Libration Amplitude Research Articles

A numerical study of the 2:1 planetary resonance

We numerically explore the long-term stability of planetary orbits locked in a 2:1 mean motion resonance for a wide range of planetary mass ratios and orbital parameters. Our major tool is Laskar's frequency map analysis. Regions of low diffusion rate are outlined in a phase space defined by the two planetary eccentricities and the libration amplitude of a critical resonance argument. Resonant systems that are dynamically stable on a long timescale must lie within these regions. The resonance locking between planets in high eccentric orbits may be destroyed by mutual close encounters. We discuss various dynamical protection mechanisms related to the resonant configuration, among which is the well-known apsidal corotation. In the case of moderate-to-low eccentricities, we find that apsidal circulators, little discussed till now, are very common among stable orbits. We also map the different types of resonant behaviour predicted by analytical theories in the phase space.

On the 2 : 1 Orbital Resonance in the HD 82943 Planetary System

We present an analysis of the HD 82943 planetary system based on a radial velocity data set that combines new measurements obtained with the Keck telescope and the CORALIE measurements published in graphical form. We examine simultaneously the goodness of fit and the dynamical properties of the best-fit double-Keplerian model as a function of the poorly constrained eccentricity and argument of periapse of the outer planet's orbit. The fit with the minimum ? is dynamically unstable if the orbits are assumed to be coplanar. However, the minimum is relatively shallow, and there is a wide range of fits outside the minimum with reasonable ?. For an assumed coplanar inclination i = 30? (sin i = 0.5), only good fits with both of the lowest order, eccentricity-type mean-motion resonance variables at the 2?:?1 commensurability, ?1 and ?2, librating about 0? are stable. For sin i = 1, there are also some good fits with only ?1 (involving the inner planet's periapse longitude) librating that are stable for at least 108 yr. The libration semiamplitudes are about 6? for ?1 and 10? for ?2 for the stable good fit with the smallest libration amplitudes of both ?1 and ?2. We do not find any good fits that are nonresonant and stable. Thus, the two planets in the HD 82943 system are almost certainly in 2?:?1 mean-motion resonance, with at least ?1 librating, and the observations may even be consistent with small-amplitude librations of both ?1 and ?2.

Guest—host complexes of spin-labeled indoles with cyclodextrins in the solid phase: an esr study

Inclusion complexes of spin-labeled pyrrolidine-(1) and piperidine-containing (2) indole derivatives with β-cyclodextrin and γ-cyclodextrin (CD) were prepared in the solid phase and studied by ESR in a wide temperature interval. For all complexes and free spin probes in solvents of different polarity, local environment polarities of the NO group of the guest molecules were determined from the outer extrema separations in the ESR spectra measured at 77 K. From analysis of the Saturation Transfer (ST) ESR spectra and temperature dependences of linear ESR spectra of the complexes it follows that both guest molecules in γ-CD undergo rapid librations. The libration amplitude of the p-orbit axis of the NO group in molecule 2 increases with temperature and reaches about 16° at 333 K. The ESR lineshape of the β-CD complexes depends on the spin probe/β-CD molar ratio (ρ) even at ρ < 0.01. Lineshape analysis of the spectra recorded at different ρ showed that they consist of two components, one of them corresponding to strong spin-spin interaction between guest molecules and the other corresponding to almost absence of this interaction. The spectral components can be attributed to microphases of the complexes and to isolated complexes in the β-CD matrix, respectively. Simulation of the ST ESR and linear ESR spectra of the magnetically diluted complexes showed that the guest motion inside the CD cavity is better described by rotational jumps rather than Brownian diffusion in an orientation potential. In the temperature range 238—333 K, the rotational frequencies of 1 and 2 are in intervals 1.8·107−6·107 s−1 and 4·107−1.3·108 s−1, respectively. The rotation occurs over the whole solid angle. Significant differences in the character of molecular dynamics in the γ-CD and β-CD complexes can be explained by different stoichiometry, namely, 1: 1 for the former and 2: 1 for the latter and by different orientation of guest molecules in the complexes. In both cyclodextrins the rotational mobility of molecules 2 is higher than that of 1 owing to intramolecular conformational transitions in the piperidine ring of 2 and steric hindrances produced by the methyl group in 1.

