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Through the art of applying assumptions which restrict the orbital shape and relative inclination of celestial bodies such as the Earth and Moon, trajectory prediction of small bodies of comparatively negligible mass (i.e., a satellite) within a multi-body gravitational system is possible. In this research, Lagrangian analytical methods are applied to formulate succinct derivations of the circular restricted three-body problem (CR3BP), the elliptical restricted three-body problem (ER3BP), and the bicircular restricted four-body problem (BCR4BP). The presence of, or lack thereof, equilibrium points within each dynamical model is discussed and presented in both graphical and tabular form. In terms of application, a form of periodic trajectories within the Earth-Moon system, identified herein as cislunar periodic orbits, is propagated using each of the presented dynamical models. The dynamical variations in these cislunar periodic orbits when transitioning between dynamical models are analyzed and discussed. The methodology behind cislunar periodic orbit generation is also discussed with 33 cislunar periodic orbits presented. Finally, through means of differential correction, it is shown how much error in Δ V ( Δ e V ), the BCR4BP dynamics introduce on the CR3BP solutions for a given number of patchpoints. Results of this analysis show the ER3BP to have a significantly higher perturbative effects than the BCR4BP on cislunar periodic orbits which are closed in the CR3BP. Based on the 33 orbits analyzed, correlation was also observed between the Jacobi constant and the dynamical variations present during the transition of cislunar periodic orbits to higher fidelity models, with larger Jacobi constants being associated with more dynamical variations as an orbit transitions from the CR3BP to both the ER3BP and BCR4BP.

In the framework of the spatial circular Hill three-body problem, we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbits families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite g , f , the libration (Lyapunov) a ,  c , and collision$${\\mathscr {B}}_0$$B 0 families. Since the Conley–Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.

We consider the problem of n points with positive masses interacting pairwise with forces inversely proportional to the distance between them. In particular, it is the classical gravitational, Coulomb or photo-gravitational n -body problem. Under this general form of interaction, we investigate the integrability problem of three bodies. We show that the system is not integrable except in one case when two among three interaction constants vanish. In our investigation, we use the Morales–Ramis theorem concerning the integrability of a natural Hamiltonian system with a homogeneous potential and its generalization.

The geometry of the weak stability boundary region for the planar restricted three-body problem about the secondary mass point has been an open problem. Previous studies have conjectured that it may have a fractal structure. In this paper, this region is studied for infinitely many cycles about the secondary mass point, instead of a finite number studied previously. It is shown that in this case the boundary consists of a family of infinitely many Cantor sets and is thus fractal in nature. It is also shown that on two-dimensional surfaces of section, it is the boundary of a region only having bounded cycling motion for infinitely many cycles, while the complement of this region generally has unbounded motion. It is shown that this shares many properties of a Mandelbrot set. Its relationship to the non-existence of KAM tori is described, among many other properties. Applications are discussed.

Comet 39P/Oterma is known to make fast transitions between heliocentric orbits outside the orbit of Jupiter and heliocentric orbits inside that of Jupiter. In this paper, the dynamics of comet Oterma is modelled and fitted in the Planar Elliptic RTBP. Using the computations presented in Duarte(Celest. Mech. Dyn. Astron 136:26, 2024), we look for the invariant objects around L1 and L2, which in the case of the Planar Elliptic RTBP is invariant tori and their stable and unstable manifolds that are the skeleton that guides Oterma in its rapid transition.

Generalized restricted three body problems consist of adding some extra hypotheses to the Restricted three body problem (RTBP) in order to have a new problem, not very different of the original RTBP. However, not any additional hypothesis is allowed; it must satisfy the laws of Physics. Among the several generalizations found in literature, we prove that at least there are two hypotheses that cannot be used, namely: 1) Perturbation in Coriolis and/or centrifugal forces, and 2) primaries are spheroids moving on elliptical orbits.

Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to efficiently determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet. In particular, connections between the periodic orbits of such two different moons are achieved. The strategy is applicable for any type of direct transfers that satisfy the analytical constraints. Case studies are presented for the Jovian and Uranian systems. The transition of the transfers into higher-fidelity ephemeris models confirms the validity of the MMAT method as a fast tool to provide possible transfer options between two consecutive moons.

This study considers the framework of a planar Restricted Three-Body Problem (RTBP) in which both the primaries are oblate spheroids with their equatorial planes coincident with the plane of motion while the more massive primary is a source of radiation. The nonlinear stability of the triangular equilibrium points is investigated with the help of the mean motion derived by Sharma et al. [15] which is more accurate than the one used in earlier works. The Lagrangian of a third infinitesimally small body when it is near either of the triangular equilibrium points L 4 /L 5 is formulated with respect to a synodic coordinate frame located at the respective equilibrium point. From the second order part of the Lagrangian, extracted using Taylor series expansion, second order part of the Hamiltonian of the system is derived and the equations of motion of the third body near the equilibrium points are obtained. Analyzing the characteristic equation of the system, which is similar for both the equilibrium points L 4 /L 5 , critical mass value that marks the linear stability of the triangular equilibrium points is obtained. Furthermore, Moser’s conditions are employed to find the two exceptional values of the mass parameter in the stable range at which stability cannot be guaranteed. By comparing with the results available in the literature, all three critical mass values are found to be lower than the values obtained for an ideal RTBP framework without oblateness or photo-gravitational effect. In addition to that, critical mass values are also affected due to change in the mean motion expression.

A novel, compact formalism of rigid body motion dynamics is presented in a general reference frame based on the geometric mechanics framework. This formalism, proposed on the special Euclidean space (SE(3)) of the Lie group, naturally accounts for the orbit/attitude coupling due to the gravitational moments and forces. It is demonstrated that the structure of the rigid body dynamics equations is preserved in different coordinate frames. Then, using the expression of gravitational potential, this global framework is applied to the circular restricted full three-body problem (CRF3BP) of a near-rectilinear halo orbit (NRHO) similar to that of Gateway, where the equations are uniquely provided in the body frame of the spacecraft and the barycentric rotating frame. For the sake of propagation and consistency with the classical circular restricted three-body problem (CR3BP), the equations of the CRF3BP are also presented in a non-dimensional form. Trajectories computed with different inertia tensors are compared with those obtained using the traditional equations of motion of a point-mass spacecraft subject to the gravitational fields of two larger primaries. The comparison between CRF3BP and CR3BP suggests: (a) the need for computation of customized families of halo orbits considering inertia properties of each spacecraft; and (b) the use of attitude-only control for station-keeping in future NRHO missions to reduce station-keeping costs which would otherwise be relatively large if the point-mass approximation were used.

We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill’s lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden’s method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of “Copenhagen problem” and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill’s lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined.