Planar Circular Restricted Three-body Problem Research Articles

A dynamical study of Hilda asteroids in the Circular and Elliptic RTBP.

The Hilda group is a set of asteroids whose mean motion is in a 3:2 orbital resonance with Jupiter. In this paper, we use the planar Circular Restricted Three-Body Problem (CRTBP) as a dynamical model and we show that there exists a family of stable periodic orbits that are surrounded by islands of quasi-periodic motions. We have computed the frequencies of these quasi-periodic motions and we have shown how the Hilda family fits inside these islands. We have compared these results with the ones obtained using the Elliptic Restricted Three-Body Problem and they are similar, showing the suitability of the CRTBP model. It turns out that, to decide if a given asteroid belongs to the Hilda class, it is much better to look at its frequencies in the planar CRTBP rather than to use two-body orbital elements as it is commonly done today.

Feasibility of orbital capture of near-earth asteroids based on the planar-circular restricted three-body problem

With the goal of efficiently extracting samples or even materials from the surface of an asteroid, this study proposed and investigated a method to change the velocity vector of a near-Earth asteroid and place it into an orbit where it is captured by the Earth’s gravitational field. The change in the orbit of an asteroid is not directly discussed in relation to the change in the velocity vector but is indirectly considered by the change in the Jacobi integral, which is the first integral of the circular-planar restricted three-body problem. In addition, the distribution of the smaller alignment index (SALI) is investigated to find a capture point where the asteroid is not put into a chaotic orbit. The proposed method is numerically demonstrated for fictional asteroid capture missions. The results show that several asteroids can be put into stable captured orbits. Additionally, we propose a method to optimize the value of the Jacobi integral, aiming to stabilize periodic captured orbits. Numerical integration confirms that when the Jacobi integral is optimized, the orbital lifetime of the captured orbit exceeds 500 years.

Investigating temporary capture in the Sun-Jupiter three-body system via Lagrangian coherent structures

Abstract The temporary capture (TC) of Jupiter-family objects has long been a pivotal focus in celestial mechanics research. This study investigates the temporary capture of objects near Jupiter within the context of the planar circular restricted three-body problem (PCRTBP), employing Lagrangian coherent structures (LCSs) and periapsis Poincaré maps. Initially, LCSs are identified via periapsis Poincaré maps and applied to segment the phase space. Parameter scanning enables a detailed analysis, classifying the orbital behaviors of objects in the proximity of Jupiter into three distinct categories: temporary capture, low-energy flyby, and collision, each designating specific regions in phase space. Subsequently, a novel method for screening potential TC objects within the Jupiter system is proposed and validated, informed by the dynamic characteristics of TC motions. The efficacy of this method is illustrated by the reidentification of six known TC comets and the prediction of a prospective TC asteroid, 2002 GV28. Within the framework of the PCRTBP, analogous TC trajectories for these comets and asteroids are identified, offering novel insights into the dynamics of temporary capture events.

A study of the nonlinear dynamics inside the exoplanetary system Kepler-22 using MATLAB® software

Kepler is a discovery-class mission designed to determine the frequency of Earth-radius planets in and near the habitable zone of solar-type stars. A habitable zone of a star is defined as a range of orbits within which a rocky planet can support liquid water on its surface. The most intriguing question driving the search for habitable planets is whether they host life.
 The aim of this paper is to study the motion of a “test particle” inside the exoplanetary system Kepler-22. This system consists of a sun-like star, Kepler-22, and a terrestrial exoplanet, Kepler-22b. This exoplanet is situated in the habitable zone of its star. Kepler-22b is located about 180 pc from Earth in the constellation of Cygnus. It was discovered by NASA’s Kepler Space Telescope in December 2011 and the planet is about 2.4 times the radius of Earth. Scientists don't yet know if Kepler-22b has a rocky, gaseous or liquid composition.
 In this study, let’s derive Lagrange points and perform several numerical tests to discover different possible orbits around the star Kepler-22. From many numerical tests performed, it is also possible to found two tadpole orbits around the Lagrange points L4 and L5 and a tadpole orbit around the exoplanet Kepler-22b, which encircles the two Lagrange points L1, and L2. Some of these orbits are found in the habitable zone and others outside. We have also examined the possibility of the existence of an exomoon around the terrestrial exoplanet Kepler-22b. In this case we have considered the mass of this exomoon.
 The Circular Restricted Three-Body Problem is used in this study. If it is further assumed that the third body (for example a planet, satellite, an asteroid or just a “test particle”) travels in the same plane as the two larger bodies, then there is the Planar Circular Restricted Three-Body Problem

