Families of Halo-like invariant tori around $$L_2$$ in the Earth-Moon Bicircular Problem

Abstract

The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around L 2 in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that there are at least two different families of Halo-like quasi-periodic orbits around L 2 .

Transfers from the Earth to $$L_2$$ Halo orbits in the Earth–Moon bicircular problem

“…In Rosales et al (2021), the authors study the effect of the Sun's gravity on the neighborhood of the translunar point of the BCP. In particular, the most relevant families (horizontal and vertical Lyapunov families and the Halo one) are considered.…”

“…where x is a parameterization of the invariant curve, ψ s is the eigenfunction associated to the stable eigenvalue λ s (see Rosales et al 2021 for the computation of ψ s and λ s ). The value h is selected such that |h| < h 0 /|λ s |, for a fixed value h 0 ∈ R + such that h 0 /|λ s | is small (for example, on the order of 10 −6 or 10 −7 ), so that the error of this approximation is O(h 2 0 ).…”

“…In this context, the classical Halo orbits gain, generically, the frequency of the perturbation becoming two-dimensional invariant tori. Following the nomenclature of Rosales et al (2021), we have labeled these two-dimensional tori as Halo orbits of Type I. On the other hand, the two-dimensional quasiperiodic Halo orbits of the RTBP whose frequency is resonant with the one of the Sun, remain two-dimensional tori once the perturbation is turned on.…”

“…On the other hand, the two-dimensional quasiperiodic Halo orbits of the RTBP whose frequency is resonant with the one of the Sun, remain two-dimensional tori once the perturbation is turned on. Those orbits are named, also according to Rosales et al (2021), as Halo orbits of Type II. The strategy has been to compute the stable manifold of several Halo orbits of Type I and Type II of the BCP noticing that there are some parts of those manifolds that intersect with the LEO sphere of parking orbits around the Earth.…”

“…See (Simó et al 1995;Castellà and Jorba 2000;Jorba 2000) for results regarding specifically the triangular points, for the case of L 1 and Jorba andNicolás (2020, 2021) for the case of L 3 . Moreover, the case of the translunar point has been tackled in Alessi et al (2012), Rosales et al (2021).…”