Rodney S. Gomes

Chaotic capture of Jupiter's Trojan asteroids in the early Solar System

Jupiters Trojans are asteroids that follow essentially the same orbit as Jupiter, but lead or trail the planet by an angular distance of ∼60 degrees (co-orbital motion). They are hypothesized to be planetesimals that formed near Jupiter and were captured onto their current orbits while Jupiter was growing, possibly with the help of gas drag and/or collisions. This idea, however, cannot explain some basic properties of the Trojan population, in particular its broad orbital inclination distribution, which ranges up to ∼40 degrees (ref. 8). Here we show that the Trojans could have formed in more distant regions and been subsequently captured into co-orbital motion with Jupiter during the time when the giant planets migrated by removing neighbouring planetesimals. The capture was possible during a short period of time, just after Jupiter and Saturn crossed their mutual 1:2 resonance, when the dynamics of the Trojan region were completely chaotic. Our simulations of this process satisfactorily reproduce the orbital distribution of the Trojans and their total mass.

Origin of the Structure of the Kuiper Belt during a Dynamical Instability in the Orbits of Uranus and Neptune

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 35 AU 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 ∼ 35 AU to the Kuiper belt without excessive inclination excitation. In particular, we show that during the phase when Neptunes 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 ∼ 27 AU and an eccentricity of ∼0.3, and that subsequently Neptunes eccentricity damped in ∼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.

The origin of the Kuiper Belt high-inclination population

I simulate the orbital evolution of the four major planets and a massive primordial planetesimal disk composed of 104 objects, which perturb the planets but not themselves. As Neptune migrates by energy and angular momentum exchange with the planetesimals, a large number of primordial Neptune-scattered objects are formed. These objects may experience secular, Kozai, and mean motion resonances that induce temporary decrease of their eccentricities. Because planets are migrating, some planetesimals can escape those resonances while in a low-eccentricity incursion, thus avoiding the return path to Neptune close encounter dynamics. In the end, this mechanism produces stable orbits with high inclination and moderate eccentricities. The population so formed together with the objects coming from the classical resonance sweeping process, originates a bimodal distribution for the Kuiper Belt orbits. The inclinations obtained by the simulations can attain values above 30° and their distribution resembles a debiased distribution for the high-inclination population coming from the real classical Kuiper Belt.

Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture

We study the orbital evolution of the four giant planets of our solar system in a gas disk. Our investigation extends the previous works by Masset & Snellgrove and Morbidelli & Crida, which focused on the dynamics of the Jupiter-Saturn system. The only systems we found to reach a steady state are those in which the planets are locked in a quadruple mean-motion resonance (i.e., each planet is in resonance with its neighbor). In total, we found six such configurations. For the gas-disk parameters found in Morbidelli & Crida, these configurations are characterized by a negligible migration rate. After the disappearance of the gas, and in the absence of planetesimals, only two of these six configurations (the least compact ones) are stable for a time of hundreds of millions of years or more. The others become unstable on a timescale of a few Myr. Our preliminary simulations show that, when a planetesimal disk is added beyond the orbit of the outermost planet, the planets can evolve from the most stable of these configurations to their current orbits in a fashion qualitatively similar to that described in Tsiganis et al.

Evidence from the asteroid belt for a violent past evolution of Jupiter's orbit

We use the current orbital structure of large (>50 km) asteroids in the main asteroid belt to constrain the evolution of the giant planets when they migrated from their primordial orbits to their current ones. Minton & Malhotra showed that the orbital distribution of large asteroids in the main belt can be reproduced by an exponentially decaying migration of the giant planets on a timescale of τ ∼ 0.5 Myr. However, self-consistent numerical simulations show that the planetesimal-driven migration of the giant planets is inconsistent with an exponential change in their semi-major axes on such a short timescale. In fact, the typical timescale is τ 5 Myr. When giant planet migration on this timescale is applied to the asteroid belt, the resulting orbital distribution is incompatible with the observed one. However, the planet migration can be significantly sped up by planet–planet encounters. Consider an evolution where both Jupiter and Saturn have close encounters with a Neptune-mass planet (presumably Uranus or Neptune itself) and where this third planet, after being scattered inward by Saturn, is scattered outward by Jupiter. This scenario leads to a very rapid increase in the orbital separation between Jupiter and Saturn which we show here to have only mild effects on the structure of the asteroid belt. This type of evolution is called a “jumping-Jupiter” case. Our results suggest that the total mass and dynamical excitation of the asteroid belt before migration were comparable to those currently observed. Moreover, they imply that, before migration, the orbits of Jupiter and Saturn were much less eccentric than their current ones.

