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The orbits of Pluto's four small satellites (Styx, Nix, Kerberos, and Hydra) are nearly circular and coplanar with the orbit of the large satellite Charon, with orbital periods nearly in the ratios 3:1, 4:1, 5:1, and 6:1 with Charon's orbital period. These properties suggest that the small satellites were created during the same impact event that placed Charon in orbit and had been pushed to their current positions by being locked in mean-motion resonances with Charon as Charon's orbit was expanded by tidal interactions with Pluto. Using the Pluto-Charon tidal evolution models developed by Cheng et al. (2014), we show that stable capture and transport of a test particle in multiple resonances at the same mean-motion commensurability is possible at the 5:1, 6:1, and 7:1 commensurabilities, if Pluto's zonal harmonic $J_{2P} = 0$. However, the test particle has significant orbital eccentricity at the end of the tidal evolution of Pluto-Charon in almost all cases, and there are no stable captures and transports at the 3:1 and 4:1 commensurabilities. Furthermore, a non-zero hydrostatic value of $J_{2P}$ destroys the conditions necessary for multiple resonance migration. Simulations with finite but minimal masses of Nix and Hydra also fail to yield any survivors. We conclude that the placing of the small satellites at their current orbital positions by resonant transport is extremely unlikely., Comment: 22 pages, including 7 figures; accepted for publication in Icarus

Both Pluto and its satellite Charon have rotation rates synchronous with their orbital mean motion. This is the theoretical end point of tidal evolution where transfer of angular momentum has ceased. Here we follow Pluto's tidal evolution from an initial state having the current total angular momentum of the system but with Charon in an eccentric orbit with semimajor axis $a \approx 4R_P$ (where $R_P$ is the radius of Pluto), consistent with its impact origin. Two tidal models are used, where the tidal dissipation function $Q \propto$ 1/frequency and $Q=$ constant, where details of the evolution are strongly model dependent. The inclusion of the gravitational harmonic coefficient $C_{22}$ of both bodies in the analysis allows smooth, self consistent evolution to the dual synchronous state, whereas its omission frustrates successful evolution in some cases. The zonal harmonic $J_2$ can also be included, but does not cause a significant effect on the overall evolution. The ratio of dissipation in Charon to that in Pluto controls the behavior of the orbital eccentricity, where a judicious choice leads to a nearly constant eccentricity until the final approach to dual synchronous rotation. The tidal models are complete in the sense that every nuance of tidal evolution is realized while conserving total angular momentum - including temporary capture into spin-orbit resonances as Charon's spin decreases and damped librations about the same., Comment: 36 pages, including 18 figures; accepted for publication in Icarus

Tsiganis et al. (2005) have proposed that the current orbital architecture of the outer solar system could have been established if it was initially compact and Jupiter and Saturn crossed the 2:1 orbital resonance by divergent migration. The crossing led to close encounters among the giant planets, but the orbital eccentricities and inclinations were damped to their current values by interactions with planetesimals. Brunini (2006) has presented widely publicized numerical results showing that the close encounters led to the current obliquities of the giant planets. We present a simple analytic argument which shows that the change in the spin direction of a planet relative to an inertial frame during an encounter between the planets is very small and that the change in the obliquity (which is measured from the orbit normal) is due to the change in the orbital inclination. Since the inclinations are damped by planetesimal interactions on timescales much shorter than the timescales on which the spins precess due to the torques from the Sun, especially for Uranus and Neptune, the obliquities should return to small values if they are small before the encounters. We have performed simulations using the symplectic integrator SyMBA, modified to include spin evolution due to the torques from the Sun and mutual planetary interactions. Our numerical results are consistent with the analytic argument for no significant remnant obliquities., Comment: 14 pages, including 2 figures; uses AASTeX; a slightly expanded Section 2; accepted for publication in Icarus

