Spatial Circular Restricted Three-body Problem Research Articles

Fully numerical computation of heteroclinic connection families in the spatial three-body problem

Heteroclinic connections in the spatial circular restricted three-body problem play an important role in astrodynamics. This paper presents a fully numerical methodology for the computation of the two-parameter families of connections that exist between families of invariant tori in the three-body problem. The computation of connections is presented as a two-point boundary value problem in which the initial and final states belong to unstable and stable manifolds of two respective normally hyperbolic invariant manifolds. A flow map torus computation procedure is modified to compute particular trajectories within these spaces, and a robust two-parameter continuation scheme is paired with a shooting technique to compute the entire connection family. Results are provided in the Sun–Earth and Earth–Moon representations of the spatial three-body problem. The Sun–Earth connection family works to verify the methodology with previously developed semi-analytical connection computation techniques. The Earth–Moon example demonstrates the methodology in a setting where fully numerical techniques are required to study the connection family.

Averaged Spatial Circular Restricted Three-Body Problem: Internal Case and New Results

Averaged Spatial Circular Restricted Three-Body Problem: Internal Case and New Results

Symplectic splitting of Hamiltonian structures and reduced monodromy matrices

In Hamiltonian systems monodromy and reduced monodromy matrices of periodic orbits are symplectic. A symmetry is a symplectic or anti-symplectic involution which leaves the Hamiltonian invariant. A natural class to study are periodic orbits which are invariant under such involutions. Depending on whether the orbit is invariant under a symplectic or anti-symplectic symmetry, its monodromy matrix satisfies special properties. In the symplectic case, the monodromy matrix admits a symplectic decomposition, depending on the symplectic involution's fixed point set. In the anti-symplectic case, the monodromy matrix is of special symmetric form, which was constructed by Frauenfelder–Moreno. Such monodromy matrices are very beneficial in applied problems. The goal of this paper is to design a general construction behind the reduced monodromy's symplectic splitting, by using the notion of Hamiltonian manifolds, and to give a general framework for monodromy and reduced monodromy matrices of invariant periodic orbits in Hamiltonian systems with symmetries. In addition, we show an application related to contact forms, especially to the spatial circular restricted three-body problem for energy level sets of contact type.

Pseudo-holomorphic dynamics in the restricted three-body problem

AbstractIn this paper, we identify the five dimensional analogue of the finite energy foliations introduced by Hofer–Wysocki–Zehnder for the study of three dimensional Reeb flows, and show that these exist for the spatial circular restricted three-body problem (SCR3BP) whenever the planar dynamics is convex. We introduce the notion of a fiberwise-recurrent point, which may be thought of as a symplectic version of the leafwise intersections introduced by Moser, and show that they exist in abundance for a perturbative regime in the SCR3BP. We then use this foliation to induce a Reeb flow on the standard 3-sphere, via the use of pseudo-holomorphic curves, to be understood as the best approximation of the given dynamics that preserves the foliation. We discuss examples, further geometric structures, and speculate on possible applications.

On the semi-analytical construction of halo orbits and halo tubes in the elliptic restricted three-body problem

The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1−L2. In this paper, by implementing a non-linear Floquet–Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-body problem and generalize the halo orbits of the circular problem. Since the libration amplitude of such halo orbits is large (comparable to the distance of L1−L2 from the secondary body), and the Birkhoff normal forms are obtained through series expansions at the Lagrangian points, we provide also an error analysis of the method with respect to the orbits of the genuine elliptic restricted three-body problem.

Global hypersurfaces of section in the spatial restricted three-body problem

We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the two-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-1 map of a positive reparameterization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.

A gravity assist mapping for the circular restricted three-body problem using Gaussian processes

Inspired by the Keplerian Map and the Flyby Map, a Gravity Assist Mapping using Gaussian Process Regression for the fully spatial Circular Restricted Three-Body Problem is developed. A mapping function for quantifying the flyby effects over one orbital period is defined. The Gaussian Process Regression model is established by proper mean and covariance functions. The model learns the dynamics of flyby’s from training samples, which are generated by numerical propagation. To improve the efficiency of this method, a new criterion is proposed to determine the optimal size of the training dataset. We discuss its robustness to show the quality of practical usage. The influence of different input elements on the flyby effects is studied. The accuracy and efficiency of the proposed model have been investigated for different energy levels, ranging from representative high- to low-energy cases. It shows improvements over the Kick Map, an independent semi-analytical method available in literature. The accuracy and efficiency of predicting the variation of the semi-major axis are improved by factors of 3.3, and 1.27×104, respectively.

