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Role of secular resonances in the history of Trojans
No full textWe 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°. (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 ν16 secular resonance, even if this resonance is effective in pumping up inclinations for orbits with comparatively large libration amplitudes of about 60° (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 ν16 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° (Marzari and Scholl 1998b). (b) After the growth of Jupiter and Saturn has been completed: Trojans with large libration amplitudes near the ν16 secular resonance increase their orbital inclinations up to 20° while keeping their large libration amplitudes. We show in this paper that the ν16 is effective also at initial inclinations lower than 4°. (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 2ωS-ωJ-ωT (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 ν6 resonance may contribute to the instability of low-libration Saturn Trojan orbits on a much shorter time scale
Planetesimal accretion in binary star systems
Full text linkPlanetesimal accretion in close binary systems is a complex process for the gravitational perturbations of the companion star on the planetesimal orbits. These perturbations excite high eccentricities that can halt the accumulation process of planetesimals into planets also in those regions around the star where stable planetary orbits would eventually be possible. However, the evolution of a planetesimal swarm is also affected by collisions and gas drag. In particular, gas drag combined with the secular perturbations of the secondary star forces a strong alignment of all the planetesimal periastra. Since periastra are also coupled to eccentricities via the secular perturbations of the companion, the orbits of the planetesimals, besides all being aligned, also have very close values of eccentricity. This orbital ``phasing'' strongly reduces the contribution of the eccentricity to the relative velocities between planetesimals, and the impact speeds are dominated by the Keplerian shear: accretion becomes possible. This behavior is not limited to small planetesimals but also affects bodies as large as 100 km in diameter. The effects of gas drag are in fact enhanced by the presence of the constant forced component in the orbital eccentricity of the planetesimals. We describe analytically the periastron alignment by using the secular equations developed by Heppenheimer, and we test the prediction of the theory with a numerical code that integrates the orbits of a swarm of planetesimals perturbed by gas drag and collisions. The gas density is assumed to decrease outward, and the collisions are modeled as inelastic. Our computations are focused on the α Centauri system, which is a good candidate for terrestrial planets as we will show. The impact velocities between planetesimals of different sizes are computed at progressively increasing distances from the primary star and are compared with estimates for the maximum velocity for accretion. According to our simulations in the α Centauri system, the formation of a planet within 2 AU of the primary star is possible because of the orbital phasing forced by gas drag
On the Instability of Jupiter's Trojans
No full textWe have numerically explored the mechanisms that destabilize Jupiter's Trojan orbits outside the stability region defined by Levison et al. (1997, Nature385, 42-44). Different models have been exploited to test various possible sources of instability on timescales on the order of ~108 years. In the restricted three-body model, only a few Trojan orbits become unstable within 108 years. This intrinsic instability contributes only marginally to the overall instability found by Levison et al. In a model where the orbital parameters of both Jupiter and Saturn are fixed, we have investigated the role of Saturn and its gravitational influence. We find that a large fraction of Trojan orbits become unstable because of the direct nonresonant perturbations by Saturn. By shifting its semimajor axis at constant intervals around its present value we find that the near 5:2 mean motion resonance between the two giant planets (the Great Inequality) is not responsible for the gross instability of Jupiter's Trojans since short-term perturbations by Saturn destabilize Trojans, even when the two planets are far out of the resonance. Secular resonances are an additional source of instability. In the full six-body model with the four major planets included in the numerical integration, we have analyzed the effects of secular resonances with the node of the planets. Trojan asteroids have relevant inclinations, and nodal secular resonances play an important role. When a Trojan orbit becomes unstable, in most cases the libration amplitude of the critical argument of the 1:1 mean motion resonance grows until the asteroid encounters the planet. Libration amplitude, eccentricity, and nodal rate are linked for Trojan orbits by an algebraic relation so that when one of the three parameters is perturbed, the other two are affected as well. There are numerous secular resonances with the nodal rate of Jupiter that fall inside the region of instability and contribute to destabilize Trojans, in particular the ν16. Indeed, in the full model the escape rate over 50 Myr is higher compared to the fixed model. Some secular resonances even cross the stability region delimited by Levison et al. and cause instability. This is the case of the 3:2 and 1:2 nodal resonances with Jupiter. In particular the 1:2 is responsible for the instability of some clones of the L4 Trojan (3540) Protesilaos
Long term stability of Earth Trojans
No full textWe explore the long-term stability of Earth Trojans by using a chaos indicator, the Frequency Map Analysis. We find that there is an extended stability region at low eccentricity and for inclinations lower than about even if the most stable orbits are found at . This region is not limited in libration amplitude, contrary to what found for Trojan orbits around outer planets. We also investigate how the stability properties are affected by the tidal force of the Earth-Moon system and by the Yarkovsky force. The tidal field of the Earth-Moon system reduces the stability of the Earth Trojans at high inclinations while the Yarkovsky force, at least for bodies larger than 10 m in diameter, does not seem to strongly influence the long-term stability. Earth Trojan orbits with the lowest diffusion rate survive on timescales of the order of years but their evolution is chaotic. Their behaviour is similar to that of Mars Trojans even if Earth Trojans appear to have shorter lifetimes
