Transits close to the Lagrangian solutions L ₁, L ₂ in the elliptic restricted three-body problem

Abstract

In the last decades a peculiar family of solutions of the circular restricted three body problem has been used to explain the temporary captures of small bodies and spacecrafts by a planet of the Solar System. These solutions, which transit close to the Lagrangian points L 1, L 2 of the CRTBP, have been classified using the values of approximate local integrals and of the Jacobi constant. The use for small bodies of the Solar System requires to consider a hierarchical extension of the model, from the CRTBP to the full N planetary problem. The elliptic restricted three body, which is the first natural extension of the CRTBP, represents already a challenge, since global first integrals such as the Jacobi constant are not known for this problem. In this paper we extend the classification of the transits occurring close to the Lagrangian points L 1, L 2 of the ERTBP using a combination of the Floquet theory and Birkhoff normalizations. Provided that certain non-resonance conditions are satisfied, we conjugate the Hamiltonian of the problem to an integrable normal form Hamiltonian with remainder, which is used to define approximate local first integrals and to classify the transits of orbits through a neighbourhood of the Lagrange equilibria according to the values of these integrals. We provide numerical demonstrations for the Earth–Moon ERTBP.

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

“…We believe that the results herein contribute significantly to the state-of-the-art in the literature. As implied in the introduction to this paper, exploring the dynamical properties of perturbations of the CR3BP has lately become a popular area of investigation in astrodynamics (refer to Jorba & Nicolás, 2020;Paez & Guzzo, 2021;Kumar et al, 2021) for just a few recent examples). This study, by outlining a simple and straightforward method for delineating transit and nontransit behavior within perturbed models, elegantly fills an important niche in this emerging topic.…”

“…Prior investigations into models more complicated than the CR3BP have successfully found periodic and quasiperiodic orbits in the vicinity of former Lagrange points by employing single shooting or multiple shooting algorithms (Gómez et al, 2003;. Studies have found quasi-periodic orbits on the center manifolds of these dynamical replacements and have numerically demonstrated associated transit phenomena (Jorba & Nicolás, 2020;Paez & Guzzo, 2021).…”

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

“…The ERTBP is conveniently represented as a non-autonomous Hamiltonian system having the Lagrangian solutions L 1 , L 2 but without a global first integral, such as the Jacobi constant, which is used to define the Poincaré sections and label the halo orbits in the CRTBP. In the paper [28] we have introduced Floquet-Birkhoff normal forms for the ERTBP which allowed us to generalize and compute the families of planar and vertical Lyapunov orbits generating at L 1 , L 2 , as well as the low-energy transits from one side to the other of the secondary body. In this paper we compute halo orbits for the ERTBP from resonant Birkhoff-Floquet normal forms.…”

“…In the CR3BP the result is achieved by computing a resonant Birkhoff normal form of large order N : by neglecting the large order remainder, one remains with and integrable Hamiltonian system which is used to compute the Poincaré section of the Hamiltonian flow on the center manifold, and consequently the halo orbits. We extend these methods to the ERTBP by providing a definition of halo orbits in the elliptic problem, and a method of computation based on the resonant version of the Floquet-Birkhoff normal forms which were introduced in [28]. The resonant Floquet-Birkhoff normal forms will be used to define also the manifolds tubes and the transit motions associated to halo orbits in the ERTBP.…”

“…Remark. In our paper [28] we constructed non-resonant Floquet-Birkhoff normal forms, so that all the terms K j (Q, P; e) were integrable in the sense that, when represented using the variables q, p, they depended on the variables only through the combinations iq 1 p1 , iq 2 p2 (the actions of the elliptic motions expressed in Birkhoff complex variables ) and q3 p3 (the action of the hyperbolic motion). This type of normal form, which is specifically designed for the efficient analytic computation of the planar and vertical Lyapunov orbits, as well as their manifold tubes, can be constructed if the three frequencies describing the motion are strictly non resonant up to order N .…”