Equilibrium Points L4 Research Articles

New Transient Co-orbital Asteroids of Venus

Venus has no known natural satellites but has 5 known co-orbitals. These are objects trapped in a 1:1 mean-motion resonance with Venus. Co-orbital configurations include retrograde satellite orbits (RS), tadpole orbits (T) around the Lagrangian equilibrium points L4 or L5, and horseshoe orbits around both L4 and L5 (H). At high eccentricity or inclination, co-orbital configurations may also involve compounds of T and RS (T-RS, T-RS-T), H and RS (H-RS) orbits, or transitions between different co-orbital modes. Here we identify asteroids in 2 RS, 1 L4-tadpole, 2 H-RS, and 2 T-RS orbits, as well as 8 additional asteroids in possible temporary co-orbital status. Although the majority of these objects do not yet have well-characterized orbits, 2020 CL1, and 2020 SB do and are very likely to be new co-orbital asteroids. With the new candidates, Venus would have a population of 20 co-orbital asteroids, comparable to those of Mars and Earth.

Periodic changes in the morphology of the Galactic resonance rings

ABSTRACT We study the periodic enhancement of either trailing or leading segments of the resonance elliptical rings in the dynamical model of the Galaxy which reproduces distributions of observed velocities derived from Gaia DR3 (EDR3) data along the Galactocentric distance. The model disc forms a nuclear ring, an inner combined ring, and outer resonance rings R1 and R2. The backbone of the inner combined ring is banana-type orbits around the Lagrange equilibrium points L4 and L5. Orbits associated with the unstable equilibrium points L1 and L2 also support the inner ring. We have found the changes of the morphology of the inner ring with a period of P = 0.57 ± 0.02 Gyr, which is close to the period of revolution along the long-period orbits around the points L4 and L5. A possible explanation of these morphological changes is the formation of an overdensity which then begins circulating along the closed contour. In the region of the Outer Lindblad Resonance (OLR), we have found the changes of the morphology of the outer rings with a period of P = 2.0 ± 0.1 Gyr. Probably, the morphological changes of the outer rings are due to the orbits trapped by the OLR. These orbits exhibit librations of the direction of orbital elongation with respect to the minor axis of the bar as well as the long-term variations in the stellar angular momentum, energy, average radius of the orbit, and eccentricity. Among many librating orbits, we discovered orbits with the libration period of P = 1.91 ± 0.01 Gyr, which may cause the morphological changes of the outer rings.

NUMERICAL INVESTIGATIONS OF THE ORBITAL DYNAMICS AROUND A SYNCHRONOUS BINARY SYSTEM OF ASTEROIDS

In this article, equilibrium points and families of periodic orbits in the vicinity of the collinear equilibrium points of a binary asteroid system are investigated with respect to the angular velocity of the secondary body, the mass ratio of the system and the size of the secondary. We assume that the gravitational fields of the bodies are modeled considering the primary as a mass point and the secondary as a rotating mass dipole. This model allows to compute families of planar and halo periodic orbits that emanate from the equilibrium points L1 and L2. The stability and bifurcations of these families are analyzed and the results are compared with the results obtained with the restricted three-body problem (RTBP). The results provide an overview of the dynamical behavior in the vicinity of a binary asteroid system.

The dynamics of co-orbital giant exomoons – applications for the Kepler-1625 b and Kepler-1708 b satellite systems

ABSTRACT Exomoons are a missing piece of exoplanetary science. Recently, two promising candidates were proposed, Kepler-1625 b-I and Kepler-1708 b-I. While the latter still lacks a dynamical analysis of its stability, Kepler-1625 b-I has already been the subject of several studies regarding its stability and origin. Moreover, previous works have shown that this satellite system could harbour at least two stable massive moons. Motivated by these results, we explored the stability of co-orbital exomoons using the candidates Kepler-1625 b-I and Kepler-1708 b-I as case studies. To do so, we performed numerical simulations of systems composed of the star, planet, and the co-orbital pair formed by the proposed candidates and another massive body. For the additional satellite, we varied its mass and size from a Mars-like to the case where both satellites have the same physical characteristics. We investigated the co-orbital region around the Lagrangian equilibrium point L4 of the system, setting the orbital separation between the satellites from θmin = 30° to θmax = 90°. Our results show that stability islands are possible in the co-orbital region of Kepler-1708 b-I as a function of the co-orbital companion’s mass and angular separation. Also, we identified that resonances of librational frequencies, especially the 2:1 resonance, can constrain the mass of the co-orbital companion. On the other hand, we found that the proximity between the host planet and the star makes the co-orbital region around Kepler-1625 b-I unstable for a massive companion. Finally, we provide TTV profiles for a planet orbited by co-orbital exomoons.

