Large Libration Amplitudes Research Articles

Dynamic simulation of orientational disorder in organic crystals: methyl groups, trifluoromethyl groups and whole molecules.

Large amplitude librations of atomic groups or of entire molecules in their crystals are simulated using optimized intermolecular potentials and crystal structures deposited in the Cambridge Structural Database. The analysis proceeds by a simple static model in which reorientations take place in a fixed environment, or by Monte Carlo (MC) simulation of equilibria dotted by rotational defects, or eventually by full Molecular Dynamics (MD). The simplest approach provides a valuable qualitative preview, but MC and MD are becoming easily accessible to the general solid-state chemist thanks to the facilities of the newly developed Milano Chemistry Molecular Simulation (MiCMoS) platform. Their combined results offer a wealth of information on the behaviour of phenyl-methyl and phenyl-trifluoromethyl groups, almost invariably affected by rotational flipping, whose nature and consequences are discussed with respect to disorder modelling in the refinement of X-ray structures. Whole-body reorientation takes place in flat molecules, benzene being the well-known prototype, but also in a very large molecule like coronene. Molecular dynamics of rotations in the cyclohexa-1,4-diene crystal offer a spectacular picture of the energetic profiles with jumping times. The dynamic oscillations described here are seldom considered in the formulation of crystal `bonds' or of `synthon' stability.

The Great Inequality and the Dynamical Disintegration of the Outer Solar System

Abstract Using an ensemble of N-body simulations, this paper considers the fate of the outer gas giants (Jupiter, Saturn, Uranus, and Neptune) after the Sun leaves the main sequence and completes its stellar evolution. Due to solar mass loss—which is expected to remove roughly half of the star’s mass—the orbits of the giant planets expand. This adiabatic process maintains the orbital period ratios, but the mutual interactions between planets and the width of mean-motion resonances (MMR) increase, leading to the capture of Jupiter and Saturn into a stable 5:2 resonant configuration. The expanded orbits, coupled with the large-amplitude librations of the critical MMR angle, make the system more susceptible to perturbations from stellar flyby interactions. Accordingly, within about 30 Gyr, stellar encounters perturb the planets onto the chaotic subdomain of the 5:2 resonance, triggering a large-scale instability, which culminates in the ejections of all but one planet over the subsequent ∼10 Gyr. After an additional ∼50 Gyr, a close stellar encounter (with a perihelion distance less than ∼200 au) liberates the final planet. Through this sequence of events, the characteristic timescale over which the solar system will be completely dissolved is roughly 100 Gyr. Our analysis thus indicates that the expected dynamical lifetime of the solar system is much longer than the current age of the universe, but is significantly shorter than previous estimates.

Vibrational Response of Felodipine in the THz Domain: Optical and Neutron Spectroscopy Versus Plane-Wave DFT Modeling

We present a joint experimental and computational terahertz (THz) spectroscopy study of the most stable polymorph (form I) of an antihypertensive pharmaceutical solid, felodipine (FLD). The vibrational response has been analyzed at room temperature by combining optical (THz-TDS, FT-IR, THz-Raman) and neutron (INS) terahertz spectroscopy. With the challenging example of a large and flexible molecular solid, we illustrate the complementarity of the experimental techniques. We show how the results can be understood by employing ab initio modeling and discuss current progress in the field. To this end, we employ plane wave formulation of density functional theory (plane wave DFT) along with harmonic lattice dynamics calculations (HLD) and ab initio molecular dynamics (AIMD) simulations. Based on a comprehensive theoretical analysis, we discover an inconsistency in the commonly accepted structural model, which can be linked to a distinct librational dynamics of the side ester chains. As a result, only a moderate agreement with the experimental spectra can be achieved. We, therefore, propose an alternative structural model, effectively accounting for the influence of the large-amplitude librations and allowing for a comprehensive analysis of the vibrational resonances up to 4.5 THz. In that way, we illustrate the applicability of the computationally supported THz spectroscopy to detect subtle structural issues in molecular solids. While the provided structural model can be treated as a guess, the problem calls for further revision by means of high-resolution crystallography. The problem also draws a need of extending the THz experiments toward low-temperature conditions and single-crystal samples. On the other hand, the studied system emerges as a challenge for the DFT modeling, being extremely sensitive to the level of the theory used and the resulting description of the intermolecular forces. FLD form I can be, hence, considered as a testbed for the use of more sophisticated theoretical approaches, particularly relying on an advanced treatment of the van der Walls forces and going beyond zero-temperature conditions and harmonic approximation.