(Abridged) The orbits of the recently discovered satellites of Pluto, S/2005 P2 and S/2005 P1, are significantly non-Keplerian, even if P2 and P1 have negligible masses, because the mass ratio of Charon-Pluto is ~0.1. We present an analytic theory with P2 and P1 treated as test particles. This analytic theory shows that the azimuthal periods of P2 and P1 are shorter than the Keplerian orbital periods and that the periapse and ascending node of each of the satellites precess at nearly equal rates in opposite directions. The deviation from Kepler's third law is already detected in the unperturbed Keplerian fit of Buie and coworkers. We also present direct numerical orbit integrations with different assumed masses for P2 and P1 within the ranges allowed by the albedo uncertainties. If the albedos are as high as that of Charon, the masses of P2 and P1 are sufficiently low that their orbits are well described by the analytic theory. There is at present no evidence that P2 has any significant epicyclic eccentricity. However, the orbit of P1 has a significant epicyclic eccentricity, and its prograde periapse precession with a period of 5300 days should be easily detectable. If the albedos are as low as that of comets, the large inferred masses induce significant variations in the epicyclic eccentricities and/or periapse longitudes on the 400-500-day timescales, due to the proximity of P2 and P1 to the 3:2 mean-motion commensurability. In fact, for the maximum inferred masses, P2 and P1 may be in the 3:2 mean-motion resonance, with the resonance variable involving the periapse longitude of P1 librating. Observations that sample the orbits of P2 and P1 well on the 400-500-day timescales should provide strong constraints on the masses of P2 and P1 in the near future., Comment: 25 pages, including 11 figures; uses AASTeX; minor changes; accepted for publication in Icarus

In determining Mercury's core structure from its rotational properties, the value of the normalized moment of inertia, $C/MR^2$, from the location of Cassini 1 is crucial. If Mercury's spin axis occupies Cassini state 1, its position defines the location of the state. The spin might be displaced from the Cassini state if the spin is unable to follow the changes in the state position induced by the variations in the orbital parameters and the geometry of the solar system. The spin axis is expected to follow the Cassini state for orbit variations with time scales long compared to the 1000 year precession period of the spin about the Cassini state because the solid angle swept out by the spin axis as it precesses is an adiabatic invariant. Short period variations in the orbital elements of small amplitude should cause displacements that are commensurate with the amplitudes of the short period terms. By following simultaneously the spin position and the Cassini state position during long time scale orbital variations over past 3 million years (Quinn {\it et al.}, 1991) and short time scale variations from JPL Ephemeris DE 408 (Standish, 2005) we show that the spin axis will remain within one arcsec of the Cassini state after it is brought there by dissipative torques. We thus expect Mercury's spin to occupy Cassini state 1 well within the uncertainties for both radar and spacecraft measurements, with correspondingly tight constraints on $C/MR^2$., Comment: 23 pages,12 figures, In press in Icarus

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/MR^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, Comment: 34 pages, 6 figures, AASTEX In press in Icarus

(Abridged) We investigate the dynamical evolution of coplanar hierarchical two-planet systems where the ratio of the orbital semimajor axes alpha=a_1/a_2 is small. The orbital parameters obtained from a multiple Kepler fit to the radial velocity variations of a star are best interpreted as Jacobi coordinates and Jacobi coordinates should be used in any analyses of hierarchical planetary systems. An approximate theory that can be applied to coplanar hierarchical two-planet systems with a wide range of masses m_j and orbital eccentricities e_j is the octupole-level secular perturbation theory (OSPT). The OSPT shows that if the ratio of the maximum orbital angular momenta, lambda \approx (m_1/m_2) alpha^{1/2}, for given a_j is approximately equal to a critical value lambda_{crit}, then libration of the difference in the longitudes of periapse, w_1-w_2, about either 0 or 180 deg. is almost certain, with possibly large amplitude variations of both e_j. We establish that the OSPT is highly accurate for systems with alpha<0.1 and reasonably accurate even for systems with alpha as large as 1/3, provided that alpha is not too close to a significant mean-motion commensurability or above the stability boundary. The HD 168443 system is not in a secular resonance and its w_1-w_2 circulates. The HD 12661 system is the first extrasolar planetary system found to have w_1-w_2 librating about 180 deg. The libration of w_1-w_2 and the large-amplitude variations of both e_j in the HD 12661 system are consistent with the analytic results on systems with lambda \approx lambda_{crit}. The HD 12661 system with the best- fit orbital parameters and sin i = 1 is affected by the close proximity to the 11:2 commensurability, but small changes in the outer orbital period can result in configurations that are not affected by mean-motion commensurabilities., Comment: 32 pages, including 8 figures; uses AASTeX v5.0; accepted for publication in ApJ