Trajectories Derived from Periodic Orbits around the Lagrangian Point L1 and Lunar Swing-Bys: Application in Transfers to Near-Earth Asteroids

To present a set of trajectories derived from the retrograde periodic orbits around the Lagrangian equilibrium point L1, this paper considers the Circular Restricted Three-body Problem with Earth-Moon masses (CR3BP), the Restricted Bicircular, and Full Four-Body Sun-Earth-Moon-spacecraft Problems (BCR4BP and FR4BP, respectively). These periodic orbits are predicted by the dynamics of the CR3BP. To generate the trajectories of this set, first, slightly different increments of velocity (∆Vs) from those needed to generate periodic orbits around L1 are applied to a spacecraft in circular low Earth orbits in the same direction of their motion when the Earth, the spacecraft, and the Moon are aligned in this order. Thus, translunar trajectories derived from the periodic orbits are obtained and they will lead the spacecraft to the vicinity of the Moon. Depending on the values of the |∆Vs|, which are also functions of the relative positioning between the Sun, the Earth, and the Moon, three types of trajectories of interest are found: Collision with the Moon, escape, and geocentric orbits with large semi-major axes. For a well-defined interval of the |∆Vs|, the trajectories accomplish swing-bys with the Moon and obtain energy to escape from the Earth–Moon system and reach Near-Earth Asteroids (NEAs) between the orbits of Venus and Mars. This procedure reduces the costs of inserting spacecraft into transfer trajectories to a set of NEAs in terms of the required |∆V| by up to 5% when compared to Lambert’s problem, for example. This work also presents analyses of examples of transfers to the NEAs 3361 Orpheus, 99942 Apophis, and 65803 Didymos, from 2025 on.

Science orbits in the Saturn–Enceladus circular restricted three-body problem with oblate primaries

This contribution investigates the properties of a category of orbits around Enceladus. The motivation is the interest in the in situ exploration of this moon following Cassini’s detection of plumes of water and organic compounds close to its south pole. In a previous investigation, a set of heteroclinic connections were designed between halo orbits around the equilibrium points L1 and L2 of the circular restricted three-body problem with Saturn and Enceladus as primaries. The kinematical and geometrical characteristics of those trajectories makes them good candidates as science orbits for the extended observation of the surface of Enceladus: they are highly inclined, they approach the moon and they are maneuver-free. However, the low heights above the surface and the strong perturbing effect of Saturn impose a more careful look at their dynamics, in particular regarding the influence of the polar flattening of the primaries. Therefore, those solutions are here reconsidered by employing a dynamical model that includes the effect of the oblateness of Saturn and Enceladus, individually and in combination. The dynamical equivalents of the halo orbits around the equilibrium points L1 and L2 and their stable and unstable hyperbolic invariant manifolds are obtained in the perturbed models, and maneuver-free heteroclinic connections are identified in the new framework. A systematic comparison with the corresponding solutions of the unperturbed problem shows that qualitative and quantitative features are not significantly altered when the oblateness of the primaries is taken into account. Furthermore, it is found that the J2 coefficient of Saturn plays a larger role than that of Enceladus. From a mission perspective, the results confirm the scientific value of the solutions obtained in the classical circular restricted three-body problem and suggests that this simpler model can be used in a preliminary feasibility analysis.

