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To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.icarus.2007.11.035 Byline: Harold F. Levison (a), Alessandro Morbidelli (b), Christa VanLaerhoven (c), Rodney Gomes (d), Kleomenis Tsiganis (e) Keywords: Origin; Solar System; Kuiper belt; Trans-neptunian objects; Planets; migration; Celestial mechanics Abstract: We explore the origin and orbital evolution of the Kuiper belt in the framework of a recent model of the dynamical evolution of the giant planets, sometimes known as the Nice model. This model is characterized by a short, but violent, instability phase, during which the planets were on large eccentricity orbits. It successfully explains, for the first time, the current orbital architecture of the giant planets [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461], the existence of the Trojans populations of Jupiter and Neptune [Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465], and the origin of the late heavy bombardment of the terrestrial planets [Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A., 2005. Nature 435, 466-469]. One characteristic of this model is that the proto-planetary disk must have been truncated at roughly 30 to 35AU so that Neptune would stop migrating at its currently observed location. As a result, the Kuiper belt would have initially been empty. In this paper we present a new dynamical mechanism which can deliver objects from the region interior to [approximately equal to]35AU to the Kuiper belt without excessive inclination excitation. In particular, we show that during the phase when Neptune's eccentricity is large, the region interior to its 1:2 mean motion resonance becomes unstable and disk particles can diffuse into this area. In addition, we perform numerical simulations where the planets are forced to evolve using fictitious analytic forces, in a way consistent with the direct N-body simulations of the Nice model. Assuming that the last encounter with Uranus delivered Neptune onto a low-inclination orbit with a semi-major axis of [approximately equal to]27AU and an eccentricity of [approximately equal to]0.3, and that subsequently Neptune's eccentricity damped in [approximately equal to]1 My, our simulations reproduce the main observed properties of the Kuiper belt at an unprecedented level. In particular, our results explain, at least qualitatively: (1) the co-existence of resonant and non-resonant populations, (2) the eccentricity-inclination distribution of the Plutinos, (3) the peculiar semi-major axis -- eccentricity distribution in the classical belt, (4) the outer edge at the 1:2 mean motion resonance with Neptune, (5) the bi-modal inclination distribution of the classical population, (6) the correlations between inclination and physical properties in the classical Kuiper belt, and (7) the existence of the so-called extended scattered disk. Nevertheless, we observe in the simulations a deficit of nearly-circular objects in the classical Kuiper belt. Author Affiliation: (a) Southwest Research Institute, Department of Space Studies, 1050 Walnut Street, Suite 400, Boulder, CO 80302, USA (b) Observatoire de la CA[acute accent]te d' Azur, BP 4229, 06304 Nice Cedex 4, France (c) Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver BC, V6T 1Z1, Canada (d) Observatorio Nacional/MCT, Rua General Jose Cristino 77, 20921-400 Rio de Janeiro, RJ, Brazil (e) Section of Astrophysics, Astronomy & Mechanics, Department of Phyiscs, Aristotle University of Thessaloniki, GR 54 124 Thessaloniki, Greece Article History: Received 30 December 2006; Revised 13 November 2007

We have shown, in previous publications, that stable chaos is associated with medium/high-order mean motion resonances with Jupiter, for which there exist no resonant periodic orbits in the framework of the elliptic restricted three-body problem. This topological 'defect' results in the absence of the most efficient mechanism of eccentricity transport (i.e., large-amplitude modulation on a short time scale) in three-body models. Thus, chaotic diffusion of the orbital elements can be quite slow, while there can also exist a nonnegligible set of chaotic orbits which are semiconfined (stable chaos) by 'quasi-barriers' in the phase space. In the present paper we extend our study to all mean motion resonances of order q [less than or equal to] 9 in the inner main belt (1.9-3.3 AU) and q [less than or equal to] 7 in the outer belt (3.3-3.9 AU). We find that, out of the 34 resonances studied, only 8 possess resonant periodic orbits that are continued from the circular to the elliptic three-body problem (regular families), namely, the 2/1, 3/1, 4/1, and 5/2 in the inner belt and the 7/4, 5/3, 11/7, and 3/2 in the outer belt. Numerical results indicate that the 7/3 resonance also carries periodic orbits but, unlike the aforementioned resonances, 7/3-periodic orbits belong to an irregular family. Note that the five inner-belt resonances that carry periodic orbits correspond to the location of the main Kirkwood gaps, while the three outer-belt resonances correspond to gaps in the distribution of outerbelt asteroids noted by Holman and Murray (1996, Astron. J. 112, 1278-1293), except for the 3/2 case where the Hildas reside. Fast, intermittent eccentricity increase is found in resonances possessing periodic orbits. In the remaining resonances the time-averaged elements of chaotic orbits are, in general, quite stable, at least for times t ~ 250 Myr. This slow diffusion picture does not change qualitatively, even if more perturbing planets are included in the model. Key Words: asteroids, dynamics; celestial mechanics; chaos.

In a previous publication (Tsiganis et al. 2000, Icarus 146, 240-252), we argued that the occurrence of stable chaos in the 12/7 mean motion resonance with Jupiter is related to the fact that there do not exist families of periodic orbits in the planar elliptic restricted problem and in the 3-D circular problem corresponding to this resonance. In the present paper we show that nonexistence of resonant periodic orbits, both for the planar and for the 3-D problem, also occurs in other jovian resonances--namely the 11/4, 22/9, 13/6, and 18/7--where cases of real asteroids on stable-chaotic orbits have been identified. This property may provide a 'protection mechanism,' leading to semiconfinement of chaotic orbits and extremely slow migration in the space of proper elements, so that diffusion is practically unrelated to the value of the Lyapunov time, [T.sub.L], of chaotic orbits. However, we show that, in more complicated dynamical models, the long-term evolution of chaotic orbits initiated in the vicinity of these resonances may also be governed by secular resonances. Finally, we find that stable-chaotic orbits have a characteristic spectrum of autocorrelation times: for the action conjugate to the critical argument the autocorrelation time is of the order of the Lyapunov time, while for the eccentricity- and inclination-related actions the autocorrelation time may be longer than [10.sup.3][T.sub.L]. This behavior is consistent with the trajectory being sticky around a manifold of lower-than-full dimensionality in phase space (e.g., a 4-D submanifold of the 5-D energy manifold in a three-degrees-of-freedom autonomus Hamiltonian system) and reflects the inability of these 'flawed' resonances to modify secular motion significantly, at least for times of the order of 200 Myr. Key Words: asteroids; dynamics; celestial mechanics; chaos.