An Integrable Model for the Dynamics of Planetary Mean-motion Resonances

Abstract I consider the dynamics of mean-motion resonances (MMRs) between pairs of coplanar planets and derive a new integrable Hamiltonian model for the planets’ resonant motion. The new model generalizes integrable Hamiltonians previously derived for first-order resonances to the case of higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such truncated expansion is necessary to derive an integrable model. This enables the new model to accurately capture the dynamics of both first- and higher-order resonances for eccentricities up to orbit crossing. I demonstrate that predictions of the new integrable model agree well with numerical integrations of resonant planet pairs. Finally, I explore the secular evolution of resonant planets’ eccentricities. I show that the secular dynamics are governed by conservation of an angular-momentum-deficit-like quantity. I also demonstrate that secular frequencies depend on planets’ resonant libration amplitude and this generally gives rise to a secular resonance inside the MMR at large libration amplitudes. The integrable model derived in this work can serve as a framework for analyzing the dynamics of planetary MMRs in a wide variety of contexts.

Mean flow generation by an intermittently unstable boundary layer over a sloping wall

Direct numerical simulations (DNS) of the flow in various rotating annular confinements have been conducted to investigate the effects of wall inclination on secondary fluid motions due to an unstable boundary layer. The inner wall resembles a truncated cone (frustum) whose apex half-angle is varied from$18^{\circ }$to$0^{\circ }$(straight cylinder). The large inner radius$r_{1}$, the mean rotation rate$\unicode[STIX]{x1D6FA}_{0}$and the kinematic viscosity$\unicode[STIX]{x1D708}$were kept constant resulting in the constant Ekman number$E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{0}r_{1}^{2})=4\times 10^{-5}$. Flows were excited by time-harmonic modulation of the inner wall’s rotation rate (so-called longitudinal libration) by prescribing the amplitude$\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{0}$and the forcing frequency$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FA}_{0}$. By steepening the inner wall and hence reducing the effect of the local Coriolis force in the boundary layer three different flow regimes can be realized: a rotation-dominated, a libration-dominated and an intermediate regime. In the present study we focus on the libration-dominated regime. For small libration amplitudes (here$\unicode[STIX]{x1D700}=0.2$), a laminar Ekman–Stokes boundary layer (ESBL) is realized at the librating wall. With the aid of laminar boundary layer theory and DNS we show that the ESBL exhibits an oscillatory mass flux along the librating wall (Ekman property) and an oscillatory azimuthal velocity, which resembles a radially damped wave (Stokes property). For large libration amplitudes (here$\unicode[STIX]{x1D700}=0.8$), the DNS results exhibit an intermittently unstable ESBL, which turns centrifugally unstable during the prograde (faster) part of a libration period. This instability is due to the Stokes property and gives rise to Görtler vortices, which are found to be tilted with respect to the azimuth when the librating wall is at a finite angle relative to the axis of rotation. We show that this tilt is related to the Ekman property of the ESBL. This suggests that linear and nonlinear dynamics are equally important for this intermittent instability. Our DNS results indicate further that the Görtler vortices propagate into the fluid bulk where they generate an azimuthal mean flow. This mean flow is notably different from the mean flow driven in the case of the stable ESBL. A diagnostic analysis of the Reynolds-averaged Navier–Stokes (RANS) equations in the unstable flow regime hints at a competition between the radial and axial turbulent transport terms which act as generating and destructing agents for the azimuthal mean flow, respectively. We show that the balance of both terms depends on the wall inclination, that is, on the wall-tangential component of the Coriolis force.