Image classification of retrograde resonance in the planar circular restricted three-body problem

Image classification of retrograde resonance in the planar circular restricted three-body problem

Ballistic Capture Analysis Using the Energy Transition Domain

In the framework of the planar Circular Restricted Three-Body Problem (CR3BP) and for a given value of the Jacobi constant, a necessary condition for a ballistic capture is a passage through zero two-body energy, which defines a region in the synodic plane called energy transition domain (ETD). The latter is the locus of points where the zero-energy condition and selected Jacobi constant are simultaneously fulfilled. This paper shows how an ETD can be described analytically and exploited for the efficient computation and insightful classification of the complete set of ballistic capture trajectories in the planar problem. The new methodology is applied to different three-body systems and highlights the influence of the three-body energy on key aspects of the capture mechanism, such as prograde to retrograde transition. Other interesting results concern three-body energy limits, collisions, long-duration captures, altitude range throughout the capture process, and the origin of ballistic capture orbits.

Transverse foliations in the rotating Kepler problem

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee’s results on homoclinics [23] and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic–heteroclinic chain in the planar circular restricted three-body problem [20], using pseudoholomorphic curves.

Study the Structure of Phase Space in the Frame Work of Planar Circular Restricted Three-Body Problem

The numerical technique of Poincare surface of sections is used to create the surface of sections and to investigate the stability of the phase plane in various systems. This includes investigating the dependence of mass ratio and amplitude of stability areas in the Sun-Jupiter, Sun-Mercury, Sun-Io, and Sun-Uranus systems on the Jacobi constant. The amplitude of the stability areas diminishes with decreasing mass ratio. The phase plane structure differs for smaller mass ratios than it does for larger mass ratios.

On the equilibrium dynamics of a binary system with two Kerr-like bodies

In this study, we investigate a system comprising of two Kerr-like bodies within the context of the planar circular restricted three-body problem. Our main objective is to explore how the equilibrium dynamics of the system are influenced by the free parameters present in the potential. To achieve this, we employ a combination of semi-analytical and numerical methods. Specifically, we numerically compute the coordinates, linear stability, and types of the libration points on the (x,y) plane. Additionally, we extend the analysis beyond a previously published paper by considering the general case where the main bodies of the system have different transition parameter values. In other words, the parameters that control the ratio between classical Newtonian and post-Newtonian gravity in each main body vary. Our findings reveal that this new setup, in contrast to the scenario with equal transition parameter values, exhibits a minimum of five equilibrium points, while the maximum number of equilibria remains unchanged at thirteen.

Studying the Equilibrium Points of the Modified Circular Restricted Three-body Problem: The Case of Sun–Haumea System

We intend to study a modified version of the planar Circular Restricted Three-Body Problem (CRTBP) by incorporating several perturbing parameters. We consider the bigger primary as an oblate spheroid and emitting radiation while the small primary has an elongated body. We also consider the perturbation from a disk-like structure encompassing this three-body system. First, we develop a mathematical model of this modified CRTBP. We have found there exist five equilibrium points in this modified CRTBP model, where three of them are collinear and the other two are non-collinear. Second, we apply our modified CRTBP model to the Sun–Haumea system by considering several values of each perturbing parameter. Through our numerical investigation, we have discovered that the incorporation of perturbing parameters has resulted in a shift in the equilibrium point positions of the Sun–Haumea system compared to their positions in the classical CRTBP. The stability of equilibrium points is investigated. We have shown that the collinear equilibrium points are unstable and the stability of non-collinear equilibrium points depends on the mass parameter μ of the system. Unlike the classical case, non-collinear equilibrium points have both a maximum and minimum limit of μ for achieving stability. We remark that the stability range of μ in non-collinear equilibrium points depends on the perturbing parameters. In the context of the Sun–Haumea system, we have found that the non-collinear equilibrium points are stable.

