseshoe and quasi-satellite co-orbitals have been identified, which only briefly (<1,000 yr) exhibit co-orbital motions. Via an extensive numerical study, we identify for the first time some Trojans that are certainly only “metastable”; instead of being primordial, they are recent captures from heliocentric orbits into moderately long-lived (10 kyr–100 Myr) metastable states that will escape back to the scattering regime. We have also identified (1) the first two Jovian horseshoe co-orbitals that exist for many resonant libration periods and (2) eight Jovian quasi-satellites with metastable lifetimes of 4–130 kyr. Our perspective on the Trojan population is thus now more complex as Jupiter joins the other giant planets in having known metastable co-orbitals that are in steady-state equilibrium with the planet-crossing Centaur and asteroid populations; the 27 identified here are in agreement with theoretical estimates.

Influence of non-gravitational forces on the co-orbital motion

ABSTRACT In the Solar system, there exist many non-gravitational perturbations for co-orbital objects, such as the solar radiation pressure, Yarkovsky effect, and so forth. Their effects play important roles in the dynamics of co-orbital objects as they lead to long-term perturbations accumulating. The motivation of this paper is to investigate the general mechanism of the non-gravitational force on the co-orbital motion in the circular restricted three-body problem. We propose an effective method for perturbed co-orbital motions by analysing the locus of the co-orbital objects in a two-dimensional map. Several expressions derived uncover how the non-gravitational force acts on orbital parameters. Taking the Sun–Jupiter system as an example, we implement numerical computations to demonstrate the validity of our results. Numerical computation shows that most of loci of co-orbital motions are in agreement with our conclusions. Some interesting phenomena of perturbed co-orbital motion, such as the co-orbital transition and escape, are found and explained. The results obtained from this paper provide an efficient approach to analyse the evolution of perturbed co-orbital motions.

On the Co-orbital Motion of Any Inclination

The co-orbital motion occurs when a celestial body (e.g. an asteroid) shares the same semimajor axis with the perturbing object (e.g. a planet), and thus they are in a 1:1 mean motion resonance. Trojan asteroids in the tadpole orbits around planets in the Solar System are these co-orbital objects. The motion and origin of some Trojan asteroids, particularly those on high-inclination orbits, are still not fully understood. In this paper, a newly developed perturbation function expansion, which is applicable to the 1:1 resonance, is used to investigate the co-orbital motion in three-dimensional space. The position of the resonance center and the resonance width are calculated for different initial orbital elements, and the relationship between the orbital types and the initial orbital elements is analyzed. The results obtained by the analytical method are compared with and verified by the results from numerical simulations. A panorama of the co-orbital motion in the wide orbital elements space is obtained.

Asteroids co-orbital motion classification based on Machine Learning

ABSTRACT In this work, we explore how to classify asteroids in co-orbital motion with a given planet using Machine Learning. We consider four different kinds of motion in mean motion resonance with the planet, nominally Tadpole at L4 and L5, Horseshoe and Quasi-Satellite, building three data sets defined as Real (taking the ephemerides of real asteroids from the JPL Horizons system), Ideal and Perturbed (both simulated, obtained by propagating initial conditions considering two different dynamical systems) for training and testing the Machine Learning algorithms in different conditions. The time series of the variable θ (angle related to the resonance) are studied with a data analysis pipeline defined ad hoc for the problem and composed by: data creation and annotation, time series features extraction thanks to the tsfresh package (potentially followed by selection and standardization) and the application of Machine Learning algorithms for Dimensionality Reduction and Classification. Such approach, based on features extracted from the time series, allows to work with a smaller number of data with respect to Deep Learning algorithms, also allowing to define a ranking of the importance of the features. Physical interpretability of the features is another key point of this approach. In addition, we introduce the SHapley Additive exPlanations for Explainability technique. Different training and test sets are used, in order to understand the power and the limits of our approach. The results show how the algorithms are able to identify and classify correctly the time series, with a high degree of performance.

