Let us start by assuming that the Earth and Moon are point masses that revolve in circular orbits around their common centre of mass, and consider a particle with infinitesimal mass under the gravitational attraction of the Earth and Moon (in this context, the Earth and Moon are sometimes called primaries). The study of the dynamics of this third particle is the so-called RTBP. It is usual to take a system of reference with the origin at the centre of mass, and with rotating axes such that the primaries are fixed on the x axis, the y axis is contained in the plane of motion of the primaries and orthogonal to the x axis, and the z axis is orthogonal to the plane. This system of reference is called synodical. The unit of distance is the distance between primaries, the unit of mass is the total mass of the primaries, and the unit of time is such that the gravitational constant is one. In these units, the period of the primaries around their common centre of mass is equal to . In this system of reference, the Earth and Moon have coordinates and , respectively. It turns out that the equations of motion of the infinitesimal particle can be written in Hamiltonian form, with Hamiltonian function given by
where is the distance of the particle to the Earth, and the distance of the particle to the Moon. It is well known (Szebehely 1967; Meyer & Hall 1992) that, in synodical coordinates, the RTBP has five equilibrium points, three of them lie on the x axis (they are also called Euler points, collinear points or simply , and ) and two of them are the third vertex of an equilateral triangle that has the primaries as the other vertices (they are also called Lagrangian points, triangular points or and ).
We start focusing on the dynamics around the equilateral point of the RTBP (the same results will hold for due to the symmetries of this model). For the value of the mass corresponding to the Earth-Moon case, is an elliptic equilibrium point. Under very general conditions, standard results from KAM theory (Arnold et al. 1988) predict that, around an elliptic equilibrium point of a Hamiltonian system, there exist many quasi-periodic solutions. In suitable coordinates (the so-called action-angle variables), these solutions correspond to linear motions taking place on invariant tori, whose dimension coincides with the number of basic frequencies of the quasi-periodic motion. For this reason, in this context it is usual to refer to quasi-periodic solutions as invariant tori. Due to the constraints imposed by the Hamiltonian structure, the dimension of these tori (i.e., the number of basic frequencies of the quasi-periodic motions) cannot be greater than half the dimension of the phase space. In this sense, we will distinguish between the maximal dimensional tori (tori whose dimension equals half the dimension of the phase space) and lower dimensional tori. It is well known that the set filled by the maximal dimensional tori has positive measure and empty interior. Obviously, initial conditions inside this set give rise to trajectories that are always close to the equilibrium point. So, to study the instability of an elliptic point we should look at the dynamics on the complementary of the tori set. The (possible) diffusion inside this set is usually called Arnold diffusion (Arnold 1964). The main results in this direction are only upper bounds on the diffusion speed (Giorgilli et al. 1989; Simo 1989; Celletti & Giorgilli 1991; Jorba & Villanueva 1997a; Giorgilli & Skokos 1997; Benettin et al. 1998; Niederman 1998). One of the main conclusions of these works is that, under general conditions, the time needed to escape from a neighbourhood of the equilibrium point is (at least) exponentially large with respect to the distance from the initial condition to the equilibrium point. Hence, the application of these results to the RTBP corresponding to the Earth-Moon case results in very large stability times. This kind of stability is usually called effective stability.
On the other hand, we can consider the motion of a small particle in the real Solar system, in a neighbourhood of the triangular points of the Earth and Moon. Here, we will use as "real Solar system" the model defined by the JPL ephemeris (http://ssd.jpl.nasa.gov/horizons.html ). This is a numerically defined vector field, that can be numerically integrated for the time span for which the positions of the main bodies are known. Note that, for the real system, the triangular points are no longer equilibrium points since the hypotheses used to derive the RTBP are not satisfied. It is known that arbitrary trajectories starting near the (geometrically defined) triangular points move away after a short time (Schutz & Tapley 1970;iDez et al. 1991) and hence, the RTBP is not a good model to study this problem because it displays a qualitative behaviour that is quite different.
For this reason, we will start using the so called Bicircular Problem (from now on, BCP). This model can be seen as a time dependent periodic perturbation of the RTBP, that includes the main effect coming from the Sun. We believe that this model was first introduced by Cronin et al. (1964), and it has some of the main features of the real model (for instance, the neighbourhood of the equilateral points is unstable). Our purpose is to use the BCP as a first model to describe some properties of the real system. In particular, it is known that the BCP has several families of quasi-periodic solutions near the equilateral points (Simo et al. 1995; Jorba 1998; Castella & Jorba 2000). It turns out that some of these tori are hyperbolic, while some others are elliptic (Jorba 1998, 2000), and that these elliptic tori are found at some distance from the equilateral points. It is known that lower-dimensional normally elliptic tori give rise, under general conditions, to regions of effective stability around them (Jorba & Villanueva 1997a,b). Here, we will estimate the size and shape of these regions by means of numerical simulations, to show that they are relevant for this problem.
The last step will be to show, by means of numerical simulations, that some of the stability regions found in the BCP model subsist in the real system, at least for time spans of 1 000 years. As will be discussed later (in Sect. 5), these regions are a suitable place to look for Trojan asteroids in the Earth-Moon system.
This kind of stability was previously observed (Gomez et al. 1993; Simo et al. 1995), but with smaller regions and much shorter time intervals (around 60 years). The main improvements in this work come from the computation of the vertical families of quasi-periodic solutions for the BCP and the use of initial conditions around them. It seems that these families of tori are the skeleton that supports these stability regions. A preliminary version of these results was presented in a conference paper (Jorba 1998).
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000