Origin of the structure of the Kuiper belt during a dynamical instability in the orbits of Uranus and Neptune

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 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 ∼27 AU and an eccentricity of ∼0.3, and that subsequently Neptune's 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.     

Mass of the Kuiper belt

The Kuiper belt includes tens of thousand of large bodies and millions of smaller objects. The main part of the belt objects is located in the annular zone between 39.4 and 47.8 au from the Sun; the boundaries correspond to the average distances for orbital resonances 3:2 and 2:1 with the motion of Neptune. One-dimensional, two-dimensional, and discrete rings to model the total gravitational attraction of numerous belt objects are considered. The discrete rotating model most correctly reflects the real interaction of bodies in the Solar system. The masses of the model rings were determined within EPM2017—the new version of ephemerides of planets and the Moon at IAA RAS—by fitting spacecraft ranging observations. The total mass of the Kuiper belt was calculated as the sum of the masses of the 31 largest trans-Neptunian objects directly included in the simultaneous integration and the estimated mass of the model of the discrete ring of TNO. The total mass is \((1.97 \pm 0.35)\times 10^{-2} \ m_{\oplus }\). The gravitational influence of the Kuiper belt on Jupiter, Saturn, Uranus, and Neptune exceeds at times the attraction of the hypothetical 9th planet with a mass of \(\sim 10 \ m_{\oplus }\) at the distances assumed for it. It is necessary to take into account the gravitational influence of the Kuiper belt when processing observations and only then to investigate residual discrepancies to discover a possible influence of a distant large planet.     

Transformation of Trojans into quasi-satellites during planetary migration and their subsequent close-encounters with the host planet

We use numerical integrations to investigate the dynamical evolution of resonant Trojan and quasi-satellite companions during the late stages of migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. Our migration simulations begin with Jupiter and Saturn on orbits already well separated from their mutual 2:1 mean-motion resonance. Neptune and Uranus are decoupled from each other and have orbital eccentricities damped to near their current values. From this point we adopt a planet migration model in which the migration speed decreases exponentially with a characteristic timescale τ (the e-folding time). We perform a series of numerical simulations, each involving the migrating giant planets plus test particle Trojans and quasi-satellites. We find that the libration frequencies of Trojans are similar to those of quasi-satellites. This similarity enables a dynamical exchange of objects back and forth between the Trojan and quasi-satellite resonances during planetary migration. This exchange is facilitated by secondary resonances that arise whenever there is more than one migrating planet. For example, secondary resonances may occur when the circulation frequencies, f, of critical arguments for the Uranus-Neptune 2:1 mean-motion near-resonance are commensurate with harmonics of the libration frequency of the critical argument for the Trojan and quasi-satellite 1:1 mean-motion resonance . Furthermore, under the influence of these secondary resonances quasi-satellites can have their libration amplitudes enlarged until they undergo a close-encounter with their host planet and escape from the resonance. High-resolution simulations of this escape process reveal that ≈80% of jovian quasi-satellites experience one or more close-encounters within Jupiter’s Hill radius (R_H) as they are forced out of the quasi-satellite resonance. As many as ≈20% come within R_H/4 and ≈2.5% come within R_H/10. Close-encounters of escaping quasi-satellites occur near or even below the 2-body escape velocity from the host planet. Finally, the exchange and escape of Trojans and quasi-satellites continues to as late as 6-9τ in some simulations. By this time the dynamical evolution of the planets is strongly dominated by distant gravitational perturbations between the planets rather than the migration force. This suggests that exchange and escape of Trojans and quasi-satellites may be a contemporary process associated with the present-day near-resonant configuration of some of the giant planets in our Solar System.     