Dynamics of dust and meteoroids due to electromagnetic transport in the heliosphere

ABSTRACT Observations of dust in the Solar system have indicated the existence of structures at higher ecliptic latitudes, the origin of which is still an ongoing debate. In a previous study, we studied how the interplanetary magnetic field affects the orbital motion of charged dust particles that are moving in co-orbital motion with Jupiter. Our findings revealed that the Lorentz force causes oscillations in orbital inclinations that lead to electromagnetic transport of the dust particles to higher ecliptic latitudes. In this work, using numerical simulations, we investigate how this transportation depends on orbital lifetime, strength of the background magnetic field, planetary mass, and distance from the Sun. In addition, we study the dynamics also outside resonance. We present our findings using the saturation curve, which gives a relation between the maximum amplitude in inclination with respect to the particle size ranging from 1 to 501 $\mu$m. We further study the influence of the solar radiation pressure, the Poynting–Robertson, and the solar wind effects on the shape of the saturation curve and find that a stronger gravitational influence of the planet leads to a steeper curve, decreasing the strength of the electromagnetic transport. The radiative forces lead to a gradual dampening of the latitudinal oscillations of particles inside resonance, while they are unchanged for objects outside of resonance. We argue that the dynamics of dust and meteoroids in the Solar system can only be understood by including space weathering effects.

A semi-analytical model for coorbital motion

ABSTRACT A globally valid analytically averaged Hamiltonian model for the coorbital motion is hard to construct because the analytical expansions of the disturbing function usually diverge in the quasi-satellite domain that is close to collision singularity. In this paper, an analytically averaged model for the coorbital motion is proposed in case of the circular restricted three-body problem, which can describe properly the transitions that occur at small eccentricities and inclinations, such as the transition between the horseshoe orbit and the quasi-satellite orbit. With the help of the numerical averaging method, numerical experiments are carried out to show the validity and accuracy of the analytically averaged model. The averaged model proposed here can be easily extended to more complicated cases such as the elliptic three-body problem or the planetary three-body problem.

Analytical Study of the Co-orbital Motion in the Circular Restricted Three-body Problem

In the restricted three-body problem (RTBP), if a small body and a planet stably orbit around a central star with almost exactly the same semimajor axis, and thus almost the same mean motion, this phenomenon is called the co-orbital motion, or equivalently, the 1:1 mean motion resonance. The classical expansion of the disturbing function is divergent when the semimajor axis ratio of the small body to the planet is close to unity. Thus, most of the previous studies on the co-orbital dynamics were carried out through numerical integrations or semi-analytical approaches. In this work, we construct an analytical averaged model for the co-orbital motion in the framework of the circular RTBP. This model is valid in the entire coorbital region except in the vicinity of the collision singularity. The results of the analytical averaged model are in good agreement with the numerical averaged model even for moderate eccentricities and inclinations. The analytical model can reproduce the tadpole, horseshoe and quasi-satellite orbits common in the planar problem. Furthermore, the asymmetry of 1:1 resonance and the compound orbits (Icarus 137:293–314) in the general spatial problem can also be obtained from the analytical model.

On the retrograde planar co-orbital asteroid motion with Jupiter

AbstractWe study the motion of an asteroid being in retrograde 1/1 resonance with Jupiter (co-orbital motion). We consider the planar case (i=180°) and Jupiter is on a circular or elliptic orbit (e′ = 0.048). In the cirular model we compute families of symmetric periodic orbits and their stability type. In the elliptic model we have isolated periodic orbits which affect the orbital modes of motion as it is shown by the FLI dynamical maps.

On the co-orbital asteroids in the solar system: medium-term timescale analysis of the quasi-coplanar objects

The focus of this work is the current distribution of asteroids in co-orbital motion with Venus, Earth and Jupiter, under a quasi-coplanar configuration and for a medium-term timescale of the order of 900 years. A co-orbital trajectory is a heliocentric orbit trapped in a 1:1 mean-motion resonance with a given planet. As such, to model it this work considers the Restricted Three-Body Problem in the planar circular case with the help of averaging techniques. The domain of each co-orbital regime, that is, the quasi-satellite motion, the horseshoe motion and the tadpole motion, can be neatly defined by means of an integrable model and a simple two-dimensional map, that is invariant with respect to the mass parameter of the planet, and turns out to be a remarkable tool to investigate the distribution of the co-orbitals objects of interest. The study is based on the data corresponding to the ephemerides computed by the JPL Horizons system for asteroids with a sufficient low orbital inclination with respect to the Sun–planet orbital plane. These objects are cataloged according to their current dynamics, together with the transitions that occur in the given time frame from a given type of co-orbital motion to another. The results provide a general catalog of co-orbital asteroids in the solar system, the first one to our knowledge, and an efficient mean to study transitions.