LATE ORBITAL INSTABILITIES IN THE OUTER PLANETS INDUCED BY INTERACTION WITH A SELF-GRAVITATING PLANETESIMAL DISK

We revisit the issue of the cause of the dynamical instability during the so-called Nice model, which describes the early dynamical evolution of the giant planets. In particular, we address the problem of the interaction of planets with a distant planetesimal disk in the time interval between the dispersal of the proto-solar nebula and the instability. In contrast to previous works, we assume that the inner edge of the planetesimal disk is several AUs beyond the orbit of the outermost planet, so that no close encounters between planets and planetesimals occur. Moreover, we model the disk’s viscous stirring, induced by the presence of embedded Pluto-sized objects. The four outer planets are assumed to be initially locked in a multi-resonant state that most likely resulted from a preceding phase of gas-driven migration. We show that viscous stirring leads to an irreversible exchange of energy between a planet and a planetesimal disk even in the absence of close encounters between the planet and disk particles. The process is mainly driven by the most eccentric planet, which is the inner ice giant in the case studied here. In isolation, this would cause this ice giant to migrate inward. However, because it is locked in resonance with Saturn, its eccentricity increases due to adiabatic invariance. During this process, the system crosses many weak secular resonances—many of which can disrupt the mean motion resonance and make the planetary system unstable. We argue that this basic dynamical process would work in many generic multi-resonant systems—forcing a good fraction of them to become unstable. Because the energy exchange proceeds at a very slow pace, the instability manifests itself late, on a timescale consistent with the epoch of the late heavy bombardment (∼700 Myr). In the migration mechanism presented here, the instability time is much less sensitive to the properties of the planetesimal disk (particularly the location of its inner edge) than in the classic Nice model mechanism.

Constructing the secular architecture of the solar system II: the terrestrial planets

We investigate the dynamical evolution of the terrestrial planets during the planetesimal-driven migration of the giant planets. A basic assumption of this work is that giant planet migration occurred after the completion of terrestrial planet formation, such as in the models that link the former to the origin of the late heavy bombardment. The divergent migration of Jupiter and Saturn causes the g5 eigenfrequency to cross resonances of the form g5 = gk with k ranging from 1 to 4. Consequently these secular resonances cause large-amplitude responses in the eccentricities of the terrestrial planets if the amplitude of the g5 mode in Jupiter is similar to the current one. We show that the resonances g5 = g4 and g5 = g3 do not pose a problem if Jupiter and Saturn have a fast approach and departure from their mutual 2:1 mean motion resonance. On the other hand, the resonance crossings g5 = g2 and g5 = g1 are more of a concern: they tend to yield a terrestrial system incompatible with the current one, with amplitudes of the g1 and g2 modes that are too large. We offer two solutions to this problem. The first solution states that a secular resonance crossing can also damp the amplitude of a Fourier mode if the latter is large originally. We show that the probability of the g5 = g2 resonance damping a primordially excited g2 mode in the Earth and Venus is approximately 8%. Using the same mechanism to also ensure that the g5 = g1 resonance keeps the amplitude of the g1 mode in Mercury within 0.4 reduces the overall probability to approximately 5%. However, these numbers may change for different initial excitations and migration speeds of the giant planets. A second scenario involves a “jumping Jupiter” in which encounters between an ice giant and Jupiter, without ejection of the former, cause the latter to migrate away from Saturn much faster than if migration is driven solely by encounters with planetesimals. In this case, the g5 = g2 and g5 = g1 resonances can be jumped over, or occur very briefly. We show that, in this case, the terrestrial system can have dynamical properties comparable to what is exhibited today. In the framework of the Nice model, we estimate that the probability that Jupiter had this kind of evolution is approximately 6%.

Constructing the secular architecture of the solar system - I. The giant planets

Using numerical simulations, we show that smooth migration of the giant planets through a planetesimal disk leads to an orbital architecture that is inconsistent with the current one: the resulting eccentricities and inclinations of their orbits are too low. The crossing of mutual mean motion resonances by the planets would excite their orbital eccentricities but not their orbital inclinations. Moreover, the amplitudes of the eigenmodes characterising the current secular evolution of the eccentricities of Jupiter and Saturn would not be reproduced correctly, and only one eigenmode is excited by resonance-crossing. We show that, at the very least, encounters between Saturn and one of the ice giants (Uranus or Neptune) need to have occurred to reproduce the current secular properties of the giant planets, in particular the amplitude of the two strongest eigenmodes in the eccentricities of Jupiter and Saturn.

Explaining why the uranian satellites have equatorial prograde orbits despite the large planetary obliquity

We show that the existence of prograde equatorial satellites is consistent with a collisional tilting scenario for Uranus. In fact, if the planet was surrounded by a proto-satellite disk at the time of the tilting and a massive ring of material was temporarily placed inside the Roche radius of the planet by the collision, the proto-satellite disk would have started to precess incoherently around the equator of the planet, up to a distance greater than that of Oberon. Collisional damping would then have collapsed it into a thin equatorial disk, from which the satellites eventually formed. The fact that the orbits of the satellites are prograde requires Uranus to have had a non-negligible initial obliquity (comparable to that of Neptune) before it was finally tilted to 98°.

A mapping for nonconservative systems

We present a method to construct a mapping for perturbed systems, in which the perturbations do not need to be conservative. We use a variation of Wisdom and Holmans method, where the dissipative term is placed together with the other perturbative terms. The method is applied for two dissipative systems: one including gas drag and the other including Poynting-Robertson drag. We compare the results with those obtained by Malhotras mapping. Because the dissipative part in our method is treated as a regular perturbative term, there is no need for analytical developments of the nonconservative terms. This is a great advantage in itself and this also allows for a fast performance of the integrator.