Structure of the centre manifold of the $$L_1,L_2$$ collinear libration points in the restricted three-body problem

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

“…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 . As pointed in [29], while for the Earth-Moon system (µ = 0.0123) indeed no exact low order resonance takes place, the linear frequencies lay very close to a 1-1 resonance opening the door, for the CRTBP, to the appearance of the halo orbits at suitable large values of the Hamiltonian. Therefore, while very efficient in the context of planar Lyapunov orbits and their manifold tubes, the construction presented in [28] obviously excludes the computation of Halo orbits.…”

“…We identify the halo orbits of the ERTBP as the fixed points of the Poincaré section of the Hamiltonian system defined by K CM (which appear in addition to the central one identified by (Q 1 , P 1 ) = (0, 0)) for all the larger values of the local energy κ = 0.025,0.050, 0.077. As for the CRTBP (see [24,2,29]), the halo orbits are better described by introducing normal form variables which are adapted to the 1-1 resonance. First, we introduce on the center manifold the action-angle variables θ 1 , θ 2 , I 1 , I 2 defined in (8), and then the action-angle variables φ, χ, J φ , J χ adapted to the 1-1 resonance defined by:…”

“…The halo orbits are defined in the Circular Restricted Three-Body Problem (CRTBP) from the computation of large order resonant Birkhoff normal forms at the Lagrangian point L 1 or L 2 [16,25,24,4,2,29]. Precisely, the Birkhoff normal forms are used to compute analytically all the orbits in the center manifold of the selected Lagrangian point L 1 ; therefore, from the Poincaré section of the dynamics restricted to the center manifold one defines the halo orbits.…”

“…The halo orbits of the CRTBP have been analytically computed from computer assisted implementations of Hamiltonian perturbation theory as well as from numerical methods (see for example [16,25,4,29] for the analytic methods and [8,15,12,31,30] for the numerical ones). In this paper we consider the analytic computations based on Hamiltonian perturbation theory, which allow not only to compute the halo orbits, but also the orbits in their neighbourhoods, including their stable and unstable manifolds and transit orbits.…”

Transits close to the Lagrangian solutions $L_1,L_2$ in the Elliptic Restricted Three-body Problem

“…In [9] only the linear approximation and the planar problem were considered (i.e. N = 2 and Q 3 = P 3 = 0 in (4)), but later higher non-linear normal forms were computed, as in [41,26,15,36,16,33,27,13,29,23,38,37], etc. In these papers the dynamics of transits is obtained by approximating a Birkhoff normal form (4) of large order N with the integrable Hamiltonian (we refer to [39,34] for an introduction to polynomial normal forms):…”