2. The bicircular model
The bicircular model is a Restricted Four-Body Problem that can be seen as a modification of the RTBP in order to take into account (in an approximate way) the effect of the Sun. We refer to Cronin et al. (1964) for a detailed derivation of the equations of motion.
To define this model, we impose that: a) the Earth and Moon revolve in circular orbits around their centre of mass (as in the RTBP); and b) the Earth-Moon barycentre and the Sun also move in circular orbits around the Earth-Moon-Sun centre of mass, that is fixed at the origin. Of course, these two circular motions satisfy Kepler's laws. Then, it is not difficult to derive the equations of motion of a fourth (infinitesimal) particle moving under the attraction of these three bodies. The study of the motion of this fourth particle is the so-called BCP. Note that, in this model, the motion of the Earth, Moon and Sun is not coherent, since it does not follow a true solution of the Three-Body Problem.
To write the equations of motion of the BCP, we use the same reference frame as in the RTBP: the origin is taken at the centre of mass of Earth and Moon, the x axis is on the Earth-Moon line, the y axis is contained in the plane of motion of Earth and Moon, and the z axis is orthogonal to the plane (see Fig. 1). In this system of reference, the Sun is turning (clockwise) around the origin in a periodic way. As in the RTBP, we use normalized units such that the Earth-Moon distance is 1, the total mass of the Earth-Moon system is 1, and the period of Earth and Moon around their barycentre is . In these units, we denote by the distance from the Earth-Moon barycenter to the Sun, is the mass of the Sun, and the frequency of the motion of the Sun around the origin is denoted by . In these coordinates, is the sidereal frequency of the Sun minus 1.
Let be the coordinates of the infinitesimal particle. Then, defining the corresponding momenta as , and , the motion of the fourth particle can be described by a Hamiltonian system that depends on time in a periodic way:
where , , , , , and . We stress that we can look at the BCP as a periodic time dependent perturbation of the RTBP:
and it is clear that , and that . In Fig. 1 we have marked the Eulerian and Lagrangian points by using their corresponding coordinates in the RTBP. As the vector field of the perturbation does not vanish at these points, they are no longer equilibrium solutions.
Now, we focus on the point (due to the symmetries of the BCP, the same results will be valid for ; of course, this is not true for the real system). A straightforward application of the Implicit Function Theorem shows that, under a generic non-resonance condition (satisfied in this case) and assuming small enough, is replaced by a periodic orbit with the same period as the perturbation. This orbit tends to when tends to zero.
To reduce the number of degrees of freedom of the problem, we introduce the Poincaré section , where is the period of the perturbation, and let be the corresponding Poincaré map. Note that is an autonomous6-D (symplectic) map whose fixed points correspond to periodic orbits (of period ) for the flow, and vice versa.
Now, by means of a continuation process, we compute the fixed points of (for ranging between 0 and 1) that correspond to the periodic orbit that replaces . The results are displayed in Fig. 2 (left), where the horizontal axis shows the x coordinate of the fixed point and the vertical axis refers to the value of . Note that, for , there are three fixed points of that are close to . The projection of the corresponding periodic orbits for the flow are displayed in Fig. 2 (right). By computing the eigenvalues of the differential of at the fixed points, one can see that orbit PO1 is unstable, and that orbits PO2 and PO3 are linearly stable. Moreover, these three periodic orbits have an elliptic mode contained in the plane. In what follows, we will refer to this mode as the vertical mode of the corresponding periodic orbit. More details about the BCP can be found in the literature (Gomez et al. 1993; Simo et al. 1995).
2.1. The vertical families of 2-D tori of the BCP
Let us now consider the vertical mode of one of the periodic orbits. It is known that, under generic conditions (Jorba & Villanueva 1997a,b), there exists a Cantor family of 2-D invariant tori that extend this linear mode into the complete (i.e., nonlinear) system. This is very similar to the well-known Lyapunov centre theorem, but instead of obtaining a (smooth) family of periodic orbits, one obtains a (Cantor) family of quasi-periodic solutions, with two (linearly independent) basic frequencies. In suitable coordinates, each quasi-periodic solution fills densely a two dimensional torus. Hence, for each periodic orbit PO1, PO2 and PO3, there is a Cantor family of 2-D tori (let us call them VF1, VF2 and VF3, respectively) that "grows up" in the vertical direction.
In order to display these families, we use again the map defined before. In this way the 2-D tori become invariant curves of a 6-D symplectic map. As all these curves cross transversally the hyperplane , we select, for each curve, the only point on the invariant curve that has and . Thus, each curve is represented by a single point. We have used this representation in Fig. 3. The horizontal axis displays the coordinate while the vertical axis contains the rotation number of the invariant curve. Note that families VF1 and VF2 are connected (as suggested by the connection of the periodic orbits PO1 and PO2 in Fig. 2), while F3 reaches high amplitudes in the direction. In fact, as the projection of these quasi-periodic motions in the plane is close to an harmonic oscillator (see also Sect. 3.2), the value of when is a good approximation to the maximum value reached by the z coordinate. The details about these computations can be found in Jorba (1998) and Castellà & Jorba (2000).