A spatially resolved high spectral resolution study of Neptune’s stratosphere

Using TEXES, the Texas Echelon cross Echelle Spectrograph, mounted on the Gemini North 8-m telescope we have mapped the spatial variation of H₂, CH₄, C₂H₂ and C₂H₆ thermal-infrared emission of Neptune. These high-spectral-resolution, spatially resolved, thermal-infrared observations of Neptune offer a unique glimpse into the state of Neptune’s stratosphere in October 2007, L_S = 275.4° just past Neptune’s southern summer solstice (L_S = 270°). We use observations of the S(1) pure rotational line of molecular hydrogen and a portion of the ν₄ band of methane to retrieve detailed information on Neptune’s stratospheric vertical and meridional thermal structure. We find global-average temperatures of 163.8 ± 0.8, 155.0 ± 0.9, and 123.8 ± 0.8 K at the 7.0 × 10⁻³-, 0.12-, and 2.1-mbar levels with no meridional variations within the errors. We then use the inferred temperatures to model the emission of C₂H₂ and C₂H₆ in order to derive stratospheric volume mixing ratios (hence forth, VMR) as a function of pressure and latitude. There is a subtle meridional variation of the C₂H₂ VMR at the 0.5-mbar level with the peak abundance found at −28° latitude, falling off to the north and south. However, the observations are consistent within error to a meridionally constant C₂H₂ VMR of at 0.5 mbar. We find that the VMR of C₂H₆ at 1-mbar peaks at the equator and falls by a factor of 1.6 at −70° latitude. However, a meridionally constant VMR of at the 1-mbar level for C₂H₆ is also statistically consistent with the retrievals. Temperature predictions from a radiative-seasonal climate model of Neptune that assumes the hydrocarbon abundances inferred in this paper are lower than the measured temperatures by 40 K at 7 × 10⁻³ mbar, 30 K at 0.12 mbar and 25 K at 2.1 mbar. The radiative-seasonal model also predicts meridional temperature variations on the order of 10 K from equator to pole, which are not observed. Assuming higher stratospheric CH₄ abundance at the equator relative to the south pole would bring the meridional trends of the inferred temperatures and radiative-seasonal model into closer agreement.We have also retrieved observations of C₂H₄ emission from Neptune’s stratosphere using TEXES on the NASA Infrared Telescope Facility (IRTF) in June 2003, L_S = 266°. Using the observations from the middle of the planet and an average of the middle three latitude temperature profiles from the 2007 observations (9.5° of L_S later, the seasonal equivalent of 9.5 Earth days within Earth’s seasonal cycle), we infer a C₂H₄ VMR of at 1.5 × 10⁻³ mbar, a value that is 3.25 times that predicted by global-average photochemical models.     

On the origin of the Kuiper belt

I review the work that has been done so far aiming at the understanding of the origin of the Kuiper belt. Three peculiar characteristics of the Kuiper belt are used as constraints for the formation models. These are the unexpected dynamical excitation of the orbits, the Kuiper belt outer edge near the 1:2 resonance with Neptune and the mass paucity of the belt. Among the various scenarios proposed, those based on a primordial planetary migration give the best results. In particular, the Nice model is analyzed with respect to its coherence with the present characteristics of the belt. Special attention is given to the controversy on the origin of the Kuiper belt cold population.     

The Effect of Sweeping Secular Resonances on the Classical Kuiper Belt

Abstract The Kuiper Belt is a disk of small icy objects orbiting the Sun beyond Neptune. The region between 40-48AU in this disk is supposed to consist of dynamical “cold” objects on low-inclination orbits and is called the “Classical Kuiper Belt”. Recent observations reveal that there is a “hot” population with inclinations being as large as 30? residing in this region. Secular resonance sweeping, which took place in the late stage of formation of the planetary system when the residual nebula gas was dispersing, is a possible mechanism that can excite the orbits in this region. In this paper, we investigate in detail the excitation of orbital inclination by this mechanism. It is shown that the excitation depends sensitively on the angle δ between the midplane of the nebula gas and the invariable plane of the solar system. The excitation is very small when δ = 0?, but if the gas midplane coincides with the ecliptic, i.e. if δ ≈ 1.6?, then objects in the region of classical Kuiper belt can be excited to orbital inclinations as high as 30?, provided the nebula gas has the proper initial density and disperses at a proper rate. We also considered the orbital excitation by secular resonance sweeping with Jupiter on an inclined orbit and with migrating Jovian planets, and found the excitation is only slightly affected.     

Simulations for the Original Distribution of KBOs

Using the N-body dynamical model that includes the sun, the 8 planets, Pluto, UB313 and massless particles, we simulate the orbital evolution of 551 Kuiper Belt Objects (KBOs) with known parameters. The initial conditions of the simulations are the currently observed orbital parameters. The integration backtracks from now to -10×10⁸ yr. The results show that about 10×10⁸ years ago, more than 1/3 of the presently observed KBOs resided in the region of the present Kuiper main belt, a few were located inside the Neptune orbit, and the rest were beyond 50AU; and that about 4.5×10⁸ years ago, all the objects in the Kuiper main belt exhibited a rather good normal distribution, without so many objects concentrated in the Neptune's 3:2 resonance region, as at present time.     