NEPTUNE’S ORBITAL MIGRATION WAS GRAINY, NOT SMOOTH

ABSTRACT The Kuiper Belt is a population of icy bodies beyond the orbit of Neptune. The complex orbital structure of the Kuiper Belt, including several categories of objects inside and outside of resonances with Neptune, emerged as a result of Neptune’s migration into an outer planetesimal disk. An outstanding problem with the existing migration models is that they invariably predict excessively large resonant populations, while observations show that the non-resonant orbits are in fact common (e.g., the main belt population is ≃2–4 times larger than Plutinos in the 3:2 resonance). Here we show that this problem can be resolved if it is assumed that Neptune’s migration was grainy, as expected from scattering encounters of Neptune with massive planetesimals. The grainy migration acts to destabilize resonant bodies with large libration amplitudes, a fraction of which ends up on stable non-resonant orbits. Thus, the non-resonant-to-resonant ratio obtained with the grainy migration is higher, up to ∼10 times higher for the range of parameters investigated here, than in a model with smooth migration. In addition, the grainy migration leads to a narrower distribution of the libration amplitudes in the 3:2 resonance. The best fit to observations is obtained when it is assumed that the outer planetesimal disk below 30 au contained 1000–4000 Plutos. We estimate that the combined mass of Pluto-class objects in the original disk represented 10%–40% of the estimated disk mass ( ). This constraint can be used to better understand the accretion processes in the outer solar system.

Mechanisms for pressure-induced crystal-crystal transition, amorphization, and devitrification of SnI4.

The pressure-induced amorphization and subsequent recrystallization of SnI4 have been investigated using first principles molecular dynamics calculations together with high-pressure (119)Sn nuclear resonant inelastic x-ray scattering measurements. Above ∼8 GPa, we observe a transformation from an ambient crystalline phase to an intermediate crystal structure and a subsequent recrystallization into a cubic phase at ∼64 GPa. The crystalline-to-amorphous transition was identified on the basis of elastic compatibility criteria. The measured tin vibrational density of states shows large amplitude librations of SnI4 under ambient conditions. Although high pressure structures of SnI4 were thought to be determined by random packing of equal-sized spheres, we detected electron charge transfer in each phase. This charge transfer results in a crystal structure packing determined by larger than expected iodine atoms.

Inertial waves in a spherical shell induced by librations of the inner sphere: experimental and numerical results

We begin with an experimental investigation of the flow induced in a rotating spherical shell. The shell globally rotates with angular velocity Ω. A further periodic oscillation with angular velocity 0 ⩽ ω ⩽ 2Ω, a so-called longitudinal libration, is added on the inner sphere's rotation. The primary response is inertial waves spawned at the critical latitudes on the inner sphere, and propagating throughout the shell along inclined characteristics. For sufficiently large libration amplitudes, the higher harmonics also become important. Those harmonics whose frequencies are still less than 2Ω behave as inertial waves themselves, propagating along their own characteristics. The steady component of the flow consists of a prograde zonal jet on the cylinder tangent to the inner sphere and parallel to the axis of rotation, and increases with decreasing Ekman number. The jet becomes unstable for larger forcing amplitudes as can be deduced from the preliminary particle image velocimetry observations. Finally, a wave attractor is experimentally detected in the spherical shell as the pattern of largest variance. These findings are reproduced in a two-dimensional numerical investigation of the flow, and certain aspects can be studied numerically in greater detail. One aspect is the scaling of the width of the inertial shear layers and the width of the steady jet. Another is the partitioning of the kinetic energy between the forced wave, its harmonics and the mean flow. Finally, the numerical simulations allow for an investigation of instabilities, too local to be found experimentally. For strong libration amplitudes, the boundary layer on the inner sphere becomes unstable, triggering localized Görtler vortices during the prograde phase of the forcing. This instability is important for the transition to turbulence of the spherical shell flow.

Mechanism of order–disorder phase transition in phenanthrene crystals: an interpretation based on the rotational potential energy surface for a phenanthrene molecule in a phenanthrene crystal

Crystals of phenanthrene undergo an order–disorder phase change at 72°C. High temperature phase was found to be disordered with two possible molecular positions. In order to understand the fundamental mechanism of the phase transition, we have modeled the molecular movement within the phenanthrene crystal and estimated the rotational potential energy surface for a phenanthrene molecule, as it undergoes this flip-flop motion. The situation in phenanthrene crystal is explained by a model where the molecules exhibit large amplitude librations having been trapped in the potential wells. Fluctuations in the orientation of the neighboring molecules result in changes in the shape of the potential well and consequently giving rise to angular motions of molecules with large amplitude. The calculations show that the two minima of the potential energy surface are not equivalent, as a result, the occupancy of the two possible molecular positions will not be the same, hence the space group of the disorder phase will remain P21 and not change to P21/a.

