ON LYAPUNOV FAMILIES AROUND COLLINEAR LIBRATION POINTS

Evolution details of the planar and vertical Lyapunov families around the three collinear libration points in the restricted three-body problem were studied. Researches before were generally restricted to be within the colliding orbits with the primaries and for fixed mass parameters μ. In this paper, members after colliding orbits were computed. With increasing μ, how these families evolve was studied. The circular restricted three-body problem (CRTBP) is a simplified model describing the motion of a massless small body around two massive bodies (called primaries) which revolve each other in a circular orbit. Usually, we study the motion of the small body in a coordinate rotating system with the two primaries, which is called the synodic coordinate system, as shown in Figure 1. P1 and P2 are the two primaries, and P is the small body. The mass parameter μ is defined as m2/(m1 + m2), where m1 and m2 are masses of the two primaries. Five equilibrium points called libration points exist in the system, all in the motion plane (x–y plane in Figure 1) of the two primaries. Two of them form equilateral triangles with the two primaries. They are called triangular libration points, denoted as L4 and L5 in Figure 1. The other three lie in the line connecting the two primaries. They are called collinear libration points, denoted as L1, L2, and L3 in Figure 1. Periodic families around the five libration points are important for dynamical studies of this system. Out of them, the periodic families guaranteed by Lyapunov’s theorem are fundamental. When μ is smaller than Routh’s critical value μRouth = 0.03852 ... , the linearized model shows that the triangular libration points are of the center × center × center type. According to Lyapunov’s theorem, there are three Lyapunov families when μ μ Routh, the two planar families combine to form one single family. Many works have been done on global evolution of these families (for example, Rabe & Schanzle 1962; Deprit & Henrard 1968; Henrard 1970; Murray & Dermott 1999 ;G ´ et al. 2001b, 2001d ;H ou & Liu2008a, 2008b). For the collinear libration points, the linearized model shows that they are of the saddle × center × center type for μ ∈ (0, 1). So there are just two Lyapunov families around them. One is in the x–y plane, with the name of planar Lyapunov family. The other one moves in space, with limiting member as infinite harmonic motion along the z-axis, and is called the vertical Lyapunov family. About the dynamics and periodic families around the collinear libration points, many works have also been done (for example, Broucke 1968 ;H ´