A derivation of the luminosity function of the Kuiper belt from a broken power-law size distribution

We have derived a model of the Kuiper belt luminosity function exhibited by a broken power-law size distribution. This model allows direct comparison of the observed luminosity function to the underlying size distribution. We discuss the importance of the radial distribution model in determining the break diameter. We determine a best-fit break-diameter of the Kuiper belt size-distribution of 30<Db<90 km via a maximum-likelihood fit of our model to the observed luminosity function. We also confirm that the observed luminosity function for m(R)∼21-28 is consistent with a broken power-law size distribution, and exhibits a break at .     

Resonant and Secular Families of the Kuiper Belt

We review ongoing efforts to identify occupants of mean-motion resonances(MMRs) and collisional families in the Edgeworth–Kuiper belt. Directintegrations of trajectories of Kuiper belt objects (KBOs) reveal the 1:1(Trojan), 5:4, 4:3, 3:2 (Plutino), 5:3, 7:4, 9:5, 2:1 (Twotino), and 5:2 MMRsto be inhabited. Apart from the Trojan, resonant KBOs typically have largeorbital eccentricities and inclinations. The observed pattern of resonanceoccupation is consistent with resonant capture and adiabatic excitation bya migratory Neptune; however, the dynamically cold initial conditions priorto resonance sweeping that are typically assumed by migration simulationsare probably inadequate. Given the dynamically hot residents of the 5:2 MMRand the substantial inclinations observed in all exterior MMRs, a fraction ofthe primordial belt was likely dynamically pre-heated prior to resonancesweeping. A pre-heated population may have arisen as Neptune gravitationallyscattered objects into trans-Neptunian space. The spatial distribution of Twotinosoffers a unique diagnostic of Neptune's migration history. The Neptunian Trojanpopulation may rival the Jovian Trojan population, and the former's existence isargued to rule out violent orbital histories for Neptune. Finally, lowest-order seculartheory is applied to several hundred non-resonant KBOs with well-measured orbitsto update proposals of collisional families. No convincing family is detected.     

Exploring the 7:4 mean motion resonance—II: Scattering evolutionary paths and resonance sticking

In our preliminary study, we have investigated basic properties and dynamical evolution of classical TNOs around the 7:4 mean motion resonance with Neptune (a∼43.7 AU), motivated by observational evidences that apparently present irregular features near this resonance (see [Lykawka and Mukai, 2005a. Exploring the 7:4 mean motion resonance—I. Dynamical evolution of classical trans-Neptunian objects (TNOs). Space Planet. Sci. 53, 1175-1187]; hereafter “Paper I”). In this paper, we aim to explore the dynamical long-term evolution in the scattered disk (but not its early formation) based on the computer simulations performed in Paper I together with extra computations. Specifically, we integrated the orbital motion of test particles (totalizing a bit more than 10,000) placed around the 7:4 mean motion resonance under the effect of the four giant planets for the age of the Solar System. In order to investigate chaotic diffusion, we also conducted a special simulation with on-line computation of proper elements following tracks in phase space over 4-5 Gyr. We found that: (1) A few percent (1-2%) of the test particles survived in the scattered disk with direct influence of other Neptunian mean motion resonances, indicating that resonance sticking is an extremely common phenomenon and that it helps to enhance scattered objects longevity. (2) In the same region, the so-called extended scattered TNOs are able to form via very long resonance trapping under certain conditions. Namely, if the body spends more than about 80% of its dynamical lifetime trapped in mean motion resonance(s) and there is the action of a k+1 or (k+2)/2 mean motion resonance (e.g., external mean motion resonances with Neptune described as (j+k)/j with j=1 and 2, respectively). According to this hypothetical mechanism, 5-15% of current scattered TNOs would possess thus probably constituting a significant part of the extended scattered disk. (3) Moreover, considering hot orbital initial conditions, it is likely that the trans-Neptunian belt (or Edgeworth-Kuiper belt) has been providing members to the scattered disk, so that scattered TNOs observed today would consist of primordial scattered bodies mixed with TNOs that came from unstable regions of the trans-Neptunian belt in the past.Considering the three points together, our results demonstrated that the scattered disk has been evolving continuously since early times until present.     