Observational properties of low-energy orbits around icy moons

The moons of the giant planets are getting into the focus of the next exploration missions to the outer solar system. The reason for this interest resides in their dynamical role within their respective systems and the signs of habitability suggested in some cases by surface and subsurface features. Flyby missions offer adequate recognition scenarios for the initial exploration phase of a celestial body, but they must be followed by orbiters if more extended observations are required. The dynamical structures of the Circular Restricted Three-Body Problem pave the way to such opportunities. In particular, libration point orbits around the two collinear equilibrium points L1 and L2 of a planet–moon-spacecraft system are characterized by low speeds relative to the moon. Heteroclinic and homoclinic connections built from these orbits allow close-up views of the moons. The aim of this work is to determine the observational performance of heteroclinic and homoclinic connections of planar Lyapunov orbits around L1 and L2 for three systems of scientific relevance: Jupiter–Europa, Uranus–Titania and Uranus–Oberon. Lunar surface coverage, repeat patterns, lunar distance ranges and speed profiles are derived and discussed for each case.

Transit regions and ejection/collision orbits in the RTBP

In this paper we analyse the global behaviour of the whole set of ejection orbits in the planar circular RTBP. We consider ejection from the big or the small primary, that is we take the mass parameter μ, the mass traditionally associated with the small primary, in a range of values μ ∈ (0, 1) (the other primary has mass 1−μ). A discussion on the relation between the Lyapunov periodic orbit around the collinear equilibrium point L1 and the ejection orbits is carried out in the range of values of the Jacobi constant such that the associated Hill regions permit only a bounded movement. In particular a chaotic infinity of heteroclinic connections between a primary and the LPO1 are obtained. As a consequence a chaotic infinity of ejection-collision orbits is also derived. Finally, 2D plots, called colour code diagrams, allow to describe the global dynamics of the ejection orbits given a range of time. Such diagrams provide a very accurate understanding of the dynamics of the orbits under discussion.

Dust grains in mean motion orbital resonances with a planet

The orbital evolution of interplanetary dust particles in the Solar System is governed by gravitational and non-gravitational forces. Non-gravitational effects become more significant with decreasing size of dust grains and the effects are relevant on long-time scales. This work considers action of the solar corpuscular radiation in the form of solar wind, solar electromagnetic radiation in the form of the Poynting-Robertson (P-R) effect and gravitational attraction of the Sun and planets. The solar wind influences the evolution of dust grains more significantly than the P-R effect.The paper focuses on mean motion orbital resonances of a dust particle with a planet. We deal with the effect of non-gravitational forces in the restricted three body problem and we also investigate the stability of the equilibrium solution. The motion of the dust grain near the equilibrium points L4 and L5 can be stable under certain circumstances. The analytical results are also supported by numerical calculations.

Effect of perturbations in the Coriolis and centrifugal forces in the Robe‐finite straight segment model with arbitrary density parameter

AbstractThe present article is a study of the effect of small perturbations ϵ and ϵ′ in the Coriolis and centrifugal forces, respectively, on the existence and stability of the equilibrium points in the Robe's restricted three‐body problem when the smaller primary is a finite straight segment. The effect of small perturbations in the centrifugal force has a substantial effect on the location of the equilibrium points, but a small perturbation in the Coriolis force has no impact on them. The present model has two collinear, two out‐of‐plane, and an infinite number of noncollinear equilibrium points. Furthermore, we have discussed the stability of the equilibrium points analytically. The collinear equilibrium points L1 and L2 are found to be conditionally stable for the parameters μ, k, l, ϵ, and ϵ′. It is perceived that, for any ϵ with ∣ϵ ∣ ≪ 1 and ϵ′ ≤ 0.2, L1 is stable, and for ϵ′ ≥ 0.23, it becomes unstable. Furthermore, for any ϵ with |ϵ| ≪ 1 and ϵ′ ≤ −0.025, L2 is stable, and for ϵ′ ≥ −0.024, it becomes unstable. The noncollinear and out‐of‐plane equilibrium points are always unstable.

Road map to L4/L5 with a solar sail

This paper explores the capability of solar sails to transfer a probe from the displaced Sun-Earth L1 and L2 libration points to the region of practical stability (RPS) around the triangular equilibrium points L4 and L5. If the sailcraft arrives inside the RPS with zero synodical velocity, it will remain there with minor station keeping requirements. Moreover, the location of the RPS is ideal for space weather missions as the Sun can be observed from a different angle compared to spacecraft orbiting the L1 point. The unstable manifolds of the displaced L1 and L2 points come close to these regions providing opportunities for natural transfer trajectories. By varying the solar sail orientation along the manifold, the dynamics can be altered and simple transfer trajectories that reach the RPS with few sail maneuvers are enabled. However, these trajectories are not optimal from a transfer time perspective, but can serve as suitable initial guesses for a direct optimization method to find minimum-time transfers between the displaced L1 and L2 points and the RPS at L4 and L5.