Polymorphs of p-nitrophenol as studied by variable-temperature X-ray diffraction and calorimetry: comparison with m-nitrophenol

Crystal structures of two polymorphic forms of p-nitrophenol have been determined at several temperatures between 120 and 375 K. The thermal expansion tensor has been determined for both polymorphs. The rigid-body mean-square amplitudes of molecular translations and librations and the amplitudes of the internal torsions of the nitro group have been calculated at different temperatures. Differential scanning calorimetry was used to find the temperature and enthalpy of the polymorphic transformation. The results were compared with those recently obtained for m-nitrophenol polymorphs. Some conclusions concerning the polymorphism of p- and m-nitrophenols are presented.

The Instability of Venus Trojans

Using Laskar's frequency map analysis, we investigate the long-term stability of Venus Trojan orbits. By measuring the diffusion rate of proper frequencies, we outline in phase space the most stable region. It is located between 5° and 10° in proper inclination with libration amplitudes ranging from 50° to 110°. The proper eccentricity is lower than 0.13. The stable region is surrounded by secular resonances that destabilize orbits on short timescales. The dynamical half-life of orbits within the stable region is about 6 × 108 yr. The Yarkovsky effect further reduces the dynamical lifetime. Therefore, primordial Venus Trojans, if any, cannot have survived until the present. Transient Trojans, on the other hand, cannot be excluded.

Neutron powder diffraction study of the orientational order–disorder phase transition in calcite, CaCO3

Neutron powder diffraction studies of calcite on heating towards the orientational order–disorder phase transition show that the phase transition is not a simple analogue of an Ising-like transition, but more similar to a rotational analogue of Lindemann melting. The transition is precipitated by the librational amplitude of the carbonate molecular ions exceeding a critical value rather than a result of a statistical entropy of ‘wrong’ orientations. Using tested interatomic potentials the single-particle orientational potential and nearest-neighbour orientational interactions have been calculated.