The growth of Jupiter and Saturn and the capture of Trojans
No full textWe have studied the capture of planetesimals in Trojan-type orbits by a growing proto-planet. The change of the gravity field due to the mass growth causes a significant fraction of planetesimals orbiting nearby to be trapped as Trojans of the proto-planet. After a planetesimal is captured on a Trojan-type orbit, the libration amplitude of its critical argument is consistently reduced by the further mass growth of the proto-planet. The dynamical mechanism is discussed and the characteristics of the Trojan population captured by Jupiter during its growth are analysed. We find an interesting mechanism which could explain the observed high inclination Trojans. The synergy of a Kozai secular resonance with the growth of Jupiter's mass generates high inclination Trojans from low inclination-high eccentricity planetesimals orbiting near the growing proto-planet. The libration amplitudes of the model Trojans trapped by the mass-growth of Jupiter are higher compared to those of the observed Trojans. A possible mechanism that decreases the libration amplitudes of the model population is collisional evolution. We also show that the simultaneous formation of Jupiter and Saturn strongly inhibits the capture of planetesimals as Saturn Trojans. The interference of the 1:1 resonance with a secular resonance and, in some cases, also with the 5:2 resonance with Jupiter (Innanen and Mikkola, 1989), generates instability and causes the ejection of most Saturn Trojans out of resonance before the end of Saturn's mass growth
Clues to the origin of jupiter's trojans: the libration amplitude distribution
No full textWe model with numerical algorithms the dynamical processes that possibly lead to the trapping of Jupiter's Trojans from a primordial population of planetesimals orbiting nearby a proto-Jupiter. The predictions of models based on mutual planetesimal collisions and on the mass growth of Jupiter are compared with observations. In particular, we concentrate on the distribution of the libration amplitude. The two mechanisms for trapping reproduce closely the libration amplitude distribution of the real Trojans only when the long-term dynamical diffusion described by , Nature 385, 42-44) is taken into account
Saturn Trojans: Stability Regions in the Phase Space
No full textWe use the frequency map analysis method to identify for Trojan orbits of Saturn the regions in the proper orbital element phase space characterized by higher stability. We find that Trojan orbits with proper eccentricity around 0.05, libration amplitude of about 80°, and inclination lower than 15° show a slow diffusion in the proper frequency of the longitude of perihelion ω, which indicates long-term stability. Numerical integration of some of these stable orbits indicates a half-life of about 2.5 Gyr. Orbits with inclination of about 20° are destabilized by a secular resonance with the forcing term 2g6-g5. At higher inclinations Saturn Trojan orbits are unstable on a short timescale (a few×105 yr). Applying the frequency map analysis to the numbered Jupiter Trojans, we find that the size of the stability region is much larger for Jupiter Trojans than for Saturn Trojans. Moreover, the diffusion rate is significantly lower, suggesting that the dynamical lifetimes of Jupiter Trojans are considerably longer. The frequency analysis method allows us to separate the proper and forced components of the eccentricity of Trojans. A semianalytical model for secular motion of Saturn Trojans is presented
Formation of terrestrial planets in close binary systems: The case of alpha Centauri A
No full textAt present the possible existence of planets around the stars of a close binary system is still matter of debate. Can planetary bodies form in spite of the strong gravitational perturbations of the companion star? We study in this paper via numerical simulation the last stage of planetary formation, from embryos to terrestrial planets in the alpha Cen system, the prototype of close binary systems. We find that Earth class planets can grow around alpha Cen A on a time-scale of 50 Myr. In some of our numerical models the planets form directly in the habitable zone of the star in low eccentric orbits. In one simulation two of the final planets are in a 2:1 mean motion resonance that, however, becomes unstable after 200 Myr. During the formation process some planetary embryos fall into the stars possibly altering their metallicity
The MATROS project: Stability of Uranus and Neptune Trojans. The case of 2001 QR322
No full textWe present in this paper an analysis of the long term stability of Trojan type orbits of both Uranus and Neptune. Employing the Frequency Map Analysis (hereinafter FMA) we measure the diffusion speed in the phase space for a large sample of Trojan orbits with short numerical integrations. High resolution diffusion maps are derived for different values of initial inclination. These maps outline where the most stable orbits can be found in the Trojan clouds of the two planets.
The orbit of the newly discovered Neptune Trojan 2001 QR322 has been analysed in detail with the FMA method. In the phase space the body is located close to the border of a stable region for low inclination Neptune Trojans. Numerical integrations over 4.5 Gyr of clone orbits generated from the covariance matrix show that only 10% of the clones escape from the Trojan cloud.
The proper frequencies of the Trojan motion computed with the FMA algorithm allow us to to derive a numerical secular theory. From this theory it is possible to locate in the phase space the main secular resonances that can perturb Trojan orbits of the two planets and lead to instability
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