7 Long-term integrations and results.

In addition to the previous estimates, it is also interesting to study the dynamic evolution of our TCB orbits on a long time scale. This consents us to study the dynamic mechanisms at work, and to analyse whether the dynamic path that, according to the previous section, they took to achieve these orbits is actually completely lost as a consequence of the chaotic nature of planet-crossing orbits.

We have thus integrated the orbits of 20 bodies on the basis of their highest probability of coming from one of the four previous sources. The integrations have been carried out with an integrator based on the Bulirsch-Stoer technique (Stoer & Bulirsch 1980), and optimised for dealing accurately with planetary close encounters (cf. Michel et al. 1996). The dynamic model included all the planets except Pluto and Mercury, the mass of the latter being added to that of the Sun. The integration interval spanned at least 10 Myr backward and 10 Myr forward in time, resulting in a total timespan of 20 Myr (which was extended in some cases).

As is well known, the results of long-term integrations of planet-crossing orbits cannot be seen as deterministic reconstructions or predictions of the real evolutions. Nevertheless, they are very useful in providing qualitative and/or statistical information on the most frequent orbital behaviours, on the effectiveness of various dynamic mechanisms and the corresponding lifetimes. Moreover integrating backward and forward in time merely provides a simple way of doubling the size of the sample and thus of improving the statistics; we point out that backward integrations cannot provide information on the sources of the bodies either individually or statistically.

7.1 Bodies from the intermediate source 3:1

We have considered 4 orbits with probability P₁ in the range between 0.54 and 0.60. The most frequent end-state of particles on these initial orbits is an impact with the Sun (4 in the backward integration and 3 in the forward one). Only one body has a semi-major axis, which becomes greater than 100 AU. The median lifetime of this sample is about 2 Myr, while the mean lifetime is of the order of 3 Myr. Most bodies (7/8) collide with the Sun while they are located in a mean motion resonance (5are in the 3:1 resonance - see e.g. Fig. 8 - and 2 in the 8:3 one).

During the integration time all the orbits are temporarily located in the 3:1 mean motion resonance, and 6 orbits are affected by secular resonance with both the inner and outer planets. Note that, although they are based on a limited sample of integrated orbits, our results agree with previous studies (Gladman et al. 1997; Migliorini et al. 1998; Michel et al. 2000a).

Figure 8: Time evolution (backward and forward) of the semi-major axis a (AU), eccentricity and inclination of a TCB orbit which has the greatest probability of coming from the 3:1 source.

7.2 Bodies from the intermediate $\mathsfsl{\nu_6}$ source.

We have integrated 4 particles with probability P₂ in the range 0.81 and 0.90. All the bodies have a semi-major axis smaller than 1.2 AU, an eccentricity smaller than 0.23 and a small inclination $i <15^{\circ}$.

The 8 evolutions corresponding to the integrations both backward and forward in time, are dominated by close approaches with the terrestrial planets (Fig. 9). We found that only 2 bodies have a collision with Venus in the forward integration, at +4.6 Myr and +6.2 Myr, respectively, while up to 10 Myr backward all the bodies survive. Thus the median lifetime and the mean lifetime are larger than 10 Myr, as previously found by Michel et al. (2000b).

Figure 9: Time evolution (backward and forward) of the semi-major axis a (AU), eccentricity and inclination of a TCB orbit which has the greatest probability of coming from the $\nu _6$ source and which survives during the whole integration time span. The dashed horizontal line represents the Earth's orbital radius.

7.3 Bodies from the intermediate Mars-Crosser source.

We have considered 5 bodies which, according to criterium 1 originated in the MC region. Note that applying criterium 2, two bodies (labeled 636 and 685in Table 8) have P₃ - P₂ < 0.1 which means that they may either come from the $\nu _6$ resonance or the MC source.

Since the 5 bodies have large eccentricities, a small increase in e is sufficient to induce a collision with the Sun. All the end-states backward and forward in time are solar collisions, the median and mean lifetimes being about 2 Myr and 1.84 Myr, respectively. 8 solar collisions occur while the bodies are located either in the region where the secular resonances $\nu _2$, $\nu _5$ and $\nu _7$ overlap (e.g. Fig. 10) or in the overlapping region of the $\nu _5$ and $\nu _7$resonances. Only one impact into the Sun happens while the orbit is located in the 3:1 mean motion resonance and the last one results from the effect of the 5:1 mean motion resonance with Jupiter. As previously found semianalytically by Michel & Froeschlé (1997), our numerical results show that in the region a < 2 AU, the secular resonances with both the inner and outer planets are effective dynamic mechanisms.

Figure 10: Time evolution (backward and forward) of the semi-major axis a (AU), eccentricity and inclination of a TCB orbit which has the greatest probability of coming from the MC source. The plots labelled $\sigma _2$, $\sigma _5$ and $\sigma _7$ represent the evolutions of the critical arguments of the secular resonance $\nu _2$, $\nu _5$ and $\nu _7$, respectively. Here, $\sigma _{{\rm j}}$ is equal to $\varpi - g_{{\rm j}} t - \beta _{{\rm j}}$, where $\varpi$ is the longitude of perihelion of the particle, $g_{{\rm j}}$is the proper frequency and $\beta _{{\rm j}}$ is the phase at time t=0 of planet j.

7.4 Bodies from the intermediate JFC source

We integrated 7 bodies that are presumably of cometary origin. The median lifetime of this sample is about $4 \times 10^5$ yr. This value is similar to the $4.5 \times 10^5$ yr found by Levison & Duncan (1994), although our sample is much smaller. The mean lifetime is a little larger and equal to 0.747 Myr.

Over the 14 evolutions, 5 Sun-grazing were recorded (3 in the forward integrations and 2 backward), 5 reached a semi-major axis larger than 100 AU, and the last 4 are ejected from the Solar System (the eccentricities e > 1.0). We notice that the solar collisions always occur when the bodies are inside the chaotic region of the 3:2 mean motion resonance with Jupiter and also in the Kozai resonance with $\omega$ librating about  $90^{\circ}$ or  $270^{\circ}$ (Fig. 11).

Figure 11: Time evolution (backward and forward) of the semi-major axis a (AU), eccentricity, inclination and argument of perihelion $\omega$ of a TCB orbit which has the greatest probability to come from the JFC source.