Practical NMR relaxation for chemists

Preface. 1. How and Why Nuclei Relax. 1.1. Nucleus in the magnetic field. 1.2. Spin--lattice and spin--spin nuclear relaxation. 1.2.1. Macroscopic magnetization: relaxation times T1 and T2. 1.3. Molecular motions as reason of nuclear relaxation. 1.3.1. Correlation times and activation energies of Molecular Motions. 1.3.2. Isotropic and anisotropic molecular motions. 1.4. Bibliography for Chapter 1. 2. How to Measure the NMR Relaxation Times. 2.1. Exponential and non--exponential nuclear relaxation. 2.2. Measurements of spin--lattice relaxation times. 2.3. Measurements of selective and bi--selective T1 times. 2.4. Determinations of T1( and T2 times. 2.5. Preparation of samples for relaxation experiments. 2.6. Bibliography to Chapter 2. 3. Errors in Determinations of Relaxation Times. 3.1. Instrumental errors. 3.2. Incorrect parameters for T1, T2 measurements and T1, T2 calculations. 3.3 Coupled nuclear relaxation. 3.4. Chemical exchanges. 3.5. Bibliography to Chapter 3. 4. NMR Relaxation by Dipole--Dipole and Quadrupole Interactions. 4.1. The intramolecular dipole--dipole relaxation: homo-- and hetero--nuclear dipolar coupling and the spectral density function. 4.2. Haw to reveal the presence of the dipolar mechanisms. 4.2.1. NOE as a test for dipole--dipole nuclear relaxation. 4.2.2. Evaluations of the dipolar contributions from selective and non--selective T1 times. 4.3. Intermolecular dipole--dipole interactions. 4.4. Electric field gradients at quadrupolar nuclei. 4.5. Nuclear quadrupole coupling constant as a measure of the electric field gradient. 4.6. Quadrupole relaxation. 4.7. Bibliography to Chapter 4. 5. Relaxation by Chemical Shift Anisotropy, Spin--Rotation Relaxation, Scalar Relaxation of the Second Kind and Cross--Mechanisms. 5.1. Relaxation by chemical shift anisotropy. 5.2. Spin--rotation relaxation. 5.3. Interference mechanisms of nuclear relaxation. 5.4. The scalar relaxation of the second kind. 5.5 Bibliography to Chapter 5. 6. Nuclear Relaxation in Molecular Systems with Anisotropic Motions. 6.1. Spin--lattice nuclear relaxation in ellipsoid molecules: Temperature dependences of T1times. 6.2. How to reveal anisotropic molecular motions in solutions. 6.3. Nuclear relaxation in the presence of correlation time distributions. 6.4. Bibliography to Chapter 6. 7. 1H T1 Relaxation Diagnostics in Solutions. 7.1. Revealing weak intermolecular interactions by T1 time measurements in solutions. 7.2. T1 studies of exchanges in simple molecular systems. 7.3. Structural 1H T1 criterion. 7.4. Partially--relaxed NMR spectra. 7.5. Bibliography to Chapter 7. 8. Internuclear Distances from the 1H T1 Relaxation Measurements in Solutions. 8.1. X...H distances: metal -- hydride bond lengths. 8.1.1. How to determine metal--hydride bond lengths by standard 1H T1 measurements. 8.1.2. Metal--hydride bond lengths by 1H T1sel and 1H T1min times measurements. 8.2. Proton--proton distances by standard T1 measurements. 8.3. H--H distances from T1sel / T1bis measurements. 8.4. H--H distances in intermediates. 8.5. Analyzing the errors in 1H T1 determinations of internuclear distances. 8.6. Bibliography to Chapter 8. 9. Deuterium Quadrupole Coupling Constants from 2H T1 Relaxation Measurements in Solutions. 9.1. How to determine DQCC values. 9.2. DQCC values from the 2H T1 times measurements in solutions (fast motional regime). 9.3. DQCC values via 2H T1min measurements in solutions. 9.4. Errors in DQCC determinations. 9. 5. Bibliography to Chapter 9. 10. Spin--Lattice 1H and 2H Relaxation in Mobile Groups. 10.1. 1H T1 times and H--H distances in the presence of fast vibrations and librations. 10.2. 1H T1 times and H--H distances in the presence of fast rotational diffusion. 10.3. The spectral density function for high--amplitude librations. 10.4. 900--jumps in four--fold potential. 10. 5. Deuterium spin--lattice NMR relaxation in mobile molecular fragments. 10.6. Bibliography to Chapter 10. 11. Relaxation of Nuclei Other Than 1H and 2H) and Specific Relaxation Experiments. 11.1. Chemical shift anisotropies and nuclear quadrupole coupling constants from T1 times of heavy nuclei in solutions. 11.2. Multinuclear relaxation approaches to complexation, association and H--bonding. 11.3. Na relaxation in solutions of complex molecular systems. 11. 4. Character of molecular motions from 17O and 2H T1 relaxation in solutions. 11.5. 2D T1 and T1( NMR experiments. 11.6. Chemical exchanges in complex molecular systems from 15N nuclear relaxation in solutions. 11.7. R1/R2 method. 11.8. Cross--correlation relaxation rates and structure of complex molecular systems in solutions. 11.9. Variable -- field relaxation experiments. 11.10. Bibliography to Chapter 11. 12. Paramagnetic NMR Relaxation. 12.1. Theoretical basics of the paramagnetic relaxation enhancement. 12.2. Paramagnetic relaxation rate enhancements in the presence of chemical exchanges. 12.3. Structural applications of paramagnetic relaxation rate enhancements. 12.4. Kinetics of ligand exchanges via paramagnetic relaxation rate enhancements. 12.5. Longitudinal electron relaxation time in paramagnetic centers from variable--high field NMR experiments. Bibliography. Concluding Remarks. Index.

S/2004 S 5: A new co-orbital companion for Dione

We describe the astrometry and dynamics of S/2004 S 5 (provisionally named Polydeuces), a new satellite of Saturn, discovered in images recorded using the Imaging Science Subsystem (ISS) of the Cassini spacecraft. A total of 119 observations are presented, 101 of which were obtained with Cassini's narrow-angle camera and 18 using the wide-angle camera. Orbit determinations, based on a uniformly precessing ellipse model clearly showed S/2004 S 5 to be a possible co-orbital of Dione. By fitting a numerical solution of the full equations of motion to the available observations, we show that S/2004 S 5 is executing tadpole motion around Dione's L 5 Lagrangian point, with a libration amplitude varying between + 21.16 ± 0.03 ° and − 31.68 ± 0.06 ° over a period of 791.3 ± 0.1 days.