Generalized Analytical Results on n-Ejection–Collision Orbits in the RTBP. Analysis of Bifurcations

In the planar circular restricted three-body problem and for any value of the mass parameter mu in (0,1) and nge 1, we prove the existence of four families of n-ejection–collision (n-EC) orbits, that is, orbits where the particle ejects from a primary, reaches n maxima in the (Euclidean) distance with respect to it and finally collides with the primary. Such EC orbits have a value of the Jacobi constant of the form C=3mu +Ln^{2/3}(1-mu )^{2/3}, where L>0 is big enough but independent of mu and n. In order to prove this optimal result, we consider Levi-Civita’s transformation to regularize the collision with one primary and a perturbative approach using an ad hoc small parameter once a suitable scale in the configuration plane and time has previously been applied. This result improves a previous work where the existence of the n-EC orbits was stated when the mass parameter mu >0 was small enough. Moreover, for decreasing values of C, there appear some bifurcations which are first numerically investigated and afterward explicit expressions for the approximation of the bifurcation values of C are discussed. Finally, a detailed analysis of the existence of n-EC orbits when mu rightarrow 1 is also described. In a natural way, Hill’s problem shows up. For this problem, we prove an analytical result on the existence of four families of n-EC orbits, and numerically, we describe them as well as the appearing bifurcations.

Orbital Dynamics Landscape near the Most Distant Known Trans-Neptunian Objects

The most distant known trans-Neptunian objects (TNOs), those with perihelion distance above 38 au and semimajor axis above 150 au, are of interest for their potential to reveal past, external, or present but unseen perturbers. Realizing this potential requires understanding how the known planets influence their orbital dynamics. We use a recently developed Poincaré mapping approach for orbital phase space studies of the circular planar restricted three-body problem, which we have extended to the case of the 3D restricted problem with N planetary perturbers. With this approach, we explore the dynamical landscape of the 23 most distant TNOs under the perturbations of the known giant planets. We find that, counter to common expectations, almost none of these TNOs are far removed from Neptune’s resonances. Nearly half (11) of these TNOs have orbits consistent with stable libration in Neptune’s resonances; in particular, the orbits of TNOs 148209 and 474640 overlap with Neptune’s 20:1 and 36:1 resonances, respectively. Five objects can be ruled currently nonresonant, despite their large orbital uncertainties, because our mapping approach determines the resonance boundaries in angular phase space in addition to semimajor axis. Only three objects are in orbital regions not appreciably affected by resonances: Sedna, 2012 VP113 and 2015 KG163. Our analysis also demonstrates that Neptune’s resonances impart a modest (few percent) nonuniformity in the longitude of perihelion distribution of the currently observable distant TNOs. While not large enough to explain the observed clustering, this small dynamical sculpting of the perihelion longitudes could become relevant for future, larger TNO data sets.

Artificial Controlling of the Collinear Liberation Points Using Lorentz Force in the Restricted Three-Body Problem

This work studies the possibility of generating artificial collinear liberation points for the planar circular restricted three-body problem using Lorentz force affecting a charged spacecraft due to the magnetic field of a planet. It is considered to be a magnetic dipole inclined by angle α with the spin axis of the planet. The acceleration components for Lorentz force are first derived in an inertial planet-center coordinate system. Then, they are transformed into the rotating coordinate system of the three-body system, with the planet naturally the smaller primary in a planet-Sun system. The equations for the liberation points are derived including the charge per unit mass as the controlling parameter. Finally, the values of the charge per unit mass required for controlling the collinear liberation point positions are derived. A numerical application for the Sun-Jupiter system is introduced and the relation between the position of the artificial liberation point and the charge per unit mass is presented graphically.

On the problem of convexity for the restricted three-body problem around the heavy primary

In this note, we prove that below the first critical energy level, a proper combination of the Ligon-Schaaf and Levi-Civita regularization mappings provides a convex symplectic embedding of the double cover of the energy surfaces of the planar rotating Kepler problem into ${\mathbb R}^{4}$ endowed with its standard symplectic structure. This convex embedding extends to the bounded component of the planar circular restricted three-body problem around the heavy body outside a small neighborhood of the collisions. This opens up new approaches to attack the Birkhoff conjecture about the existence of a global surface of section in the restricted planar circular three-body problem using holomorphic curve techniques.