Latitudinal dynamics of co-orbital charged dust in the heliosphere

In recent years, observations have found evidence for dust at higher ecliptic latitudes. Different possible explanations for these signatures have been proposed, most commonly assuming that they originate from collisions of young asteroid families. In the present work, we investigate the influence of the interplanetary magnetic field causing strong latitudinal oscillations that may affect the creation and evolution of dust at these latitudes. Using numerical simulations of a charged dust particle affected by the Lorentz force, we analyse the effect of a simplified magnetic field model specifically on the long-term evolution of the orbital plane of the dust grain. Additionally, we demonstrate the significant agreement with the results of the semi-analytical secular-resonant model we have developed for charged particles in co-orbital motion with a planet. We have found that the interplanetary magnetic field determines the three-dimensional distribution of micron-sized dust grains, causing large excursions of the orbital inclination that distribute the particles to high ecliptic latitudes. The strength of these oscillations depends in particular on the particle size and on the distance to the Sun. Farther outwards in the Solar System, the particle amplitudes are larger.

Revisiting the averaged problem in the case of mean-motion resonances in the restricted three-body problem

A classical approach to the restricted three-body problem is to analyze the dynamics of the massless body in the synodic reference frame. A different approach is represented by the perturbative treatment: in particular the averaged problem of a mean-motion resonance allows to investigate the long-term behavior of the solutions through a suitable approximation that focuses on a particular region of the phase space. In this paper, we intend to bridge a gap between the two approaches in the specific case of mean-motion resonant dynamics, establish the limit of validity of the averaged problem and take advantage of its results in order to compute trajectories in the synodic reference frame. After the description of each approach, we develop a rigorous treatment of the averaging process, estimate the size of the transformation and prove that the averaged problem is a suitable approximation of the restricted three-body problem as long as the solutions are located outside the Hill’s sphere of the secondary. In such a case, a rigorous theorem of stability over finite but large timescales can be proven. We establish that a solution of the averaged problem provides an accurate approximation of the trajectories on the synodic reference frame within a finite time that depend on the minimal distance to the Hill’s sphere of the secondary. The last part of this work is devoted to the co-orbital motion (i.e., the dynamics in 1:1 mean-motion resonance) in the circular-planar case. In this case, an interpretation of the solutions of the averaged problem in the synodic reference frame is detailed and a method that allows to compute co-orbital trajectories is displayed.

OSSOS. XXIII. 2013 VZ70 and the Temporary Coorbitals of the Giant Planets

Abstract We present the discovery of 2013 VZ70, the first known horseshoe coorbital companion of Saturn. Observed by the Outer Solar System Origins Survey for 4.5 yr, the orbit of 2013 VZ70 is determined to high precision, revealing that it currently is in “horseshoe” libration with the planet. This coorbital motion will last at least thousands of years but ends ∼10 kyr from now; 2013 VZ70 is thus another example of the already-known “transient coorbital” populations of the giant planets, with this being the first known prograde example for Saturn (temporary retrograde coorbitals are known for Jupiter and Saturn). We present a theoretical steady-state model of the scattering population of trans-Neptunian origin in the giant planet region (2–34 au), including the temporary coorbital populations of the four giant planets. We expose this model to observational biases using survey simulations in order to compare the model to the real detections made by a set of well-characterized outer solar system surveys. While the observed number of coorbitals relative to the scattering population is higher than predicted, we show that the number of observed transient coorbitals of each giant planet relative to each other is consistent with a trans-Neptunian source.

