3. The JPL model
In this section, we will check the quasi-stability results for a very realistic model of Solar system. The model used here is defined by the Jet Propulsion Laboratory (JPL) ephemeris, file DE406. The JPL ephemeris comprises files containing interpolatory polynomials for the orbital data of each planet, including the Sun, Earth and Moon. These data can be easily used to derive the vector field acting on an infinitesimal particle under the attraction of the main bodies of the Solar system. Of course, this (numerically obtained) vector field is only defined for the time interval for which the ephemeris is provided.
3.1. Numerical simulations
The initial time for the numerical integration is the modified Julian day 20.1978749133 (day 0.0 corresponds to year 2000.0), that coincides with the first full Moon of year 2000. 1 This is to have an initial relative position of the Earth, Moon and Sun similar to the one of the BCP for . The simulation stops at year 3000.0. We have also used the same mesh as in the BCP model (see Sect. 2.2). Due to the lack of symmetry between and in the real system, we have carried out simulations for both cases. The results obtained are very similar, and are discussed in the following sections.
3.1.1. Stability around the family VF2
In order to compare these results with the ones for the BCP, we have used the same mesh as in Sect. 2.2. Fig. 7 shows the number of initial conditions corresponding to trajectories that do not escape (in the same sense as in Sect. 2.2) for time spans of 10 years (upper curve) and 100 years (lower curve). We do not include the results for 1 000 years because only a few trajectories subsist: in the case, the number of these points is never greater than 8, while for the case this number is slightly bigger (the more stable section corresponds to the case , that contains 85 surviving points). Taking into consideration these results and the behaviour around the family VF2, seems slightly more stable than . Moreover, if the integration time were long enough, we believe that all these trajectories would finally escape.
Note the size of the quasi-stable region for a time interval of 10 years (see Fig. 8 and Fig. 9). Although a stability time of 10 years is totally irrelevant for astronomical purposes, it could be interesting for astronautical applications, since there would be no need for any kind of control to keep an spacecraft there. The strategy to transfer a spacecraft from a parking orbit to these regions is not considered here. Roughly speaking, a transfer by means of a double lunar swing-by from the standard Ariane Geostationary Transfer Orbit to the vicinity of the triangular points can be completed in about 2 months using a little bit less than 900 m/s of (Companys et al. 1996). For a dynamical systems approach to similar transfers, see also Gómez et al. (1993).
3.1.2. Stability around the family VF3
This family seems to be responsible for the largest quasi-stability regions that we have found near to the Lagrangian points. Fig. 10 shows the number of initial conditions corresponding to non escaping trajectories (this is the same plot as in Fig. 4 but for the JPL model). As we have used the same parameters to define the mesh of initial conditions, the "units" in these plots -the vertical axis- are also the same as in the BCP. So, although the stability region for the JPL model is smaller than that of the BCP, it is remarkable that there exists a quite large stability region for a time span of 1 000 years. In order to compare with the simulations for the BCP model, note that 1 000 years is more than 13 000 revolutions of the Moon around the Earth.
Fig. 11 and Fig. 12 show the coordinates of a few slices of the quasi-stability region for a time interval of 1 000 years near the triangular points and , respectively. The selected slices correspond to places in which the region is quite large, according to Fig. 10. We recall that the unit of distance is the Earth-Moon distance for the modified Julian day 20.1978749133, that is, 384 467 km.
From these plots, it seems that the BCP model contains several features of the JPL system. This is especially true for the location of the quasi-stability regions, as has been shown by these calculations. However, the comparison between Fig. 4 (right) and Fig. 10 also shows that not all the features of JPL are contained in BCP. We believe that these differences are mainly due to resonances in JPL that are not present in BCP, due to the simplicity of the latter. The development and study of improved models is the subject of work in progress.
Obviously, the size of these regions could depend on the length of the integration time. To try to show this dependence, we have carried out the following calculation. To fix the discussion, let us focus on the point, and let be the orbit on the family VF3 for which when (). Let be the number of non escaping initial conditions after n years (we are using the same mesh as before), around the orbit . Let be , that is, denotes the total number of non escaping points around VF3. The values of , for n ranging between 100 and 1 000 years are displayed in Fig. 13. We have tried several fits to these data, trying to show the persistence of a certain number of points when n goes to infinity (Simo et al. 1995), but the results are inconclusive. It turns out that the number of remaining points strongly depends on the fitting function used, and there are no theoretical reasons for suggesting any particular type of fitting function. We think that longer numerical integrations will help to elucidate this question.
3.2. Typical trajectories in the stable region
To explain the dynamics inside the quasi-stable region, we first show a few trajectories corresponding to the slices and (see Fig. 10). In each slice, we have selected an initial condition near the "central part" of the stable region. For each initial condition, we have computed the corresponding orbit in the JPL model, for the case, and the results are displayed in Fig. 14 and Fig. 15 (similar figures are obtained for the case). To give a sense of velocity we have drawn the orbit as a polygonal line, putting a vertex for each day of the integration.
It is interesting to look at the projection of these trajectories. They show that the vertical oscillations are very close to an harmonic oscillator: there is a clear frequency that dominates this motion and the remaining frequencies appear as a perturbation. This dominating frequency is already contained in the BCP and RTBP (it is related to the vertical frequency of the points in the RTBP and to the vertical frequency of the orbits PO1, PO2 and PO3 of the BCP), and its period is a little less than a month.
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000