Neptune Trojans and Plutinos: colors, sizes, dynamics, and their possible collisions

Neptune Trojans and Plutinos are two subpopulations of trans-Neptunian objects located in the 1:1 and the 3:2 mean motion resonances with Neptune, respectively, and therefore protected from close encounters with the planet. However, the orbits of these two kinds of objects may cross very often, allowing a higher collisional rate between them than with other kinds of trans-Neptunian objects, and a consequent size distribution modification of the two subpopulations. Observational colors and absolute magnitudes of Neptune Trojans and Plutinos show that i) there are no intrinsically bright (large) Plutinos at small inclinations, ii) there is an apparent excess of blue and intrinsically faint (small) Plutinos, and iii) Neptune Trojans possess the same blue colors as Plutinos within the same (estimated) size range do. For the present subpopulations we analyzed the most favorable conditions for close encounters/collisions and address any link there could be between those encounters and the sizes and/or colors of Plutinos and Neptune Trojans. We also performed a simultaneous numerical simulation of the outer Solar System over 1 Gyr for all these bodies in order to estimate their collisional rate. We conclude that orbital overlap between Neptune Trojans and Plutinos is favored for Plutinos with large libration amplitudes, high eccentricities, and small inclinations. Additionally, with the assumption that the collisions can be disruptive creating smaller objects not necessarily with similar colors, the present high concentration of small Plutinos with small inclinations can thus be a consequence of a collisional interaction with Neptune Trojans and such hypothesis should be further analyzed.

SECULAR EVOLUTION OF HD 12661: A SYSTEM CAUGHT AT AN UNLIKELY TIME

The eccentricity evolution of multiple planet systems can provide valuable constraints on planet formation models. Unfortunately, the inevitable uncertainties in the current orbital elements can lead to significant ambiguities in the nature of the secular evolution. Integrating any single set of orbital elements inadequately describes the full range of secular evolutions consistent with current observations. Thus, we combine radial velocity observations of HD 12661 with Markov Chain Monte Carlo sampling to generate ensembles of initial conditions for direct n-body integrations. We find that any mean motion resonances are quite weak and do not significantly impact the secular evolution, and that current observations indicate circulation or large amplitude libration of the periapses. The eccentricity of the outer planet undergoes large oscillations for nearly all of the allowed two-planet orbital solutions. This type of secular evolution would arise if planet c had been impulsively perturbed, perhaps due to strong scattering of an additional planet that was subsequently accreted onto the star. Finally, we note that the secular evolution implied by the current orbital configuration implies that planet c spends ~96% of the time following an orbit more eccentric than that presently observed. Either this system is being observed during a relatively rare state, or additional planets are affecting the observed radial velocities and/or the system's secular eccentricity evolution.

Optical absorption of methoxy and carboethoxy derivatives of 1,3-diphenyl-1 H-pyrazolo[3,4- b]quinoline

Paper deals with experimental investigations and quantum chemical calculations of the optical absorption spectra of methoxy and carboethoxy 1,3-diphenyl derivatives of the pyrazoloquinoline ([PQ]): 6-methoxy-1,3-dyphenil-[PQ], 6-methoxy-1,3-( p-methoxyphenyl)-[PQ], 6-methoxy-1-( p-methoxyphenyl)-[PQ] and 6-carboethoxy-1,3-diphenyl-[PQ]. The quantum chemical calculations are performed by means of the semiempirical quantum chemical methods (AM1 or PM3) applied to: (a) the equilibrium molecular conformation in vacuo ( T = 0 K); (b) the molecular dynamic (MD) trajectory ( T = 300 K) which includes the dynamics of a certain molecular fragment (moiety) only (fragmental MD simulations); or (c) the MD trajectory obtained for most general case within the total MD simulations at T = 300 K. The results of these calculations are compared with the measured spectra of the optical absorption. The quantum chemical simulations show that the dynamics of the methoxy or carboethoxy groups practically does not influence the absorption spectrum whereas the strongest its modification ( 300 < λ ≤ 360 nm) is found to be related with dynamics of phenyl(II) [Ph2] or phenyl(II)-methoxy [Ph2-MeO] moieties which are characterized by large libration amplitudes. At the same time, the total MD reproduces evidently much better the halfwidth of most absorption bands compared to ones observed in the measured spectra. Comparing the measured and calculated spectral positions of the absorption threshold the quantum chemical method PM3 gives the best agreement for all compounds.

