3 Identification of the stable orbits

The reason for the existence of the two apparently stable regions, h1 and h2, is now studied using the technique of Poincaré surfaces of section. In the following the method and the results are presented.

In order to determine the orbital elements of a particle at any instant it is necessary to know its position (x,y) and velocity $(\dot x,\dot y)$; these correspond to a point in a four dimensional phase space. The existence of the Jacobi constant implies the existence of a three dimensional surface in the four dimensional space. For a fixed value of the Jacobi constant only three of the four quantities are needed, say x, y and $\dot x$, since the other one, $\dot y$is determined, up to a change in sign, by the Jacobi constant. By defining a plane, say y = 0, in the resulting three dimensional space, the values of x and $\dot x$ can be plotted every time the particle has y = 0. The ambiguity in the sign of $\dot y$ is removed by considering only those crossing with a fixed sign of $\dot y$. The section is obtained by fixing a plane in phase space and plotting the points when the trajectory intersects this plane in a particular direction. This technique is good at determining the regular or chaotic nature of the trajectory. In the Poincaré map, quasi-periodic orbits appear as closed well-defined curves. Periodic orbits appear as isolated single points inside such islands. Any "fuzzy'' distribution of points in the surface of section implies that the trajectory is chaotic.

This is the method of the Poincaré surface of section or the Poincaré map. It has been largely used to determine the location and size of regular and chaotic regions in the phase space of the circular restricted three-body problem.

In order to explore the region of apparent stability shown in Fig. 2 we produced about thirty Poincaré surfaces of section. The range of C_j considered coincide with the range needed to cover the white areas, h1 and h2, shown in Fig. 2 ( $3.1800\leq C_j\leq 3.2040$). A sample of the Poincaré surfaces of section generated are given in Figs. 3 and 4.

Figure 4: Set of Poincaré surfaces of section for different values of Jacobi constant, $3.183\leq C_j\leq 3.194$. The range of Cj considered in these plots coincide with the range needed to cover the area h2 given in Fig. 2. The center of the concentric islands shown in each surface of section corresponds to one periodic orbit of the family h2(Broucke 1968).

The size of the largest islands indicate the region of stability for each Jacobi constant. From the whole set of Poincaré surfaces of section considered we verified that the regions h1 and h2 (Fig. 2) are actually stable regions. These stable regions are associated with periodic orbits.

3.1 Periodic orbits

The center of the islands for each surface of section corresponds to one periodic orbit. They are members of two families of simple direct periodic orbits called h1 and h2 (Broucke 1968), also called families g1 and g2 (Hénon 1970) or S and N (Gorkavyi 1993). They were initially classified in a single family, called family g (Broucke 1962; Szebehely 1967). It is important to note that in these previous works the main goal was to study the periodic orbits. None of them paid much attention to the quasi-periodic orbits, their size and location.

A sample of trajectories of the family of periodic orbits h1 is given in Fig. 5.

Figure 5: Trajectories of the four periodic orbits of family h1, that appeared in Fig. 3 (Cj=3.197, Cj=3.198, Cj=3.199 and Cj=3.200 from left to right).

Figure 6: Trajectories of the four periodic orbits of family h2, that appeared in Fig. 4 (Cj=3.183, Cj=3.185, Cj=3.190 and Cj=3.194 from right to left).

The trajectories chosen are four periodic orbits that appeared in Fig. 3 (C_j=3.197, C_j=3.198, C_j=3.199 and C_j=3.200 from left to right). The thick black lines correspond to the zero velocity curves that touch the points L₁ or L₂. In Fig. 6 are presented trajectories of the four periodic orbits of family h2, that appeared in Fig. 4 (C_j=3.183, C_j=3.185, C_j=3.190 and C_j=3.194 from right to left). A comparison between the two families shows that the periodic orbits of family h1 have their closest approach to the Moon at opposition while for orbits of family h2 it happens at inferior conjunction. Another feature is that the periodic orbits of family h1 are more elongated than those of family h2.

3.2 Quasi-periodic orbits

The islands in the Poincaré surface of section correspond to quasi-periodic orbits around the periodic one located in the center of the respective islands. The largest of these islands are those with the maximum amplitude of oscillation that are still stable. In order to visualize the shape and evolution of such orbits we selected two of them. In Fig. 7 we present the trajectory of one quasi-periodic orbit around a periodic orbit of family h1 that appeared in Fig. 4 (largest island at C_j=3.200).

Figure 7: Trajectory of one quasi-periodic orbit around a periodic orbit of family h1 that appeared in Fig. 4 (largest island at Cj=3.200).

Figure 8: Trajectory of one quasi-periodic orbit around a periodic orbit of family h2 that appeared in Fig. 5 (largest island at Cj=3.190).

Figure 9: Stability diagram showing the location and width of the maximum amplitude of oscillation around the periodic orbit h1 and h2 as a function of the Jacobi constant. The three thick black lines indicate the Moon's location and the values of the Jacobi constant at points L1 or L2. The thin line that crosses the figure corresponds to circular (e=0) trajectories, and divide the figure into pericentric (left) and apocentric (right) initial conditions.

In Fig. 8 we present the trajectory of one quasi-periodic orbit around a periodic orbit of family h2 that appeared in Fig. 5 (largest island at C_j=3.190). Despite of the fact that these trajectories have Jacobi constant values enough to escape. In both cases the trajectories remain inside the Roche lobe and also maintain the shape of the periodic orbit associated with each one. In astrodynamics applications this kind of orbital characteristic is valuable. These stable regions could be used to keep missions around the Moon, at a far distance, based on a very low fuel consumption.