The free precession and libration of Mercury

An analysis based on the direct torque equations including tidal dissipation and a viscous core–mantle coupling is used to determine the damping time scales of O ( 10 5 ) years for free precession of the spin about the Cassini state and free libration in longitude for Mercury. The core–mantle coupling dominates the damping over the tides by one to two orders of magnitude for the plausible parameters chosen. The short damping times compared with the age of the Solar System means we must find recent or on-going excitation mechanisms if such free motions are found by the current radar experiments or the future measurement by the MESSENGER and BepiColombo spacecraft that will orbit Mercury. We also show that the average precession rate is increased by about 30% over that obtained from the traditional precession constant because of a spin–orbit resonance induced contribution by the C 22 term in the expansion of the gravitational field. The C 22 contribution also causes the path of the spin during the precession to be slightly elliptical with a variation in the precession rate that is a maximum when the obliquity is a minimum. An observable free precession will compromise the determination of obliquity of the Cassini state and hence of C / M M R 2 for Mercury, but a detected free libration will not compromise the determination of the forced libration amplitude and thus the verification of a liquid core.

Magnetic Circular Dichroism and Absorption Spectra of Phosphinidene in Noble‐Gas Matrices

AbstractFor Abstract see ChemInform Abstract in Full Text.

Formal Integrals and Nekhoroshev Stability in a Mapping Model for the Trojan Asteroids

A symplectic mapping model for the co-orbital motion (Sandor et al., 2002, Cel. Mech. Dyn. Astr. 84, 355) in the circular restricted three body problem is used to derive Nekhoroshev stability estimates for the Sun-Jupiter Trojans. Following a brief review of the analytical part of Nekhoroshev theory, a direct method is developed to construct formal integrals of motion in symplectic mappings without use of a normal form. Precise estimates are given for the region of effective stability based on the optimization of the size of the remainder of the formal series. The stability region found for t = 1010 yrs corresponds to a libration amplitude D p = 10:6°. About 30% of asteroids with accurately known proper elements (Milani, 1993, Cel. Mech. Dyn. Astron. 57, 59), at low eccentricities and inclinations, are included within this region. This represents an improvement with respect to previous estimates given in the literature. The improvement is due partly to the choice of better variables, but also to the use of a mapping model, which is a simplification of the circular restricted three body problem.

Magnetic Circular Dichroism and Absorption Spectra of Phosphinidene in Noble-Gas Matrices

Electronic magnetic circular dichroism and absorption spectra are reported for the A 3pi(i) <-- X 3sigma- transitions of phosphinidene (PH) isolated in Ar, Kr, and Xe matrices at cryogenic temperatures (approximately 1.4-20 K) and over a range of magnetic field strengths (0-5 T). The results are analyzed by the method of moments, and parameters are extracted by fitting the experimental data to a model in which the A 3pi(i) term is split by spin-orbit (SO) coupling interactions, while the X 3sigma- term is split by spin-spin and higher-order SO coupling. The analysis indicates that, unlike the equivalent imidogen (NH) systems, ground-state PH radicals isolated in noble-gas matrices do not behave as free rotors. Trends in excited-state SO coupling constants are attributed to the external heavy-atom effect and guest-host orbital mixing. It is tentatively concluded that librational amplitudes of the guest radical decrease in the order Ar > Xe > Kr, probably as a consequence of competition between stronger guest-host interactions and larger matrix sites in heavier hosts.

Librational Motion of Spin-Labeled Lipids in High-Cholesterol Containing Membranes from Echo-Detected EPR Spectra