It is also known that, under generic hypotheses of non-resonance, the tori on these families that are close to the basic periodic orbits have the same stability as these periodic orbits. So, for moderate vertical amplitudes, we expect the tori on VF1 to be hyperbolic, and most of the tori on VF2 and VF3 to be elliptic. Families VF1 and VF2 are connected through a turning point, and this is where the change of stability takes place. Family VF3 can be continued up to very high values of the vertical amplitude and, except for small intervals of instability (produced by some resonances involving internal and normal frequencies), the tori in this family are normally elliptic. For more details on the normal stability of these families see Jorba (1998; 2000).
Recent results show that normally elliptic lower dimensional tori are surrounded by a region of effective stability (Jorba & Villanueva 1997a). This means that the time needed to escape from a neighbourhood of one of those tori increases exponentially with the initial distance to the torus. A possibility for estimating the size of this region and the speed of diffusion is to compute a normal form around the torus and to derive estimates on the remainder, as has already been done for periodic orbits (Jorba & Simo 1994; Jorba & Villanueva 1998). Of course, obtaining "analytical" bounds usually requires the use of rather pessimistic inequalities that lead us to underestimate the region where the diffusion is small enough. Here, to have a realistic estimate of the size and shape of this region, we will simply use a numerical simulation to detect initial conditions for which the corresponding trajectory remains there for a sufficiently large time interval. In what follows, we will use the words "quasi-stable region" to refer to a region such that the time taken to escape from it is bigger than a large prescribed value (this value will be specified later on).
2.2. Numerical simulations
As it has been mentioned in Sect. 2.1, the families of 2-D tori VF1, VF2 and VF3 correspond to families of invariant curves for the map . Moreover, on each family, the invariant curves can be "labeled" by the value of the coordinate when and . Let be one of these curves, and let be the point on the curve such that and . The next step is to select a set of initial conditions around the point . To reduce the amount of computations, we will use a two dimensional grid: the coordinates z, , and are fixed by the corresponding values of , and the values of x and y are used to define the mesh. Due to the shape of this regions (this will be clear later on), we will use the following polar-like grid,
where and are used to select the density of the mesh and the (integer) indices i and j move on suitable ranges, such that the mesh is (approximately) centered around the coordinates of . Then, we will use each point on the grid as an initial condition for a numerical integration of the vector field of the BCP. To decide whether the trajectory escapes, after each time step of the numerical integrator (see Sect. 4), the program uses a couple of tests. First, it checks the actual distance to the Earth and Moon. If the distance to the primaries were lower than their respective radius, then the initial condition would not be considered as belonging to the quasi-stable region. The second test is to check whether the actual point has crossed the plane (we recall that we are working around the point). Numerical experiments show that this last condition is equivalent to escape. This has already been used before (sometimes with different values for the y section) in several works (McKenzie & Szebehely 1981; Szebehely & Premkumar 1982;oGmez et al. 1993; Simo et al. 1995). In this work, we have used the values and . The different sizes between these two values is to produce a nearly squared grid. For the numerical integrations, we have used a time interval of 15 000 Moon revolutions (we recall that, in the BCP model, this is equivalent to 15 000 units of time).
The results are summarized in Fig. 4 for the VF2 and VF3 families. The horizontal axis displays the same value of used in the previous section (see also the horizontal axis of Fig. 3). The vertical axis displays the number of non escaping points after 15 000 Moon revolutions. As we have used the same mesh throughout, the number of non escaping points is a good estimate of the size of the stable region. For the Family VF2 the region is not very large, and its size decreases when the value of increases; this is a natural result since Family VF2 becomes hyperbolic when it meets VF1 (Jorba 1998, 2000). Family VF3 has a bigger stability when ranges from 0.26 and 0.92 (when is below 0.20, the size of the region associated to VF3 is in fact smaller than the VF2 one).
Fig. 5 contains a few slices of the stability region around VF2. In each plot we have marked with black dots the initial conditions that do not escape after an integration time of 15 000 Moon revolutions. The large cross in the graphic denotes the intersection of the invariant curve with this slice. The region explored has been drawn with a semi-circular box.
Several slices for Family VF3 are displayed in Fig. 6. These slices correspond to the biggest parts of the region (see Fig. 4). We recall that that the unit of distance in these plots is the Earth-Moon distance, so the size of the displayed regions is quite large in the physical system. As has been mentioned before, it is worth noting that the largest part of the region corresponds to high values of the coordinate.
As has been explained before, the vertical families of quasi-periodic solutions are seen as invariant curves of the map . To reduce the number of numerical simulations, we have restricted the search of the stable region by selecting a single point on the invariant curve (the one having and ), and taking a 2-D mesh around this point. This mesh is obtained by moving the values of this point and keeping the remaining coordinates constant. Note that this produces a concrete slice of the stable region. The computation of the full region would require a 5-D mesh, obtained by varying 5 coordinates around the initial point ( should be kept fixed to avoid changing the base invariant curve). This process can be applied to every invariant curve on the family (i.e., moving ), and this will produce an estimate of the full (6-D) region for the map . The total region for the BCP still requires us to take all the points of the stable region for as initial conditions, and to integrate them for a period of the time variable t. This would produce a 7-D region inside the 7-D phase space of the (flow of the) BCP. In other words, the 2-D slice that we have computed in the previous section is the result of taking the full 7-D region for the BCP when (a 6-D section), fix (5-D section), and fix and the momenta and to suitable values to define the final 2-D slice. The computation of the 7-D quasi-stable region would require a prohibitive amount of computer time and memory. This is the main reason for restricting the simulations to a two-dimensional slice.
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000