Apsidal asymmetric-alignment of Jupiter Trojans

ABSTRACT The most distant Kuiper belt objects exhibit the clustering in their orbits, and this anomalous architecture could be caused by Planet 9 with large eccentricity and high inclination. We then suppose that the orbital clustering of minor planets may be observed somewhere else in the Solar system. In this paper, we consider the over 7000 Jupiter Trojans from the Minor Planet Center, and find that they are clustered in the longitude of perihelion ϖ, around the locations ϖJ + 60○ and ϖJ − 60○ (ϖJ is the longitude of perihelion of Jupiter) for the L4 and L5 swarms, respectively. Then we build a Hamiltonian system to describe the associated dynamical aspects for the co-orbital motion. The phase space displays the existence of the apsidally aligned islands of libration centred on Δϖ = ϖ − ϖJ ≈ ±60○, for the Trojan-like orbits with eccentricities e &amp;lt; 0.1. Through a detailed analysis, we have shown that the observed Jupiter Trojans with proper eccentricities ep &amp;lt; 0.1 spend most of their time in the range of |Δϖ| = 0°–120○, while the more eccentric ones with ep &amp;gt; 0.1 are too few to affect the orbital clustering within this Δϖ range for the entire Trojan population. Our numerical results further prove that, even starting from a uniform Δϖ distribution, the apsidal alignment of simulated Trojans similar to the observation can appear on the order of the age of the Solar system. We conclude that the apsidal asymmetric-alignment of Jupiter Trojans is robust, and this new finding can be helpful to design the survey strategy in the future.

A semi-analytic model for the study of 1/1 resonant dynamics of the planar elliptic restricted co-orbital problem

Mean motion resonances (MMRs) are widespread in our Solar System. Moreover, resonant dynamics has always been an essential topic in planetary research. Recently, the research about exoplanets and the potential Planet Nine with large eccentricity has given rise to our interests in the secular dynamics inside MMRs in the elliptic model. In this paper, we study the fixed points of the averaged Hamiltonian and the long-term stable regions of the 1/1 resonance (or co-orbital motion) in the elliptic restricted three-body problem (ERTBP) systematically. Numerical integrations prove those test particles trapped in “apsidal co-rotation”, where both the resonant angle ϕ res and the secular angle ⧍ϖ (or apsidal longitude differences) librate simultaneously, always survive the long-term simulations. Furthermore, utilizing a semianalytical method based on the adiabatic approach, three families of long-term fixed points of the averaged Hamiltonian of the planar ERTBP inside the 1/1 resonance have been found. We call them QS-points, Hpoints, and T-points here, whose values of the (ϕ res, ⧍ϖ) are (0°, 180°), (180°, 0°), and (± 60°, ± 60°), respectively. All the fixed points of the averaged Hamiltonian of the co-orbital motion in the ERTBP are presented in the e-e′ plane (′ represents the elements of the planet in this paper). We find that QS-points and T-points always exist for the arbitrary eccentricity of a planet, while H-points only exist for the cases of low e′ and very high e. Furthermore, we measure the libration width in terms of eccentricity, ⧍e, around these stable equilibrium points in the e-⧍ϖ phase-space portraits. The “apsidal co-rotation” around all the stable equilibrium points is presented in the e-e′ plane. All these results are effectively confirmed by numerical experiments. The long-term stable zones around these periodic orbits in the e-e′ plane are significant for the research of the co-orbital motion in the ERTBP. Above all, these practical approaches that we proposed can also be used to study the secular dynamics of other MMRs.

Orbital analysis of small bodies in co-orbital motion with Jupiter through the torus structure

ABSTRACT In this paper, based on the torus structure, we investigate the orbital characteristics of Jupiter Trojans and Jupiter-family comets (JFCs) in co-orbital motion with Jupiter. The motivation of this paper is to test whether the semi-analytical approach and conclusions of the torus structure proposed in the circular restricted three-body problem are still valid for real small bodies in the ephemeris model, and how long they follow the prediction of the semi-analytical approach. Based on the topological structure of the co-orbital motion in the torus space, we propose a method for estimating the libration amplitude for Trojans. 873 Jupiter Trojans with well-determined orbits are picked as examples to demonstrate the validity of our estimation method. Numerical analysis reveals that the difference between the osculating semimajor axes of the Trojan and Jupiter can influence the accuracy of our estimation method significantly. Based on the semi-analytical approach, we predict orbital behaviours of three JFCs, 85P/Boethin, P/2012 US27 (Siding Spring), and P/2019 A1 (PANSTARRS). Numerical integration in the ephemeris model indicates that their real orbital behaviours are consistent with our predictions. In particular, we find that the current quasi-satellite state of P/2012 US27 (Siding Spring) can remain for about 1.5 × 105 yr, much longer than those that correspond to other previously reported QS companions of Jupiter.