Effects of the albedo and disc on the zero velocity curves and linear stability of equilibrium points in the generalized restricted three-body problem

ABSTRACT The important aspects of a dynamical system are its stability and the factors that affect its stability. In this paper, we present an analysis of the effects of the albedo and the disc on the zero velocity curves, the existence of equilibrium points and their linear stability in a generalized restricted three-body problem (RTBP). The proposed problem consists of the motion of an infinitesimal mass under the gravitational field of a radiating-oblate primary, an oblate secondary and a disc that is rotating about the common centre of mass of the system. Significant effects of the albedo and the disc are observed on the zero velocity curves, on the positions of equilibrium points and on the stability region. A linear stability analysis of collinear equilibrium points L1, 2, 3 is performed with respect to the mass parameter μ and albedo parameter QA of the secondary, separately. It is found that L1, 2, 3 are unstable in both cases. However, the non-collinear equilibrium points L4, 5 are stable in a finite range of mass ratio μ. After analysing the individual as well as combined effects of the radiation pressure force of the primary, the albedo force of the secondary, the oblateness of both the primary and secondary and the disc, it is found that these perturbations play a significant role in the design of the trajectories in the vicinity of equilibrium points and in the analysis of their stability property. In the future, the results obtained will improve existing results and will help in the analysis of different space missions. These results are limited to the regular symmetric disc and radiation pressure, which can be extended later.

Effect of Stellar Wind and Poynting–Robertson Drag on Photogravitational Elliptic Restricted Three Body Problem

The existence and linear stability of the planar equilibrium points for photogravitational elliptical restricted three body problem is investigated in this paper. Assuming that the primaries, one of which is radiating are rotating in an elliptical orbit around their common center of mass. The effect of the radiation pressure, forces due to stellar wind and Poynting–Robertson drag on the dust particles are considered. The location of the five equilibrium points are found using analytical methods. It is observed that the collinear equilibrium points L1, L2 and L3 do not lie on the line joining the primaries but are shifted along the y-coordinate. The instability of the libration points due to the presence of the drag forces is demonstrated by Lyapunov’s first method of stability.

Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a Hamiltonian depending on one parameter, K > 0. The corresponding Hamiltonian system of ODE has two equilibrium points L1 (unstable for all K and energy value h(L1)) and L2 (a center for K < Kcrit and a complex saddle for K > Kcrit, with energy value h(L2)). We study the Hamiltonian-Hopf bifurcation phenomena that take place for K close to Kcrit around L2. First, a local analysis based on the computation of the integrable normal form up to a finite order is carried out and the steps for the computation of this (resonant) normal form are explained in a constructive manner. The analysis of the normal form obtained allows: to claim the type of the Hopf bifurcation –supercritical–; to study the local behavior of the electron in a neighborhood of the equilibrium L2 for the original non integrable Hamiltonian (as a perturbative approach from the integrable normal form); to obtain (approximations for) the parametrizations of the relevant invariant objects that take place due to the bifurcation (periodic orbits and invariant manifolds of L2). We compute numerically such objects and analyse not only the local picture of the dynamics close to L2, but also a global description of the dynamics and the effect of the Hopf bifurcation as well as other objects that organize the dynamics are discussed. We conclude that, for K close to Kcrit and the energy level h(L2), the Hopf bifurcation has essentially no effect on the dynamics from a physical point of view. However, for bigger values of K > Kcrit, the Hopf bifurcation has a dramatic effect: different kind of orbits coexist, mostly chaotic. Such orbits provide a ionization mechanism with several passages far from and close to L2 before ionizing. Surprisingly enough, also robust confinement regions (where the electron remains confined for ever), exist in the middle of chaotic areas.