Two-pulse, echo-detected (ED) electron paramagnetic resonance (EPR) spectroscopy was used to study the librational motions of spin-labeled lipids in membranes of dipalmitoylphosphatidylcholine + 50 mol % cholesterol. The temperature dependence, over the range 77–240 K, and the dependence on position of spin-labeling in the sn-2 chain ( n = 5, 7, 10, 12, and 14) of the phospholipid, were characterized in detail. The experimental ED-spectra were corrected for instantaneous spin diffusion arising from static spin-spin interactions, by using spectra recorded at 77 K, where motional contributions are negligible. Simulations according to a model of rapid, small-amplitude librations about an axis whose direction is randomly distributed are able to describe the experimental spectra. Calibrations, in terms of the amplitude-correlation time product, 〈 α 2〉 τ c, were constructed for diagnostic spectral line-height ratios at different echo delay times, and for relaxation spectra obtained from the ratio of ED-spectra recorded at two different echo delays. The librational amplitude, 〈 α 2〉, was determined for a spin label at the 14-C position of the lipid chain from the partially motionally averaged hyperfine splitting in the conventional EPR spectra. The librational correlation time, τ c, which is deduced from combination of the conventional and ED-EPR results, lies in the subnanosecond regime and depends only weakly on temperature. The temperature dependence of the ED-EPR spectra arises mainly from an increase in librational amplitude with increasing temperature, and position down the lipid chain. A gradual transition takes place at higher temperatures, from a situation in which segmental torsional librations are cumulative, i.e., the contributions of the individual segments add up progressively upon going down the chain, to one of concerted motion only weakly dependent on chain position. Such librational motions are important for glass-like states and are generally relevant to high lipid packing densities, e.g., in cholesterol-containing raft domains and condensed complexes.

Comparative rotational dynamics of the Moon, Mercury and Titan

Constant and periodic variations in the gravitational coefficients of second harmonics of gravitational potential of the Moon, Mercury and Titan caused by the attraction of their external bodies are described analytically. On the base of a plane model of translatory–rotary motion of the above-mentioned elastic bodies on elliptical orbits the main variations in the coefficients C 20, C 22 and S 22 with periods that are a multiple of the orbit period have been obtained and evaluated. Dynamic effects in resonant rotation of the Moon, Mercury and Titan considered as an unchangeable non-spherical bodies are studied on the basis of two simple models: firstly, the plane motion on the unperturbed elliptic orbit; secondly, the rotation of Titan on the precessing circular orbit. Peale's hydrostatic values of Titan's gravitational field parameters C 20 and C 22 (Peale, 1973 Peale, SJ. (1973). Rev. Geophys. Space Phys., 11: 767[Crossref], [Web of Science ®] , [Google Scholar]) and two sets of their model values are used in paper to evaluate the unperturbed, forced and resonant effects in the rotation of Titan. For Mercury we also use an average equilibrium model of elastic body rotating in resonance 3:2 with an orbital motion. Another model of Mercury is based on the modern radio-location data (Peale et al., 2004). For the Moon, well-known dynamical characteristics are used. Obtained Cassini's positional parameters, amplitudes of periodic librations in longitude, periods of resonant librations and other characteristics of studied celestial bodies are given in mutual comparison.

Nekhoroshev stability estimates for different models of the Trojan asteroids

Estimates of the region of Nekhoroshev stability of Jupiter's Trojan asteroids are obtained by a direct (i.e. without use of the normal form) construction of formal integrals near the Lagrangian elliptic equilibrium points. Formal integrals are constructed in the Hamiltonian model of the planar circular restricted three body problem (PCRTBP), and in a mapping model (Sandor et al. 2002) of the same problem for small orbital eccentricities of the asteroids. The analytical estimates are based on the calculation of the size of the remainder of the formal series by a computer program. An analysis is made of the accumulation of small divisors in the series. The most important divisors introduce competing Fourier terms with sizes growing at similar rates as the order of truncation increases. This makes impossible to improve the estimates by considering nearly resonant forms of the formal integrals for particular near-resonances. Improved estimates were obtained in a mapping model of the PCRTBP. The main source of improvement is the use of better variables (Delaunay). Our best estimate represents a maximum libration amplitude . This is a quite realistic value which demonstrates the usefulness of Nekhoroshev theory.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Estimating Mercury's 88-day libration amplitude from orbit

In this paper it is derived that the libration of Mercury can be described by γ=Φ 0 sin M+Φ 0/K sin 2M where Φ 0 is the unknown libration amplitude, M is Mercury's mean anomaly and K=−9.483. Φ 0 can be determined by comparing pairs of images of the same landmarks taken by an orbiter at different positions of Mercury. If the angle between the orbit plane of a polar orbiter and Mercury's line of periapsis is between −60° and 60° and if one landmark at the equator is imaged per day with a relative precision of 1.6 arcsec , then the libration amplitude can be determined in two Mercury years (176 days) with an accuracy of 1 arcsec or better, which is sufficient to answer the question whether Mercury has a solid or fluid core.