A Perturbative Treatment of the Retrograde Co-orbital Motion

Abstract An important area of research in celestial mechanics is the analysis of dynamics in mean-motion resonances (MMR). In this paper, we consider a retrograde MMR 1:1 corresponding to a co-orbital motion, in which the asteroid and the planet revolve around the Sun in opposite directions. The motivation was provided by the recent discovery of real celestial bodies moving in such resonances with giant planets (the most famous example being 514107 Ka‘epaoka‘awela, a retrograde co-orbital asteroid of Jupiter). Our study is conducted in the context of a spatial restricted circular three-body problem. Applying double numerical averaging, we construct the equations describing the secular evolution of the asteroid's orbit and generate phase portraits in an e − ω space. In some cases, the averaging procedure is complicated by the coexistence of several modes of resonant motion. It leads to different scenarios of the long-term behavior and, consequently, to the splitting of the phase portrait into several sheets.

A new understanding of L4 and L5 axial orbits through the torus structure

ABSTRACT In this paper, through the critical isosurface of the pseudo-Hamiltonian of co-orbital motions in the torus space, we provide a new understanding of L4 and L5 axial orbits and their invariant manifolds in the circular restricted three-body problem. The contact points on the critical isosurface of the pseudo-Hamiltonian correspond to the locations of L4 and L5 axial orbits in the torus space, and provide a set of good initial guesses of L4 and L5 axial orbits for the multiple shooting method. Furthermore, we calculate and analyse orbital behaviours of L4 and L5 axial orbit families. Based on the topological structure of the critical isosurface of the pseudo-Hamiltonian, compound dynamical motions of invariant manifolds associated with L4 and L5 axial orbits are discussed. We present an approximate estimation for libration amplitudes of different co-orbital portions of invariant manifolds. Results obtained from numerical integration demonstrate the validity of our semi-analytical approach in the torus space..

Transient Jupiter Co-orbitals from Solar System Sources

Abstract We demonstrate dynamical pathways from main-belt asteroid and Centaur orbits to those in co-orbital motion with Jupiter, including the retrograde (inclination i &gt; 90°) state. We estimate that, at any given time, there should be ∼1 km-scale or larger escaped asteroid in a transient direct (prograde) orbit with semimajor axis near that of Jupiter’s (a ≃ a J ), with proportionally more smaller objects as determined by their size distribution. Most of these objects would be in the horseshoe dynamical state, and are hard to detect due to their moderate eccentricities (spending most of their time beyond 5 au) and longitudes relative to Jupiter being spread nearly all over the sky. We also show that ≈1% of the transient asteroid co-orbital population is on retrograde orbits with Jupiter. This population, like the recently identified asteroid (514107) 2015 BZ509, can spend millions of years with a ≃ a J including tens or hundreds of thousands of years formally in the retrograde 1:-1 co-orbital resonance. Escaping near-Earth asteroids (NEAs) are thus likely the precursors of the handful of known high-inclination objects with a ≃ a J . We compare the production of Jovian co-orbitals from escaping NEAs with those from incoming Centaurs. We find that temporary direct co-orbitals are likely dominated by Centaur capture, but we only find production of (temporary) retrograde Jovian co-orbitals (including very long-lived ones) from the NEA source. We postulate that the primordial elimination of the inner solar system’s planetesimal population could provide a supply route for a metastable outer solar system reservoir for the high-inclination Centaurs.

Erratum: phase structure of co-orbital motion with jupiter

Erratum: phase structure of co-orbital motion with jupiter