Equilibrium points in the asteroid 2001SN263

In the present work we develop a study based on the Restricted Full Three-Body Problem (RFTBP), where we take into account the size and shape of the bodies studied. The equations of motion and their solutions are discussed for a circular coplanar orbit in order to find the equilibrium points. All the analysis made are based on the physical and orbital characteristics of the most massive bodies of the triple asteroid system 2001SN263. A total of nine equilibrium points were found for this system. Five of these points are collinear and the other four are coplanar. Finally, an analysis of the stability conditions of these equilibrium points are investigated, and it was concluded that only the equilibrium points L4 and L5 are linearly stable.

Three-dimensional representations of the tube manifolds of the planar restricted three-body problem

The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium points L1, L2 play a key role in the understanding of the complicated dynamics of the circular restricted three-body problem. By developing a recent technique of computation of the stable and unstable manifolds, based on the use of Fast Lyapunov Indicators modified by the introduction of a filtering window function, we compute sample three-dimensional representations of the manifolds which show an original vista about their complicated development in the phase-space.

Chaotic Dynamics in a Low-Energy Transfer Strategy to the Equilateral Equilibrium Points in the Earth–Moon System

In the frame of the equilateral equilibrium points exploration, numerous future space missions will require maximization of payload mass, simultaneously achieving reasonable transfer times. To fulfill this request, low-energy non-Keplerian orbits could be used to reach L4 and L5 in the Earth–Moon system instead of high energetic transfers. Previous studies have shown that chaos in physical systems like the restricted three-body Earth–Moon-particle problem can be used to direct a chaotic trajectory to a target that has been previously considered. In this work, we propose to transfer a spacecraft from a circular Earth Orbit in the chaotic region to the equilateral equilibrium points L4 and L5 in the Earth–Moon system, exploiting the chaotic region that connects the Earth with the Moon and changing the trajectory of the spacecraft (relative to the Earth) by using a gravity assist maneuver with the Moon. Choosing a sequence of small perturbations, the time of flight is reduced and the spacecraft is guided to a proper trajectory so that it uses the Moon's gravitational force to finally arrive at a desired target. In this study, the desired target will be an orbit about the Lagrangian equilibrium points L4 or L5. This strategy is not only more efficient with respect to thrust requirement, but also its time transfer is comparable to other known transfer techniques based on time optimization.

On the equilibrium point L1 in the photogravitational oblate relativistic R3BP with application on Sun-planet systems

In the present article the effects of the relativistic, photogravitational and oblateness perturbations in restricted three-body problem, in brief R3BP are investigated. The relativistic effects and the dynamical shape of both radiant primaries are taken into account. The perturbed location of the collinear point L 1 is computed. Series forms of this location are obtained as new analytical results. In order to introduce a semi-analytical view, a mathematica code, using the Mathematica9 package software, is constructed so as to draw the location of collinear point L 1 versus the mass ratio μ ∈ (0, 0.5), taking into account one or more of the considered perturbations. All the obtained figures are analyzed. Some specific systems of R3BP formed of the Sun as the massive primary and a planet as the less massive primary are studied.

Numerical verification of the steepness of three and four degrees of freedom Hamiltonian systems

We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev’s theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4–L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degrees-of-freedom Hamiltonian system obtained from the Fermi-Pasta-Ulam problem by setting the number of particles equal to four.

The effect of Poynting–Robertson drag on the triangular Lagrangian points

We investigate the stability of motion close to the Lagrangian equilibrium points L4 and L5 in the framework of the spatial, elliptic, restricted three-body problem, subject to the radial component of Poynting–Robertson drag. For this reason we develop a simplified resonant model, that is based on averaging theory, i.e. averaged over the mean anomaly of the perturbing planet. We find temporary stability of particles displaying a tadpole motion in the 1:1 resonance. From the linear stability study of the averaged simplified resonant model, we find that the time of temporary stability is proportional to βa1n1, where β is the ratio of the solar radiation over the gravitational force, and a1, n1 are the semi-major axis and the mean motion of the perturbing planet, respectively. We extend previous results (Murray, C.D. [1994]. Icarus 112, 465–484) on the asymmetry of the stability indices of L4 and L5 to a more realistic force model. Our analytical results are supported by means of numerical simulations. We implement our study to Jupiter-like perturbing planets, that are also found in extra-solar planetary systems.