Understanding the origin of the orbital resonances of the Galilean satellites of Jupiter will constrain the longevity of the extensive volcanism on Io, may explain a liquid ocean on Europa, and may guide studies of the dissipative properties of stars and Jupiter-like planets. The differential migration of the newly formed Galilean satellites due to interactions with a circumjovian disk can lead to the primordial formation of the Laplace relation n_1 - 3 n_2 + 2 n_3 = 0, where the n_i are the mean orbital angular velocities of Io, Europa, and Ganymede, respectively. This contrasts with the formation of the resonances by differential expansion of the orbits from tidal torques from Jupiter., Comment: 13 pages, including 4 figures; uses scicite.sty

The 2:1 orbital resonances of the GJ 876 system can be easily established by the differential planet migration due to planet-nebula interaction. Significant eccentricity damping is required to produce the observed orbital eccentricities. The geometry of the GJ 876 resonance configuration differs from that of the Io-Europa pair, and this difference is due to the magnitudes of the eccentricities involved. We show that a large variation in the configuration of 2:1 and 3:1 resonances and, in particular, asymmetric librations can be expected among future discoveries., Comment: 4 pages, including 4 figures; uses newpasp.sty; to appear in Scientific Frontiers in Research on Extrasolar Planets (ASP Conf. Series)

(Abridged) A dynamical fit has placed the two planets about the star GJ 876 in coplanar orbits deep in 3 resonances at the 2:1 mean-motion commensurability with small libration amplitudes. The libration of both lowest order mean-motion resonance variables, theta_1 and theta_2, and the secular resonance variable, theta_3, about 0 deg. differs from the familiar geometry of the Io-Europa pair, where theta_2 and theta_3 librate about 180 deg. By considering a condition for stable simultaneous librations of theta_1 and theta_2, we show that the GJ 876 geometry results because of the large orbital eccentricities e_i, whereas the very small e_i in the Io-Europa system lead to the latter's geometry. Surprisingly, the GJ 876 resonance configuration remains stable for e_1 up to 0.86 and for amplitude of libration of theta_1 approaching 45 deg. with the current e_i. We find that inward migration of the outer planet of the GJ 876 system results in certain capture into the observed resonances if initially e_1 <0.06 and e_2<0.03 and the migration rate |(da_2/dt)/a_2| < 0.03(a_2/AU)^{-3/2} yr^{-1}. The bound on the migration rate is easily satisfied by migration due to planet-nebula interaction. If there is no eccentricity damping, eccentricity growth is rapid with continued migration within the resonance, with e_i exceeding the observed values after a further reduction in the semi-major axes a_i of only 7%. With eccentricity damping (de_i/dt)/e_i = -K|(da_i/dt)/a_i|, the e_i reach equilibrium values that remain constant for arbitrarily long migration within the resonances. The equilibrium e_i are close to the observed e_i for K=100 (K=10) if there is migration and damping of the outer planet only (of both planets). It is as yet unclear that planet-nebula interaction can produce the large value of K required to obtain the observed eccentricities., Comment: 23 pages, including 8 figures; uses AASTeX v5.0; minor additions; accepted for publication in ApJ

The probability of detecting a planetary companion of a lensing star during a microlensing event toward the Galactic center, averaged over all relevant event and galactic parameters, when the planet-star mass ratio $q=0.001$ has a maximum exceeding 10% at an orbit semimajor axis $a$ near 1.5 AU for a uniform distribution of impact parameters. The maximum probability is raised to more than 20% for a distribution of source-lens impact parameters that is determined by the efficiency of event detection. The averaging procedures are carefully defined, and they determinine the dependence of the detection probabilities on several properties of the Galaxy. The probabilities scale approximately as $\sqrt{q}$. A planet is assumed detectable if the perturbation of the single lens light curve exceeds $2/(S/N)$ for at least 20 consecutive photometric points sometime during the event. Two meter telescopes with 60 second integrations in I-band with high time resolution photometry throughout the duration of an ongoing event are assumed. The probabilities are derived as a function of $a$, where they remain significant for $0.6