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.

Stability of Jupiter Trojans investigated using frequency map analysis: the MATROS project

Using the frequency map analysis (FMA) method we investigate the stability properties of Trojan-type orbits in the proximity of the L4 and L5 Lagrangian points of Jupiter. This study is part of the MATROS project. The orbits of about 2 × 104 virtual Trojans with random initial conditions have been computed numerically and for each body the diffusion rate in frequency space has been determined by spectral analysis. The diffusion portraits show where stable orbits are located in the space of proper elements for different values of inclination. For low inclined orbits we reproduce the stability region outlined by Levison, Shoemaker & Shoemaker and, due to our fast sampling capability, we find additional resonant features in the libration amplitude versus proper eccentricity space. At higher inclinations, the stability region gradually shrinks and it disappears for inclinations of about 40°. The maximal Lyapunov characteristic exponent is computed for a limited number of Trojan orbits in our sample and the predictions concerning the chaotic behaviour of each orbit are compared with those given by the FMA method. A good agreement is obtained and the value of the Lyapunov exponent may be used to tune the results of the FMA analysis. A synthetic secular theory for the proper frequencies of Jupiter Trojans is obtained by numerically fitting the outcome of the frequency map analysis.

A solar system survey of forced librations in longitude

Forced librations are periodic rotational rate variations due to gravitational interactions with an orbital partner. We have developed an analytic theory capable of calculating expected amplitudes of forced librations for nonresonant rotators as well as for bodies existing in a spin‐orbit resonance. The theory has been applied to 34 solar system bodies, including terrestrial planets, planetary satellites, and the asteroid Eros. Parameters governing libration amplitude are the body's orbital eccentricity, moment difference, and the ratio of its spin rate to its orbital rate. In each case the largest libration amplitude is associated with the forcing frequency 2 (p − 1) n, where n is the orbital mean motion and p is the spin/orbit rate ratio. This dominant frequency is simply semidiurnal as seen from the position of the torquing body. The maximum libration angular amplitude is 1.3 × 10−2 radians for Thebe, and the maximum mean equatorial displacement is 1.4 km for Mimas.

Resonance Occupation in the Kuiper Belt: Case Examples of the 52 and Trojan Resonances

As part of our ongoing Deep Ecliptic Survey (DES) of the Kuiper belt, we report on the occupation of the 1 : 1 (Trojan), 4 : 3, 3 : 2, 7 : 4, 2 : 1, and 5 : 2 Neptunian mean motion resonances (MMRs). The previously unrecognized occupation of the 1 : 1 and 5 : 2 MMRs is not easily understood within the standard model of resonance sweeping by a migratory Neptune over an initially dynamically cold belt. Among all resonant Kuiper belt objects (KBOs), the three observed members of the 5 : 2 MMR discovered by DES possess the largest semimajor axes (a ≈ 55.4 AU), the highest eccentricities (e ≈ 0.4), and substantial orbital inclinations (i ≈ 10°). Objects (38084) 1999HB12 and possibly 2001KC77 can librate with modest amplitudes of ~90° within the 5 : 2 MMR for at least 1 Gyr. Their trajectories cannot be explained by close encounters with Neptune alone, given the latter's current orbit. The dynamically hot orbits of such 5 : 2 resonant KBOs, unlike hot orbits of previously known resonant KBOs, may imply that these objects were preheated to large inclination and large eccentricity prior to resonance capture by a migratory Neptune. Our first discovered Neptunian Trojan, 2001QR322, may not owe its existence to Neptune's migration at all. The trajectory of 2001QR322 is remarkably stable; the object can undergo tadpole-type libration about Neptune's leading Lagrange (L4) point for at least 1 Gyr with a libration amplitude of 24°. Trojan capture probably occurred while Neptune accreted the bulk of its mass. For an assumed albedo of 12%–4%, our Trojan is ~130–230 km in diameter. Model-dependent estimates place the total number of Neptune Trojans resembling 2001QR322 at ~20–60. Their existence helps to rule out violent orbital histories for Neptune.