Astron. Astrophys. 354, 1091-1100 (2000) 4. ResultsWith the aforementioned integrations we aim at answering the following questions:
4.1. The Lyapunov time of ThersitesThe calculation of the maximal LCE for (1868) Thersites is graphically represented in Fig. 1. The method used here is based on the simultaneous integration of an initially nearby orbit. The limiting growth rate of their phase-space distance, i.e. the value at which saturates, is the value of the LCE. The result agrees with that of Milani et al. (1997), i.e. the Lyapunov time has the value years. Chaotic orbits lying initially in the same connected chaotic region of the phase space should have similar values of . Thus, we expect any chaotic orbits found within our sample of `neighbors' to have Lyapunov times of the order of years as well. Note, however, that a formal mathematical proof of this statement is not at hand for such complicated dynamical systems. Also, the size of the neighborhood of initial conditions which comply to this argument cannot be known a priori . It would be interesting to test the validity of this preposition by actually performing the calculations, but this lies beyond the scopes of the present work. As will be shown in the following results, most of the `asteroids' show irregular variations of their orbital elements, which reveal the chaotic nature of their orbits.
4.2. Stability of the elements - classification of orbitsAccording to the evolution of the orbital elements, three types of dynamical behavior are found within our numerical integrations:
Figs. 2, 3 and 4 show the time series of the osculating eccentricity and inclination for one representative orbit of each of the three groups, in the order mentioned above. For the escaping orbit, Fig. 2, the libration, and eventually circulation, of the critical argument is also shown ( is the mean longitude of the asteroid and the one of Jupiter). We emphasize on the fact that, for all the UNS- and ESC-orbits, large variations of the inclination (sometimes even greater than ) are present before any `visible' changes of the mean value of the eccentricity occur. Also, we should stress that for most of the orbits the eccentricity also shows signs of chaotic variations, but this cannot be taken for granted and more refined analysis should be made (see e.g. Sect. 4.4). However, the inclination time series is quite stationary for the orbits.
Out of the 21 orbits integrated, one from the C-group and three from the N-group ( of the total) escaped. Three from the C-group and five from the N-group ( of the total) are very unstable. Finally, the rest of the orbits (one from the C-group and eight from the N-group) have elements which do not show large jumps () ^{3}. The classification of the orbits in these three groups can also be seen in Table 1, along with the escape time, , for the ESC-orbits. is given by the time at which the critical argument begins to circulate. We performed a running-window averaging in the time series of the osculating elements, the window having length points (which corresponds to 1.2 Myrs). The resulting mean elements , and , behave according to the classification made above. For all the STB-orbits is almost constant for the whole 50 Myrs time interval, while presents only small-amplitude fluctuations. On the other hand, for the UNS-orbits, the variations of the mean elements are much larger. For the ESC-orbits the averaging is made for a time interval up to 1 Myrs before the escape. The amplitude of oscillations in , which is the width of libration D about (i.e. ) is also calculated, using a running-window of points. D is again almost constant for the STB-orbits, while larger variations are seen for the UNS- and ESC-orbits. Fig. 5 is a plot of the mean value of D versus the maximum value of attained by each orbit ^{4}. It is clear from this plot that the classification made in the previous paragraph is justified. Orbits for which and are grossly unstable (filled circles denote the UNS-orbits and triangles denote the ESC-orbits). These values can, in fact, be regarded as a set of escaping parameters for orbits initially librating in the vicinity of Thersites. The STB-orbits (open circles) lie below these limits. We note that a very similar picture is obtained if one uses an `average' value for instead of . However, this `average' value is only meaningful for the STB-orbits (where it is almost the same as ). We also calculated the r.m.s. values for these elements (as in Paper I), using the standard definition , and the results on the (, ) plane are very similar to that of Fig. 5.
4.3. An example of stickinessA very interesting behavior was found for one of our escaping orbits (). The time series of e, i and are shown in Fig. 6. Looking at the eccentricity and inclination plots, one can see a phase of increase of both eccentricity and inclination beginning at about 27 Myrs. This continues up to 38 Myrs, where a small jump, clearly seen in the eccentricity, occurs. Then the values of both e and i remain fairly stable for a period of 2 Myrs, before the asteroid finally escapes. The reason for this behavior is easily understood if one looks at the evolution of the critical argument, . In the beginning, librates about , since the asteroid moves around . At Myrs a brief period of circulation reveals the ejection of the asteroid from the stability region. However, changes almost immediately to a libration about the value which corresponds to the other stable fixed point of the 1:1 mean motion resonance (). This libration, which takes place at high inclinations of , lasts for a period of 2 Myrs, after which starts circulating again. Thus, it is clear from these plots that the asteroid leaves at Myrs, takes on a `horseshoe orbit', then `sticks' to the resonant island around for 2.5 Myrs and is eventually ejected from the solar system. This `trapping' of the orbit to the stability region demonstrates clearly the sticky properties of an island's boundary. ^{5} 4.4. Analysis in the time-frequency domainIn an effort to understand the mechanism by which Thersites may be ejected from the Trojan swarm, we performed an analysis which we call time-frequency analysis (TFA). Briefly, this analysis consists of determining the spectrum of the non-singular variables and (where is the longitude of perihelion) for consecutive equal time intervals. In this way we are able to monitor if and how the frequency content of these time series is changing along the orbit. In particular we want to focus on
The method is based on the frequency analysis for equidistant time series developed by Chapront (1995), who has also supplied the code. First, the non-singular variables h and p are derived from the osculating elements. The frequency analysis is then applied within a window of points (i.e. years) shifted through the data so that two consecutive windows are years ( points) apart, i.e. they overlap by a factor of 80%. The size of the window was chosen so that very long periods could also be determined. We could have chosen an even larger overlap ratio, but this would be more time consuming without giving significant information. Experimenting on the data we found that Myrs is enough to show the basic spectral characteristics. The TFA results for the element of the three representative orbits shown in Figs. 2-4 are presented in Fig. 7. The 50 Myrs interval is covered with 37 windows. All the periods involved are between 50,000 and 80,000 years, with the dominant ones being concentrated around 60,000 years. It is obvious that the frequencies and their respective amplitudes, even for the STB-orbit, are changing from one window to the next in an erratic manner, indicating possible interaction between several close-by resonances. According to Nobili et al. (1989) - Table 5 - there are about 10-20 high-order secular resonances in this frequency range resulting from linear combinations between and , , 6, 7 and 8. It seems that, although the orbital elements of this STB-orbit look rather `stable' for this time interval, the orbit may actually be wandering inside the thin chaotic strips of a `forest' of secular resonances, without being able yet to find an escape path.