Motion of a Satellite in the Circular Three-Body Problem with Light Pressure

We consider the spatial circular restricted three-body problem with light pressure. We obtain an analytic expression for the twice-averaged force function in the nonresonant case. We show that the integrals of the averaged equations written in osculating elements are the semi-major axis of the satellite orbit, the average value of the force function, and the Lidov–Kozai integral. We investigate stationary modes of oscillations and describe the set of equilibrium states of the averaged problem for different values of the coefficient of light pressure.

The contact geometry of the spatial circular restricted 3-body problem

We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range.

Lissajous and halo orbits in the restricted three-body problem by normalization method

We perform an analytical study of the Lissajous and halo orbits around collinear points L1 and L2 in a spatial circular restricted three-body problem of an arbitrary value of the mass ratio. Using a canonical transformation procedure, we generate complete and resonant normal forms through reduction to center manifolds. The coefficients in the normal forms are explicitly expressed as functions of mass ratio for the first time so that one can evaluate the energy level at which bifurcation of halo orbit takes place. Another contribution of this paper is giving the analytical solutions of Lissajous and halo orbits in the initial synodic reference system through the inverse transformation of normalization. The analytical results are the series form of normalized action-angle variables, and their coefficients are also explicitly expressed as functions of mass ratio. Finally, comparison results demonstrate that the solutions for a Lissajous orbit derived through normalization method and Lindstedt–Poincare method are completely the same, while the solutions for a halo orbit derived through these two methods are different but have the roughly equal accuracy.

An analytically treatable model of long-term dynamics in a mean motion resonance with coexisting resonant modes

We examine a 2-DOF Hamiltonian system modeling some previously unexplored dynamical effects in first-order mean motion resonances in the spatial circular restricted three-body problem “star-planet-asteroid.” In distinction from the well-known integrable model of Sessin and Ferraz-Mello (Celest Mech Dyn Astron 32:307–332, 1984) and Wisdom (Celest Mech Dyn Astron 38:175–180, 1986), in our analysis, we kept more terms in the expansion for the disturbing function. This allowed us to study the phenomena caused by coexisting resonant modes, that were lost in the integrable model. In addition, we identified the areas of dynamical chaos in the phase space of the considered system. Despite the non-integrability of our model, we were able to compute analytically many quantities characterizing its dynamical behavior. Finally, we illustrate some of our results through the examples of the known resonant Kuiper belt objects.

A lunar flyby for a tridimensional Earth-to-Earth mission

The present work formulates an orbital transfer for an Earth-to-Earth mission between non coplanar orbits with different altitudes with a special feature: the occurrence of a lunar flyby during the transfer orbit. This lunar flyby is intended to help change the plane of motion of the spacecraft without fuel consumption. Only two-impulsive trajectories are considered with the velocity increments applied at the initial and final orbits. In order to solve this problem, a 3D patched-conic approximation associated with a two-point boundary value problem is proposed. The same transfer problem is formulated considering the spatial circular restricted three-body problem (SCR3BP). The results of the patched-conic approximation is compared with the results of the SCR3BP showing a good agreement between the models. This work also determines several trajectories in order to perform a study of the fuel consumption considering several inclinations and altitudes of both initial and final orbits around the Earth. The longitude of the ascending node of the initial orbit, and, the altitude of close approach with the Moon during the flyby are also analyzed. According to the total velocity increment analysis, the changing plane assisted by a lunar flyby can be very favorable. Despite the increase of the time of flight, the saving of fuel is considerable. Indeed, the total velocity increment of this kind of maneuver is in some cases better than the velocity increment provided by the bi-parabolic transfer.

On the boundedness of motion in the spatial circular restricted three-body problem

In this paper, we consider the existence conditions of bounded motions of an infinitesimally small particle in the spatial circular restricted three-body problem. In particular, by using the Jacobi integral that corresponds to the inertial coordinate system, we prove a theorem on boundedness of motion that enables us to supplement the Hill approach in the study of motion of an infinitesimally small particle.

Geometric chaos indicators and computations of the spherical hypertube manifolds of the spatial circular restricted three-body problem

The circular restricted three-body problem has five relative equilibria L1,L2, ...,L5. The invariant stable–unstable manifolds of the center manifolds originating at the partially hyperbolic equilibria L1,L2 have been identified as the separatrices for the motions which transit between the regions of the phase-space which are internal or external with respect to the two massive bodies. While the stable and unstable manifolds of the planar problem have been extensively studied both theoretically and numerically, the spatial case has not been as deeply investigated. This paper is devoted to the global computation of these manifolds in the spatial case with a suitable finite time chaos indicator. The definition of the chaos indicator is not trivial, since the mandatory use of the regularizing Kustaanheimo–Stiefel variables may introduce discontinuities in the finite time chaos indicators. From the study of such discontinuities, we define geometric chaos indicators which are globally defined and smooth, and whose ridges sharply approximate the stable and unstable manifolds of the center manifolds of L1,L2. We illustrate the method for the Sun–Jupiter mass ratio, and represent the topology of the asymptotic manifolds using sections and three-dimensional representations.

圆型限制性三体问题中转移轨道类型的数值研究

基于空间圆型限制性三体问题的动力学方程，我们首先数值模拟了系统以质量比为参数的分岔图，发现两个主天体的质量相当(即质量比率在0.5附近)时，第三体的动力学行为较其它质量比下更为复杂。当质量比在区间(0.4, 0.6)时，我们进一步数值模拟了32,000条转移轨道，并将发现的轨道作了简要的分类。 Based on the dynamical equations of spatial circular restricted three-body problem, the bifurcation diagram of the system with the mass ratio as the bifurcation parameter is demonstrated, and it is found that when the masses of the two main bodies are considerable equivalent (i.e., the mass ratio is around 0.5), the third body’s dynamic behavior is more complicated than at other mass ratios. When the mass ratio is in the interval (0.4, 0.6), we further simulate 32,000 transfer orbits and make a brief classification of the found orbits.

Arnold’s mechanism of diffusion in the spatial circular restricted three-body problem: A semi-analytical argument

We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L1, located between Sun and Earth. Near L1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3-sphere. For an orbit confined to this 3-sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly.We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3-sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such ‘diffusing’ orbits, and numerical evidence that the premises of the theorem are satisfied in the three-body problem considered here. We provide an explicit construction of diffusing orbits.The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the ‘outer dynamics’ along homoclinic orbits, and use very little information on the ‘inner dynamics’ restricted to the 3-sphere.As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L1 from a nearly-planar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.

A Framework for Constructing Transfers Linking Periodic Libration Point Orbits in the Spatial Circular Restricted Three-Body Problem

The Earth–Moon libration points are of interest for future missions and have been proposed for both storage of propellant and supplies for lunar missions and as locations to establish space-based facilities for human missions. Thus, further development of an available transport network in the vicinity of the Moon is valuable. In this investigation, a methodology to search for transfers between periodic lunar libration point orbits is developed, and a catalog of these transfers is established, assuming the dynamics associated with the Earth–Moon circular restricted three-body problem. Maneuver-free transfers, i.e. heteroclinic and homoclinic connections, are considered, as well as transfers that require relatively small levels of [Formula: see text]. Considering the evolution of Earth–Moon transfers as the mass parameter is reduced, a relationship emerges between the available transfers in the Earth–Moon system and maneuver-free transfers that exist within the Hill three-body problem. The correlation between transfers in these systems is examined and offers insight into the existence of solutions within the catalog. To demonstrate the persistence of the catalog transfers in a higher-fidelity model, several solutions are transitioned to a Sun–Earth–Moon ephemeris model with the inclusion of solar radiation pressure and lunar gravity harmonics. The defining characteristics are preserved in the high-fidelity model, validating both the techniques employed for this investigation and the solutions computed within the catalog.

Semi-Analytical Approach for Distant Encounters in the Spatial Circular Restricted Three-Body Problem

This paper presents a three-dimensional semi-analytical formulation to model the third-body effect on a massless particle in the circular restricted three-body problem dynamical regime. The final expressions are obtained by means of the classical Lagrange planetary equations, by considering as small parameter of the perturbative approach the mass parameter of the system. The variations of the Keplerian orbital elements of the massless particle over one orbital period are computed as a function of the relative phasing between the massless particle and the secondary, and the overall behavior is described in terms of kick maps. Several applications to mission design are introduced, and the accuracy provided by the methodology is discussed on the basis of different initial conditions and physical systems.

Transition map and shadowing lemma for normally hyperbolic invariant manifolds

For a given a normally hyperbolic invariant manifold, whose stable and unstablemanifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries:the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.

Restricted Three-Body Problem in Different Coordinate Systems

In this paper of the series, the equations of motion for the spatial circular restricted three-body problem in sidereal spherical coordinates system were established. Initial value procedure that can be used to compute both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedure was illustrated by numerical example and graphical representations of the variations of the two sidereal coordinate systems.