Survival of Trojan-type companions of Neptune during primordial planet migration

We investigate the survivability of Trojan-type companions of Neptune during primordial radial migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. We adopt the usual planet migration model in which the migration speed decreases exponentially with a characteristic time scale τ (the e-folding time). We perform a series of numerical simulations, each involving the migrating giant planets plus ∼1000 test particle Neptune Trojans with initial distributions of orbital eccentricity, inclination, and libration amplitude similar to those of the known jovian Trojans asteroids. We analyze these simulations to measure the survivability of Neptune's Trojans as a function of migration rate. We find that orbital migration with the characteristic time scale τ=10⁶ years allows about 35% of preexisting Neptune Trojans to survive to 5τ, by which time the giant planets have essentially reached their final orbits. In contrast, slower migration with τ=10⁷ years yields only a ∼5% probability of Neptune Trojans surviving to a time of 5τ. Interestingly, we find that the loss of Neptune Trojans during planetary migration is not a random diffusion process. Rather, losses occur almost exclusively during discrete prolonged episodes when Trojan particles are swept by secondary resonances associated with mean-motion commensurabilities of Uranus with Neptune. These secondary resonances arise when the circulation frequencies, f, of critical arguments for Uranus-Neptune mean-motion near-resonances (e.g., f^UN_1:2, f^UN_4:7) are commensurate with harmonics of the libration frequency of the critical argument for the Neptune-Trojan 1:1 mean-motion resonance (f^NT_1:1). Trojans trapped in the secondary resonances typically have their libration amplitudes amplified until they escape the 1:1 resonance with Neptune. Trojans with large libration amplitudes are susceptible to loss during sweeping by numerous high-order secondary resonances (e.g., f^UN_1:2≈11f^NT_1:1). However, for the slower migration, with τ=10⁷ years, even tightly bound Neptune Trojans with libration amplitudes below 10° can be lost when they become trapped in 1:3 or 1:2 secondary resonances between f^UN_1:2 and f^NT_1:1. With τ=10⁷ years the 1:2 secondary resonance was responsible for the single greatest episode of loss, ejecting nearly 75% of existing Neptune Trojans. This episode occurred during the late stages of planetary migration when the remnant planetesimal disk would have been largely dissipated. We speculate that if the number of bodies liberated during this event was sufficiently high they could have caused a spike in the impact rate throughout the Solar System.     

The Common Origin of the High Inclination TNO's

Numerical integrations of the four major planets orbits inside a primordialplanetesimals disk show that a fraction of Neptune primordial scatteredobjects are deposited into the classical Kuiper Belt at Solar System age. Theseobjects exhibit inclinations as high as 40° and can account forpresent high inclinations population in the classical Kuiper Belt. The samemechanism can also originate high perihelion scattered objects like 2000 CR₁₀₅. The process that in the end produced such objects can be divided into two phases, a migration phase where nonconservative dynamics acted to producesome stable objects already at 10⁸ years and a nonmigrating phase that helped to establish some other objects as stable TNO's. Low inclination CKBO's have inprinciple an origin through the resonance sweeping process, although someresults from numerical integrations at least suggest a possible origin also fromthe primordial Neptune scattered population.     

A taxonomic survey of comet composition 1985-2004 using CCD spectroscopy

A summary is presented of our spectroscopic survey of comets extending for roughly 19 years from 1985 to 2004 comprising data for 92 comets of which 50 showed good emissions. All data were re-analyzed using consistent reduction techniques. Our observations of comets over several apparitions and comets observed over an extended period indicate no major changes in compositional classification. To our regret, no major unidentified cometary features were found in our surveyed spectral region of 5200-10400 Å. Absolute production rates for the dominant parent molecule H₂O and the daughter species C₂, NH₂ and CN are determined within the limits of the Haser model as are values for the dust continuum, Afρ. From these data, production rate ratios are calculated for C₂/H₂O, NH₂/H₂O, CN/H₂O and Afρ/H₂O. Excluding the odd Comets Yanaka (1988r), 43P/Wolf-Harrington and 19P/Borrelly, with unusual spectra, our set of comets exhibited relatively uniform composition. Detailed analyses of our data resulted in four taxonomic classes:

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Comets of typical composition (∼70%); exhibiting typical ratios with respect to water of C₂, NH₂, and CN.

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Tempel 1 type (∼22%); having a deficiency in C₂ but normal NH₂ abundance.

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G-Z type (∼6%); having both low C₂ and NH₂ ratios.

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The unusual object Yanaka (1988r) (∼2%?); no detectable C₂ or CN emission but normal NH₂.