Symmetric periodic orbits and invariant disk-like global surfaces of section on the three-sphere

We study Reeb dynamics on the three-sphere equipped with a tight contact form and an anti-contact involution. We prove the existence of a symmetric periodic orbit and provide necessary and sufficient conditions for it to bound an invariant disk-like global surface of section. We also study the same questions under the presence of additional symmetry and obtain similar results in this case. The proofs make use of pseudoholomorphic curves in symplectizations. As applications, we study Birkhoff’s conjecture on disk-like global surfaces of section in the planar circular restricted three-body problem and the existence of symmetric closed Finsler geodesics on the two-sphere. We also present applications to some classical Hamiltonian systems.

Regions without invariant tori of given class for the planar circular restricted three-body problem

A method to establish regions of phase space through which pass no invariant tori transverse to a given direction field is applied to the planar circular restricted three-body problem. Implications for the location of stable orbits for planets around a binary star are deduced. It is expected that lessons learnt from this problem will be useful for applications of the method to other contexts such as flux surfaces for magnetic fields, guiding centre motion in magnetic fields, and classical models of chemical reaction dynamics.

Variational existence proof for multiple periodic orbits in the planar circular restricted three-body problem

The restricted three-body problem (R3BP) is an important research area that deals with significant issues in celestial mechanics, such as analyzing asteroid movement behavior and orbit design for space probes. We aim to show the existence of periodic orbits in the planar circular R3BP. To find these orbits, we adapt a variational approach and symmetry.

Equilibrium Points and Networks of Periodic Orbits in the Pseudo-Newtonian Planar Circular Restricted Three-body Problem

Abstract We explore a pseudo-Newtonian planar circular restricted three-body problem in which the primaries are modeled using an approximate gravitational potential up to the second nonvanishing term of the Fodor–Hoenselaers–Perjés expansion. We aim to understand how the main free parameters of the system affect its dynamical properties. In particular, we determine how the mass of the primaries as well as the transition parameters affect not only the properties of the points of equilibrium (total number, locations, and linear stability) but also the networks of simple symmetric periodic orbits. Our results show that, under this approach, significant variations are observed in the fixed points (number and stability) and periodic orbits of the planar circular restricted three-body problem, even when small contributions of the non-Newtonian terms are considered. We also provide direct applications of the new model potential in real observable binary stellar systems.

Melnikov method for non-conservative perturbations of the restricted three-body problem

We consider the planar circular restricted three-body problem (PCRTBP), as a model for the motion of a spacecraft relative to the Earth-Moon system. We focus on the Lagrange equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, which gives the future asymptotic of a homoclinic orbit as a function of the past asymptotic. In contrast with the more customary Melnikov theory, we do not need to assume that the asymptotic orbits have a special nature (periodic, quasi-periodic, etc.). We add a non-conservative, time-dependent perturbation, as a model for a thrust applied to the spacecraft for some duration of time, or for some other effect, such as solar radiation pressure. We compute the first order approximation of the perturbed scattering map, in terms of fast convergent integrals of the perturbation along homoclinic/heteroclinic orbits of the unperturbed system. As a possible application, this result can be used to determine the trajectory of the spacecraft upon using the thrust.

Measure, dimension, and complexity of the transient motion in Hamiltonian systems

Hamiltonian systems that are either open, leaking, or contain holes in the phase space possess solutions that eventually escape the system’s domain. The motion described by such escape orbits before crossing the escape threshold can be understood as a transient behavior. In this work, we introduce a numerical method to visually illustrate and quantify the transient motion in Hamiltonian systems based on the transient measure, a finite-time version of the natural measure. We apply this method to two physical systems: the single-null divertor tokamak, described by a symplectic map, and the Earth–Moon system, as modeled by the planar circular restricted three-body problem. Our results portray how different locations for the ensemble of initial conditions may lead to different transient dynamical scenarios in both systems. We show that these scenarios can be properly quantified from a geometrical aspect, the transient correlation dimension, and a dynamical aspect, the transient complexity coefficient.