The 2-Micron All Sky Survey (2MASS) (Skrutskie et al. 1997) and the DEep Near Infrared Survey of the southern sky (DENIS) (Epchtein et al. 1997) have revealed a heretofore unknown population of free brown dwarfs that has extended the local mass function down to as small as 0.01M_sun (Reid et al. 1999). If this local proportion of brown dwarfs extends throughout the Galaxy---in particular in the Galactic bulge---one expects an increase in the predicted fraction of short time scale microlensing events in directions toward the Galactic bulge. Zhao et al.(1996) have indicated that a mass function with 30-60% of the lens mass in brown dwarfs is not consistent with empirical microlensing data. Here we show that even the much lower mass fraction (~ 10%) of brown dwarfs inferred from the new discoveries appears inconsistent with the data. The added brown dwarfs do indeed increase the expected number of short time scale events, but they appear to drive the peak in the time scale frequency distribution to time scales smaller than that observed, and do not otherwise match the observed distribution. A reasonably good match to the empirical data (Alcock et al. 1996) is obtained by increasing the fraction of stars in the range 0.08

The dependence of microlensing time scale frequency distributions and optical depth toward the galactic center on galactic model parameters is explored in detail for a distribution of stars consisting of the Zhao (1996) bar and nucleus and the Bahcall and Soneira (1980) double exponential disk. The high sensitivity of these two microlensing measures to the circular velocity model, velocity dispersions, bulge mass, direction of the line of sight, bar axis orientation, star spatial distribution and the stellar mass function means no single galaxy property can be constrained very well without constraining most of the others. The model time scale frequency distributions are compared with that determined empirically by the MACHO group throughout. Although the MACHO empirical data are matched quite well with a nominal velocity model and with a mass function only of hydrogen burning stars that varies as m^{-2.2} to m^{-2.5} in the M star region, uncertainties in galactic structure, kinematics and content together with the paucity of published microlensing data preclude any claim of the model representing the real world. A variation of the mass function \sim m^{-1} in the M star region obtained from recent star counts, both local and in the galactic bulge, fails to yield a sufficient number of short time scale events compared to the MACHO data. The high sensitivity of the microlensing measures to the direction of the line of sight may mean that sufficient microlensing data is already in hand to constrain the bar distribution of stars. The procedure developed here for determining the time scale frequency distribution is particularly convenient for rapidly incorporating model changes as data from all sources continues to accumulate., Comment: 21 pages, 15 figures, Uses aas2pp4.sty

The statistical distribution of the masses of planets about stars between the Sun and the center of the galaxy is constrained to within a factor of three by an intensive search for planets during microlensing events. Projected separations in terms of the lens Einstein ring radius yield a rough estimate of the distribution of planetary semimajor axes with planetary mass. The search consists of following ongoing stellar microlensing events involving sources in the center of the galaxy lensed by intervening stars with high time resolution, 1% photometry in two colors in an attempt to catch any short time scale planetary perturbations of the otherwise smooth light curve. It is assumed that 3000 events are followed over an 8 year period, but with half of the lenses, those that are members of binary systems, devoid of planets. The remaining 1500 lenses have solar-system-like distributions of 4 or 5 planets. The expectations from the microlensing search are extremely assumption dependent with 56, 138, and 81 planets being detected for three sets of assumptions involving how the planetary masses and separations vary with lens mass. The events can be covered from 54% to 62% of the time on average by high time resolution photometry from a system of three or four dedicated two meter telescopes distributed in longitude, so 38% to 46% of the detectable small mass planets (very short perturbations of the light curve) will be missed. But perturbations comparable to a day in length means all of the detectable Jupiters and Saturns will in fact be detected as well as a large fraction of the Uranuses. Although meaningful statistics on planetary masses and separations can be inferred from such an intensive search, they, like the inferred data set, will be dependent on the assumed nature of the systems., Comment: 26 pages, LaTeX, uses aas2pp4.sty, 11 postscript figures, Submitted to Icarus

The proposed research to determine the probability of detecting extra solar planets from the ground with microlensing was completely accomplished during the grant period. A summary of the publications and their abstracts follow below. The publications include: 1) On microlensing rates and optical depth toward the Galactic center; 2) Newly discovered brown dwarfs not seen in microlensing timescale frequency distribution? 3) Probability of detecting a planetary companion during a microlensing event; 4) Comparison of a ground-based microlensing search for planets with a search from space; 5) Secular evolution of hierarchical planetary systems.

