1 Introduction

Considering the framework of the restricted three-body problem, the existence of regions of distant stable retrograde orbits is well known (Broucke 1968; Hénon 1970; Jefferys 1971; Winter 2000; Winter & Vieira Neto 2001). Such regions have been used in order to try to explain the origin of irregular planetary satellites (Huang & Innanen 1983; Brunini 1996; Vieira Neto & Winter 2001). In the case of the Earth-Moon-particle system (Winter 2000) these stable regions can conveniently be used in spacecraft missions to the Moon. However, in many cases those regions cannot be taken into account for missions with specifications that demand a direct trajectory. In those cases it is important to know the regions of distant stable direct orbits. Therefore, in the present work we address the problem of the location and size of the regions of distant stable direct orbits around the Moon.

The zero velocity curve associated with the Lagrangian point L1 defines a physical region around the Moon (Fig. 1) known as the "Roche lobe''. This region is usually accepted as the furthest limit of stable direct orbits around the Moon. Such a limit is also defined in terms of the value of the Jacobi constant at L1, Cj(L1)=3.2010 for the Earth-Moon case. For trajectories with Cj>Cj(L1) it is not possible to occur escape from the region around the Moon or capture from the other regions. Our main goal is to find regions of stable trajectories with Cj<Cj(L1)and identify the reason for such stability.

In order to explore such regions we adopted the temporary capture time approach (Vieira Neto & Winter 2001); the technique and the results are presented in the next section. In Sect. 3 we use the Poincaré surface of section technique to study the reason for the existence of the two stable regions found. Two families of periodic orbits are identified, h1 and h2. In Sect. 4, a diagram of stability in terms of Jacobi constant and position is generated (Winter 2000). In the last section, the robustness of the stability of the larger region, h2, is tested with the inclusion of the solar perturbation.

Figure 1: Diagram indicating the region around the Moon and the Lagrangian points L1 and L2. The dashed lines correspond to the zero velocity curves that touch the points L1 or L2.