Correlated MO Study of the Low-Barrier Intramolecular Motions in Donor−Acceptor Ethenes

Correlated MP2 and MCSCF MO calculations of several model push-pull ethenes in most cases indicate no great participation of excited singlet and triplet electronic configurations in either the minima or the transition structures for the suggested facilitated intramolecular rotation about the polarized C=C bond. This situation changes significantly only in molecules with sulfur atoms in the molecule as either donors or acceptors. The outstanding contribution of sulfur atoms as either donors or acceptors is a significant increase of push-pull ethene molecular polarizabilities. Thus, within the studied small series of mostly planar push-pull ethenes, polarizability appears a better indicator of rapid intramolecular motions about the C=C bond than straight polarity. Substituents with larger steric demands around the C=C bond are shown to likely preclude its complete turnaround, thus prompting a ramification of the interpretations of dynamic NMR phenomena in sterically constrained push-pull ethenes as large-amplitude librations resulting from strong rovibrational and relatively weak electronic coupling. These librations, as shown by complete vibrational mode analysis of corresponding rotational transition structures, cover in fact certain sectors of the intuitively suggested full rotations similar to those about C-C single bonds.

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=10 6 years allows about 35% of preexisting Neptune Trojans to survive to 5 τ, by which time the giant planets have essentially reached their final orbits. In contrast, slower migration with τ=10 7 years yields only a ∼5% probability of Neptune Trojans surviving to a time of 5 τ. 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≈11 f NT 1:1). However, for the slower migration, with τ=10 7 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=10 7 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.

On the Origin of the Trojan Asteroids: Effects of Jupiter's Mass Accretion and Radial Migration

We present analytic and numerical results which illustrate the effects of Jupiter's accretion of nebular gas and the planet's radial migration on its Trojan companions. Initially, we approximate the system by the planar circular restricted three-body problem and assume small Trojan libration amplitudes. Employing an adiabatic invariant calculation, we show that Jupiter's 30-fold growth from a 10 M ⊕ core to its present mass causes the libration amplitudes of Trojan asteroids to shrink by a factor of about 2.5 to ∼40% of their original size. The calculation also shows that Jupiter's radial migration has comparatively little effect on the Trojans; inward migration from 6.2 to 5.2 AU causes an increase in Trojan libration amplitudes of ∼4%. In each case, the area enclosed by small tadpole orbits, if made dimensionless by using Jupiter's semimajor axis, is approximately conserved. Similar adiabatic invariant calculations for inclined and eccentric Trojans show that Jupiter's mass growth leaves the asteroids' eccentricities and inclinations essentially unchanged, while 1 AU of inward migration causes an increase in both of these quantities by ∼4%. Numerical integrations confirm and extend these analytic results. We demonstrate that our predictions remain valid for Trojans with small libration amplitudes even when the asteroids have low, but nonzero, eccentricities and inclinations and/or Jupiter has an eccentricity similar to its present value. The integrations also show that Trojans with large libration amplitudes, including horseshoe orbits, are even more strongly affected by Jupiter's mass growth and radial migration than simple scaling from our analytic results would suggest. Further, the numerical runs demonstrate that Jupiter's predicted mass growth is sufficient to cause the capture of asteroids initially on horseshoe orbits into stable tadpole orbits. Thus, if Jupiter captured most of its Trojan companions before or while it accreted gas, as seems probable, then Jupiter's growth played a significant role in stabilizing Trojan objects by systematically driving them to lower libration amplitudes.