We compare a space-based microlensing search for planets, with a ground based microlensing search originally proposed by D. Tytler (Beichman, et al. 1996). Perturbations of microlensing light curves when the lens star has a planetary companion are sought by one wide angle survey telescope and an array of three or four followup narrow angle telescopes distributed in longitude that follow events with high precision, high time resolution photometry. Alternative ground based programs are considered briefly. With the four 2 meter telescopes distributed in longitude in the southern hemisphere in the Tytler proposal, observational constraints on a ground-based search for planets during microlensing events toward the center of the galaxy are severe. Probably less than 100 events could be monitored per year with high precision, high time resolution photometry with only about 42% coverage on the average regardless of how many events were discovered by the survey telescope. Statistics for the occurrence and properties for Jupiter-mass planets would be meaningful but relatively meager four years after the program was started, and meaningful statistics for Earth-mass planets would be non existent. In contrast, the 14,500 events in a proposed 4 year space based program (GEST = Galactic Exoplanet Survey Telescope) would yield very sound statistics on the occurrence, masses and separations of Jupiter-mass planets, and significant constraints on similar properties for Earth-mass planets. The significance of the Jupiter statistics would be to establish the frequency of planetary systems like our own, where terrestrial planets could exist inside the orbits of the giants.

Contents include a summary of publications followed by their abstracts titeled: 1. On microlensing rates and optical depth toward the Galactic center. 2. Newly discovered brown dwarfs not seen in microlensing timescale frequency distribution? 3. Origin and evolution of the natural satellites. 4. Probing the structure of the galaxy with microlensing. 5. Tides, Encyclopedia of Astronomy and Astrophysics. 6. The Puzzle of the Titan-Hyperion 4:3 Orbital Resonance. 7. On the Validity of the Coagulation Equation and the Nature of Runaway Growth. 8. Making Hyperion. 9. The MESSENGER mission to Mercury: Scientific objectives and implementation. 10. A Survey of Numerical Solutions to the Coagulation. 11. Probability of detecting a planetary companion during a microlensing event. 12. Dynamics and origin of the 2:l orbital resonances of the GJ876 planets. 13. Planetary Interior Structure Revealed by Spin Dynamics. 14. A primordial origin of the Laplace relation among the Galilean Satellites. 15. A procedure for determining the nature of Mercury's core. 16. Secular evolution of hierarchical planetary systems. 17. Tidally induced volcanism. 18. Extrasolar planets and mean motion resonances. 19. Comparison of a ground-based microlensing search for planets with a search from space.