It is uncertain whether there is a clear separation between the comets of typical composition and those with C₂ depletion, or whether the latter consists of a group showing a continuum of decreasing C₂/CN ratios. Our spectroscopic investigations result in a visual record of the various compositional classes, which are illustrated in a number of figures. Production rate comparisons with the comet photometry program of Schleicher and A'Hearn [A'Hearn, M.F., and 4 colleagues, 1995. Icarus 118, 223-270] for 13 comets in common yielded good agreement once the different scale lengths are taken into account. An investigation into the possible origin of our compositional groups with respect to dynamical families of comets shows that the Halley family exhibits essentially no C₂ depletion. These objects were presumably formed in the region of Saturn and Uranus and scattered into the Oort cloud. Comets formed in the space near Neptune, responsible for the scattered Kuiper Belt show a mixture of “typical” and C₂ depleted objects, while we associate comets formed in-situ in the classical Kuiper belt with our C₂ depleted group.     

An Upper Limit of Mass for a Stable Minor Planet in the Main Asteroid Belt

The stability of an imaginary planet located in the present main asteroid belt is studied with a 7-body model (Sun, Mars, Jupiter, Saturn, Uranus, Neptune and the imaginary planet). The fourth-order Hermite algorithm P(EC)³ is used, which has a very small secular energy error for the integration of periodic orbits with a constant time-step. The evolution of orbits is followed up to 10⁸ years. Our numerical results show that the low-order resonances with Jupiter can enhance the stability of the imaginary planet in some cases. The survival probability of the imaginary planet decreases with the planet mass. The upper limit of the imaginary planet's mass that can survive in the main belt is around 10²⁵ kg, i.e., about the Earth's mass.     

Masses of the Main Asteroid Belt and the Kuiper Belt from the Motions of Planets and Spacecraft

Dynamicalmass estimates for the main asteroid belt and the trans-Neptunian Kuiper belt have been found from their gravitational influence on the motion of planets. Discrete rotating models consisting ofmovingmaterial points have been used tomodel the total attraction fromsmall or as yet undetected bodies of the belts. The masses of the model belts have been included in the set of parameters being refined and determined and have been obtained by processing more than 800 thousand modern positional observations of planets and spacecraft. We have processed the observations and determined the parameters based on the new EPM2017 version of the IAA RAS planetary ephemerides. The large observed radial extent of the belts (more than 1.2 AU for the main belt and more than 8 AU for the Kuiper belt) and the concentration of bodies in the Kuiper belt at a distance of about 44 AU found from observations have been taken into account in the discrete models. We have also used individual mass estimates for large bodies of the belts as well as for objects that spacecraft have approached and for bodies with satellites. Our mass estimate for the main asteroid belt is (4.008 ± 0.029) × 10^?4/m_⊕ (3σ). The bulk of the Kuiper belt objects are in the ring zone from 39.4 to 47.8 AU. The estimate of its total mass together with the mass of the 31 largest trans-Neptunian Kuiper belt objects is (1.97 ± 0.30) × 10^?2m_⊕ (3σ), which exceeds the mass of the main asteroid belt almost by a factor of 50. The mass of the 31 largest trans-Neptunian objects (TNOs) is only about 40% of the total one.     

Discovery of a New Class of Predicted Resonance Objects beyond Jupiter

In the regions of mean diurnal motions between the orbits of Jupiter and Saturn, predicted earlier by the authors, five asteroids have been discovered that move in 1:2 and 2:3 Lindblad orbital resonances with Jupiter (external orbital commensurability) and in 2:1 resonance with Saturn (internal version of commensurability). In addition to this, in the precalculated stable resonance zones between the giant planets Saturn and Uranus, three objects have been found that possess third-order (2:5) orbital commensurability with Saturn; nine objects have been discovered between the orbits of Uranus and Neptune, whose mean motions are in 1:3 and 1:4 orbital resonances with Saturn, and more than 200 libration-stable objects, linked by lower-order orbital resonances with Neptune and Uranus have been found in the Kuiper belt.     