The 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°. (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 ν 16 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 ν 16 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 ν 16 secular resonance increase their orbital inclinations up to 20° while keeping their large libration amplitudes. We show in this paper that the ν 16 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 2ω̃ S−ω̃ J−ω̃ T (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 (10 5 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 ν 6 resonance may contribute to the instability of low-libration Saturn Trojan orbits on a much shorter time scale.

On a Possible Rotation State of (433) Eros

Due to its extremely prolate figure, Asteroid (433) Eros may exhibit an unusual nonprincipal axis rotation state. As a result of a relatively small difference between the maximum and intermediate moments of inertia, a small perturbation such as the gravitational torques experienced during a close planetary encounter could have induced a rather large amplitude libration about the longest axis. Observation of such a state will permit measurement of the moment of inertia ratios; this task may be possible once the NEAR spacecraft enters orbit around Eros in early 2000 to begin an extensive study of its composition and structure.

Secular Interactions of Coorbiting Objects

We study the dynamics of the three-body problem when two objects are in the 1:1 mean motion commensurability and can experience close encounters. Three orbit families are relevant to the dynamics: horseshoe orbits, passing orbits, and retrograde satellite orbits. The latter two families are present only when the relative eccentricity of the two objects is large in comparison to the width of their coorbital region. In the planar case, the three families are disjoint because the guiding center of the motion is stationary with respect to secular perturbations; as a result, collisional orbits are unstable. Orbit transitions between different families occur in three dimensions owing to the secular variations of the eccentricity and argument of pericenter ωr. Potentially colliding horseshoe orbits transit via eccentric retrograde satellite orbits in a secular mechanism where ωrlibrates around ±90°. Coorbital passing orbits are located at the Kozai resonances with 2ωrlibrating around either 0° or 180°. The former resonance is a dominant feature of the dynamics once passing orbits exist. The latter is found only at large values of the Jacobi constant. Large amplitude librations around the Kozai resonances involve the transition from a passing orbit to an association of horseshoe and retrograde satellite orbits and the secular periodic reversal of the relative semimajor axis of the passing orbit. The stability of these structures supports the existence of coorbiting material with the planets, especially Mercury, and favors a coorbital origin of the retrograde natural satellites.

Nebular Gas Drag and Planetary Accretion: II. Planet on an Eccentric Orbit

We study the trajectories of planetesimals whose orbits decay starward as a result of gas drag and are perturbed by the gravity of a massive planet on an eccentric orbit. Each planetesimal ultimately suffers one of three possible fates: (1) trapping in a mean motion resonance with the planet, (2) accretion by the planet, or (3) passage by the planet and continued orbital decay. At moderate to large planetary eccentricity, numerical 3-body integrations of the motion of a planetesimal in the solar nebula demonstrate that migrating planetesimals can become trapped in the 1/1 resonance. These bodies initially have large libration amplitudes (approaching 2π) which decay down to 0 at the trailing Lagrange point. With some combinations of drag rate and planetary eccentricity, over 15% of the planetesimals which encounter the planet are trapped in the 1/1 resonance. Bodies trapped in the this way could be the precursors of the Trojan asteroids. Migrating planetesimals can be caught in both pure Lindblad and combined Lindblad/corotation resonances exterior to the planet's orbit. Trapping has been found in several j/( j + k) resonances with k's ranging from 1 to 4. As one considers larger planetary eccentricities, corotation resonances become more important than Lindblad resonances, and (for a given drag rate) trapping can occur at higher k's and farther from the planet. At large planetary eccentricities, planetesimals can also be caught in ( j + 1)/ j Lindblad/corotation resonances interior to the planet. Interior trapping, which is dynamically forbidden in the case of a planet on a circular orbit, requires planetary eccentricity to increase both the planetesimal's semimajor axis and its eccentricity near conjunction to counter gas drag. Provided the planetesimal's and planet's apoapses are roughly aligned, and conjunction occurs while both bodies are approaching apoapse, then the planetesimal can become trapped in an interior resonance. The probability of a planetesimal avoiding accretion while migrating past the planet can be described as the probability of not being accreted in a single conjunction to the power of the number of conjunctions that a planetesimal has with the planet while passing through its feeding zone. In the nongravitating planet limit (which is valid for a low mean density planet on a highly eccentric orbit), the accretion probability increases with planetary eccentricity due to the growth of the planet's feeding zone. When the planet's gravity is considered using the 2+2-body approximation (which is valid for a dense planet on a moderately eccentric orbit), the accretion rate is nearly independent of planetary eccentricity. Numerical simulations show that at small planetary eccentricities, the 2+2-body approximation breaks down, and the accretion probability gradually increases with decreasing eccentricity in this regime.