One of the most intriguing scientific questions about Mercury is whether the planet possesses a liquid core. One way that this can be ascertained is through the measurement of the longitudinal libration of Mercury about its rotation axis. The MESSENGER (Mercury Surface, Space, Environment, Geochemistry, and Ranging) mission to Mercury will attempt this measurement by deriving the rotation of the planet from remotely sensed observations of the Mercury gravity field and the planet's shape. For a fully fluid core the libration is expected to be an approximately 350-m-amplitude oscillation (at the equator) about the mean rotation of the planet. Thus, the shape of the planet undergoes a librational oscillation in the same way that features on the surface participate in the motion. In this sense altimetric topography can be considered analogous to imaging, with the important exception that for altimetry, the long wavelengths corresponding to low degrees and orders in a spherical harmonic expansion, can be precisely determined from orbital measurements. The rotation of the planet thus introduces a time dependence of the topographic image in an inertial (i.e. fixed) reference frame and the libration is a variation in the planetary rotation rate, manifest as an oscillation of the topographic image. Similarly, gravity measures the distribution of mass within a planet and as such can be viewed as providing a vertically integrated 'image' of the internal density structure. For a librating planet the gravitational field will also reveal a variation in the rotation rate, manifest as an oscillation of the field with respect to an inertial reference frame. While the long-wavelength gravity field contains contributions from the radial distribution of mass with depth, the long wavelength terms are most sensitive to the mass distribution of the deep interior. Thus, both the topography and the gravity independently can be used to determine the (irregular) rotation of a planet, although the topographical method measures the rotation of the crust while the gravity measures the rotation of the distributed mass, and these are not necessarily identical. For example, for a planet that contains a fluid outer core that is effectively decoupled from the mantle the gravity field will not librate in the manner of a solid body. The libration obtained from the variable rotation of the gravity field will represent a combination of contributions due to a librating mantle and a differentially rotating fluid core. Differential rotation of the Earth's inner core had been theoretically predicted from three dimensional geodynamical models and subsequently reported from seismic observations. A difference between the libration of the lithosphere as determined from altimetry and the deep interior as determined from gravity, if it could be detected, might provide further insight into the nature of core-mantle coupling. Additional information is contained in the original extended abstract.

Evidence for a global magnetic field at Mercury has raised perplexing questions regarding the existence and nature of the planet's core. The problems related to Mercury's core are of great interest because of their implications on how planets evolve thermally and on how they generate magnetic fields. Peale showed that the measurement of four quantities could place constraints on the size and state of Mercury's core. The required parameters in Peale's experiment are the C(sub 20) and C(sub 22) coefficients in the spherical harmonic expansion of the gravity field, the planet's obliquity theta, and the amplitude phi of the forced libration in longitude. While the gravitational harmonics are best determined from an orbiting spacecraft, Earth-based radar observations may provide refined estimates of the obliquity and amplitude of the libration. We describe several techniques and attempt to quantify their potential for the proposed measurements. Additional information is contained in the original extended abstract.

We review past assertions that the determinations of the four parameters, C(20), C(22), theta, phi, are sufficient to determine the size and state of Mercury's core. C(20) and C(22) are gravitational harmonics, theta is Mercury's obliquity and phi is the amplitude of the forced, 88 day period libration in longitude. The upcoming MESSENGER orbiter mission to Mercury with onboard instrumentation capable of measuring these four parameters, and the possibility of precision measurements of Mercury's spin geometry with radar interferometry techniques make a reexamination of this proposal particularly relevant. The two necessary conditions on the core-mantle interaction for the experiment to work are: 1. The core must not follow the 88 day physical librations of the mantle. 2. The core must follow the mantle on the time scale of the 250,000 year precession of the spin in Cassini state 1. We shall assume these two conditions are satisfied to develop the method and later establish the constraints on the core viscosity for which they are satisfied. Proposed mechanisms of core mantle coupling other than a viscous coupling do not frustrate the first condition. The physical libration of the mantle about the mean resonant angular velocity arises from the periodically reversing torque on the permanent deformation as Mercury rotates relative to the Sun. Additional information is contained in the original extended abstract.

The magnitude and nature of residuals expected in the times of pulsar arrival (TOA) of the millisecond pulsar PSR 1257 + 12 as a function of the observational time span is estimated so that an observing progam may be planned to optimize the detection of a perturbation signature of a possible planetary system. The numerical and analytic procedures for evaluating the residuals are described. Calculations results are given in which a recognizable signature is shown to be a modulation of the amplitude of the TOA residual differences as a function of both epoch and observational interval. A supplementary approach involving intensive closely spaced observations over selected time intervals to find the time of a particular zero crossing of the TOA residuals is also discussed.