The spectral differences between the three orbits are clearly seen. First of all, the frequency band covered by the STB-orbit (58,000-75,000 years) is narrower than that of the other two orbits (40,000-75,000 years for the UNS-orbit and 30,000-80,000 for the ESC-orbit). The dominant modes for the STB-orbit are at years, while for the UNS-orbit they are at years. For the ESC-orbit the result is very interesting. In the first 5 windows of the TFA we also find a concentration of peaks in the 50,000-70,000 years interval. However, as time progresses, we see that the spectrum shifts towards smaller periods (35,000-50,000 years) before the asteroid finally escapes. This disappearance of periods larger than 50,000 years and the appearance of smaller periods could be interpreted as the footprint of separatrix crossing. Thus, according to the relatively large inclination of Thersites, the overlap of many high-order secular resonances may shift this asteroid between the resonant islands until it finds its way out of this `sticky' region. We note that all the orbits of the respective groups show a very similar behavior with the one presented in Fig. 7. The high-frequency band of the spectrum is shown in Fig. 8. The periods are in the range of 3,200-4,000 years. These short periods reflect the proper motion of the asteroid's longitude of perihelion. The characteristic `drift' of the spectrum, this time towards larger periods, of the ESC-orbit is again present. Also, note that modes with years that are present during the first windows, disappear quickly. The possibility that mixed mean motion resonances, with periods of this order, are also responsible for this frequency drift cannot be confirmed by the present results. The behaviour of the high-frequency band of the UNS-orbit is very similar. A slow `drift' of this band of the spectrum towards larger periods, as well as the disappearance of the smallest periods (but also their re-appearance during the last windows), is also seen. The variations of the amplitudes are not as significant as in the previous plots. The STB-orbit, on the other hand, has a high-frequency band which is more `well-defined'. The size of the band is not changing with time and there is no `drift' visible in this plot. Moreover, the shortest periods do not vanish as in the other two cases. Again the amplitudes are not varying significantly. We note here that in the medium- and low-frequency band of the spectrum the dominant modes are close to and respectively. Differences between the three classes of orbits are seen (in what concerns amplitude/frequency variations) but the spectrum does not show a `drift' or any other stricking feature.
4.5. Analysis of critical argumentsThe TFA results (in the spectrum) presented in the previous paragraph indicate possible action of secular resonances. The unstable inclination time series of most of the orbits also suggests that secular resonances involving the nodes should act in this region of the Trojan belt. As the TFA results cannot, in the present form, give definite conclusions on which resonance is responsible for the observed chaotic behavior, a different tool has to be used in order to verify the secular resonance hypothesis. Thus, we studied the evolution of the critical arguments that are formed by linear combinations of the secular angles (, and , with and 8). If a critical argument is found to be alternating between circulation and libration in an erratic manner, then chaotic motion is related to the crossing of the separatrix of the respective resonance. The arguments studied here are those corresponding to the modes of precession of the planetary orbits, as these are given by combinations of the fundamental frequencies in Tables (4)-(5) of Nobili et al. (1989) ^{6}. Checking all of the above mentioned combinations, we were able to find more than one critical arguments showing a behavior typical of chaotic evolution: these are the arguments corresponding to the linear combinations of fundamental frequencies given in rows 2, 3, 4, 6, 10, 13 and 32 of Table (5) of Nobili et al. (1989). All of these combinations involve the nodes of the outer planets (, and ). This result could explain the inclination instability, found in our results to preceed the eccentricity increase. Their periods are in the range 25,000-55,000 years, except for the one corresponding to with years! The most clear plot is obtained for the argument corresponding to years. Fig. 9 shows the evolution of this critical argument for the three representative orbits used in the previous figures. Note that for the and the orbit the evolution of the critical argument is typical of chaotic motion, while for the orbit the plot is completely different; the argument is not a slow variable. We emphasize on the fact that the classification of the orbits made in the previous paragraphs is also consistent with the study of the critical arguments: all and orbits show chaotic behavior in the plots of all the aforementioned critical arguments, while orbits do not show any indications of chaos for all the combinations that we tested ^{7}. © European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |