Role of secular resonances in the history of Trojans

We have investigated the possible role of secular resonances in the dynamical evolution of Trojans during the early phases of the Solar System. According to our previous studies (Marzari and Scholl 1998a, 1998b) a significant population of planetesimals can be captured in Jupiter and Saturn Trojan orbits by the mass growth of the two planets. If we compare the implications of our model with the present Trojan populations, two severe problems arise: (1) All the captured planetesimals have low inclinations while the observed Jupiter Trojans have significantly higher inclinations exceeding even 20\ub0. (2) No Trojan has been discovered near Saturn's Lagrangian points. In the present paper, we show that the presence of secular resonances in the Trojan regions of both Jupiter and Saturn may explain this contradiction between our model for Trojan capture and observations. We relate the high inclinations of Jupiter Trojans to the \u3bd16 secular resonance, even if this resonance is effective in pumping up inclinations for orbits with comparatively large libration amplitudes of about 60\ub0 (by ``libration amplitude'' here we mean the difference between the maximum and minimum values of the critical argument). How do we explain then the present small libration-high inclination Trojans? If we combine the effects of the \u3bd16 secular resonance with other dynamical and physical processes affecting Trojan orbits, the present dynamical structure of Jupiter Trojans may be explained as follows: (a) During the growth of Jupiter and Saturn: Trojans with large libration amplitudes and low inclinations are trapped. Libration amplitudes decrease slightly due to continuing proto-planetary growth. Eventually, synergy between Kozai resonance and proto-planetary growth may yield some Trojans with inclinations up to about 10\ub0 (Marzari and Scholl 1998b). (b) After the growth of Jupiter and Saturn has been completed: Trojans with large libration amplitudes near the \u3bd16 secular resonance increase their orbital inclinations up to 20\ub0 while keeping their large libration amplitudes. We show in this paper that the \u3bd16 is effective also at initial inclinations lower than 4\ub0. (c) Collisions reduce libration amplitudes (Marzari and Scholl 1998b) while the inclinations remain high. High-inclination Trojans are then displaced in more stable orbits. (d) Collisions and dynamical outflow (Levison et al. 1997) shape the present Trojan population. The absence of observed Saturn Trojan orbits was attributed by previous studies to instability caused by the neighboring 5 : 2 resonance with Jupiter (De la Barre et al. 1996, Mikkola and Innanen 1989, Innanen and Mikkola 1992). By integrating the trajectories of test bodies started in Saturn Trojan orbits, we show that the main source for instability is the presence of the mixed secular resonance 2\u3c9S-\u3c9J-\u3c9T (S for Saturn, J for Jupiter, and T for Trojan) inside the libration regions around L4 and L5. The crossing of this secular resonance destabilizes on a short time scale (105 years) most of the planetesimals trapped in low-libration Trojan orbits, and it is also responsible for the slow outflow of the remaining large librators. The \u3bd6 resonance may contribute to the instability of low-libration Saturn Trojan orbits on a much shorter time scale