Inner Solar System dynamical analogs of plutinos

By studying orbits of asteroids potentially in 3:2 exterior mean motion resonance with Earth, Venus, and Mars, we have found plutino analogs. We identify at least 27 objects in the inner Solar System dynamically protected from encounter through this resonance. These are four objects associated with Venus, six with Earth, and seventeen with Mars. Bodies in the 3:2 exterior resonance (including those in the plutino resonance associated with Neptune) orbit the Sun twice for every three orbits of the associated planet, in such a way that with sufficiently low libration amplitude close approaches to the planet are impossible. As many as 15% of Kuiper Belt objects share the 3:2 resonance, but are poorly observed. One of several resonance sweeping mechanisms during planetary migration is likely needed to explain the origin and properties of 3:2 resonant Kuiper Belt objects. Such a mechanism likely did not operate in the inner Solar System. We suggest that scattering by the next planet out allows entry to, and exit from, 3:2 resonance for objects associated with Venus or Earth. 3:2 resonators of Mars, on the other hand, do not cross the paths of other planets, and have a long lifetime. There may exist some objects trapped in the 3:2 Mars resonance which are primordial, with our tests on the most promising objects known to date indicating lifetimes of at least tens of millions of years. Identifying 3:2 resonant systems in the inner Solar System permits this resonance to be studied on shorter timescales and with better determined orbits than has been possible to date, and introduces new mechanisms for entry into the resonant configuration.     

Planetary migration in a planetesimal disk: why did Neptune stop at 30 AU?

We study planetary migration in a gas-free disk of planetesimals. In the case of our Solar System we show that Neptune could have had either a damped migration, limited to a few AUs, or a forced migration up to the disk's edge, depending on the disk's mass density. We also study the possibility of runaway migration of isolated planets in very massive disk, which might be relevant for extra-solar systems. We investigate the problem of the mass depletion of the Kuiper belt in the light of planetary migration and conclude that the belt lost its pristine mass well before that Neptune reached its current position. Therefore, Neptune effectively hit the outer edge of the proto-planetary disk. We also investigate the dynamics of massive planetary embryos embedded in the planetesimal disk. We conclude that the elimination of Earth-mass or Mars-mass embryos originally placed outside the initial location of Neptune also requires the existence of a disk edge near 30AU.     

Possible consequences of absence of “jupiters” in planetary systems

The formation of the gas giant planets Jupiter and Saturn probably required the growth of massive 15 Earth-mass cores on a time scale shorter than the 10⁷ time scale for removal of nebular gas. Relatively minor variations in nebular parameters could preclude the growth of full-size gas giants even in systems in which the terrestrial planet region is similar to our own. Systems containing failed Jupiters, resembling Uranus and Neptune in their failure to capture much nebular gas, would be expected to contain more densely populated cometary source regions. They will also eject a smaller number of comets into interstellar space. If systems of this kind were the norm, observation of hyperbolic comets would be unexpected. Monte Carlo calculations of the orbital evolution of region of such systems (the Kuiper belt) indicate that throughout Earth history the cometary impact flux in their terrestrial planet regions would be 1000 times greater than in our Solar System. It may be speculated that this could frustrate the evolution of organisms that observe and seek to understand their planetary system. For this reason our observation of these planets in our Solar System may tell us nothing about the probability of similar gas giants occurring in other planetary systems. This situation can be corrected by observation of an unbiased sample of planetary systems.Paper presented at the Conference onPlanetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

Resonance sticking in the scattered disk

We investigate the dynamical evolution of trans-neptunian objects (TNOs) in typical scattered disk orbits (scattered TNOs) by performing simulations using several thousand particles lying initially on Neptune-encountering orbits. We explore the role of resonance sticking in the scattered disk, a phenomenon characterized by multiple temporary resonance captures (‘resonances’ refers to external mean motion resonances with Neptune, which can be described in the form r:s, where the arguments r and s are integers). First, all scattered TNOs evolve through intermittent temporary resonance capture events and gravitational scattering by Neptune. Each scattered TNO experiences tens to hundreds of resonance captures over a period of 4 Gyr, which represents about 38% of the object's lifetime (mean value). Second, resonance sticking plays an important role at semimajor axes , where the great majority of such captures occurred. It is noteworthy that the stickiest (i.e., dominant) resonances in the scattered disk are located within this distance range and are those possessing the lowest argument s. This was evinced by r:1, r:2 and r:3 resonances, which played the greatest role during resonance sticking evolution, often leading to captures in several of their neighboring resonances. Finally, the timescales and likelihood of temporary resonance captures are roughly proportional to resonance strength. The dominance of low s resonances is also related to the latter. In sum, resonance sticking has an important impact on the evolution of scattered TNOs, contributing significantly to the longevity of these objects.