The Effect of the Nebula on the Trojan Precursors

Numerical and analytical calculations demonstrate that small planetesimals exhibit long-term stability of libration about the L4 and L5 Lagrange equilibrium points at a proto-Jupiter's distance from the Sun despite continuous dissipation of energy due to gas drag in the primordial solar nebula. This stability may mean that most of the mass of the Trojan asteroids was collisionally captured into libration and accreted into large bodies while the solar nebula was still there. If this is so, the Trojans represent samples of the material that composed Jupiter's core before the addition of the large mass of volatiles leading to the current largely gaseous planet.For a 13-M⊕. Jupiter core that is assumed not to disturb the mass distribution in the solar nebula, dissipation of orbital energy while the planetesimal is further from the Sun than the stationary point in the frame rotating with Jupiter (or the elliptic fixed point in a surface of section if Jupiter's orbit is eccentric) decreases the amplitude of libration. However, dissipation while the planetesimal is closer to the Sun than this point increases the amplitude of libration. The effects of dissipation outside the stationary point nearly cancel the effects of the dissipation inside, resulting in either a very slow increase or decrease in the amplitude of libration depending on the details of the nebular model. Those models of the nebula leading to a larger ratio of drag forces equidistant outside and inside of Jupiter's orbital distance give the librating planetesimals a higher stability. By higher stability we mean that the planetesimal is more likely to remain permanently in the region with decreasing amplitude of libration, or if the amplitude is increasing, that the rate of increase is lower. The L4 stationary points of Trojan precursors experiencing nebular drag are located more than 60° in front of Jupiter, whereas the L5 stationary points are located less than 60° behind Jupiter. The separation of these stationary points from ±60° increases as the particle diameter d decreases or the drag acceleration is otherwise increased. For a 13-M⊕ Jupiter and a nominal nebular model the stationary points at the L4 equivalent reach a maximum angular separation of about 108° in front of Jupiter for d ≈ 30 m, where smaller planetesimals do not librate in front of Jupiter. In contrast, the stationary points may be located only a few degrees behind Jupiter for d ≈ 1 m. If Jupiter has a nonzero orbital eccentricity, the stability of the trailing, L5 planetesimals is increased, but the stability of the leading, L4 planetesimals is decreased relative to the zero eccentricity case. Even if libration amplitudes are increasing slowly, the planetesimals librate around their particular stationary points in the frame rotating with Jupiter for times which are significant fractions of the nebular lifetime. This means that as Jupiter's core grows, the gentle collisions among the accumulating, librating planetesimals will induce the growth of larger Trojans.When Jupiter's mass approaches its current value, an annular gap in the distribution of nebular material is believed to form. The reversal of the radial pressure gradient on the outside of the gap means that the planetesimals are accelerated from behind when on the outside of their libration trajectories. This reverses the trend toward smaller libration amplitudes on this part of the trajectory, and the libration amplitude is increased on all parts of the libration trajectory. Small planetesimals are quickly lost, although for d > 3 km lifetimes are probably longer than the remaining life of the nebula.The striking asymmetry in the response of the planetesimals to the nebular drag between the L4 and L5 points implies that there should be an asymmetry in the numbers or total mass of Trojans remaining trapped near each stability point after the nebula is dissipated. However, these results favor more material being trapped at the trailing Trojan point, which is apparently contrary to observation. Still, the collisional evolution of the Trojans over 4.6 × 109 years may have skewed the mass distributions in such a way as to mask what was there initially, and current observational uncertainties indicate that the current apparent dominance of the L4 Trojans is not definitive.A possible history of the Trojan swarms of asteroids would have most of the mass being accumulated in libration before the nebula was dispersed, with the elimination of large libration amplitudes and dispersion in orbital inclinations occurring afterward. The dominance of high-velocity collisions after the nebula was dispersed has probably resulted in a drastic change in the size distribution and a net loss in librating mass.