After the corotation resonance with an exterior satellite proved inapplicable to the Neptune ring arc confinement, a search for other mechanisms settled on the possible influence of Neptune's magnetic field. The areas of greater optical depth around the ring are much dustier than the low optical depth regions. These particles reside in a plasma; therefore, they must carry some charge. The components of Neptune's magnetic field on the equator at the radius of the ring arcs as a function of Neptunian longitude are shown. The components are those of an offset tilted dipole model. Although the dipole model is probably not a good approximation so close to the planet, the magnitude of the field that is given is probably close to the actual value. The possible importance of the magnetic field on the smallest particles in the ring is indicated by the ratio of the magnetic field on the smallest particles in the ring is indicated by the ratio of the magnetic force to the central gravitation attraction with the field strength of B = 0.01 gauss at the ring distance. A preferred position in the orbit for magnetically perturbed particles seems to require a commensurability between the rotation of the planet and the motion of the particle in the orbit. The period of rotation is assumed to be that of the radio bursts at 16.11 hours. However, without a model for the radio emission, one cannot be absolutely sure. Jupiter's decametric radiation depends on Io's orbital position as well as the rotation, so a synodic periodicity might be appropriate. But the latter radiation is highly directed, whereas Neptune's was seen all along the spacecraft trajectory on the 16.11 hour schedule, i.e., with no shifts in phase relative to a fixed longitude on the planet. The ring orbital period is 10.536 hours which is not commensurate with the rotation period. If the 16.11 hours is interpreted as a synodic period between the rotation and a satellite motion, the closest rotation periods to 16 hours are 15.9 hours if the satellite is 1989N4 and 18.2 hours if the satellite is Triton. The former is near a 3:2 resonance with the ring particle motion. The problem deserves some more thought before a possible herding of small particles by the magnetic field is abandoned.

Properties of satellite rotation that are observable in principle, include the rotation period, the orientation of the spin axis relative to the orbit plane, precession of the spin axis due to gravitational torques, nonprincipal axis rotation or wobble, and deviations from uniform principle axis rotation or libration. Considerable order is observed in current satellite rotation states, and it is of interest to ascertain how this order came about and why some satellites do not conform to the dominant norm. There is a strong coupling between the spin and orbital motions that is primarily responsible for maintaining the ordered rotation states in most cases, but this coupling is equally responsible for destroying any chance of orderly rotation for Saturn's satellite Hyperion. Understanding the processes which constrain current rotation states as well as those of an evolutionary nature which could have brought the individual satellites to their observed rotation and orbit states allows us to sometimes infer interior properties of some satellite or even of its primary planet, although, attempts to deduce primordial rotation states are usually frustrated. The observed rotational properties of the planetary satellites are summarized, and the understanding of the processes maintaining and those leading to the observed states are outlined. Some of the inferences that can be drawn about intrinsic properties of the bodies themselves are indicated.

Considerable effort was spent investigating the applicability of a Hamiltonian averaged over high frequency terms, where long period and secular terms up to second order in eccentricity were kept. The Hamiltonian that is given from the planar, elliptic, restricted three body problem applied to Titan-Hyperion, when the Kepler terms are also expanded to second order in small quantities and several conical transformations are carried out, is presented and discussed.

The density of Comet Halley is presently determined as a function of (1) its ejected material's time of visibility; (2) the angular deviation alpha of the averaged reaction force due to jets of ejected material from the sun-comet direction, and (3) the thermal inertia for a variable alpha. The averaged gas ejection rate variation from the nucleus as a function of orbital true anomaly is estimated on the basis of the Green and Morris (1987) lightcurve. While some of the densities obtained are in reasonable agreement with the expected 1 g/cu cm, uncertainties associated with both parameters and assumptions weaken the significance of these density constraints.

Numerical simulations supported by analytical calculations are used to model the suspected condition of Comet Halley's nucleus, which has been suggested to not be in a state of principal-axis rotation. It is found easy to numerically generate lightcurves from modulated jets of material which exhibit both of the observed periodicities of 2.2 and 7.4 days, after choosing initial conditions for a representative nucleus such that the shorter period is the rotation period, and the longer period is that of precession of the spin vector in the body frame-of-reference. The improbability of exciting a spin precession about the axis of minimum moment-of-inertia, the relative instability of this state to the jet-induced torques, and the smaller probability of observing significant seasonal changes in the lightcurve in this state, all favor the model in which Halley's nucleus precesses about the axis of maximum moment-of-inertia.

The hypothetical histories presently considered for the Uranian satellite system are very sensitive to the assumed satellite masses, within the errors of their determinations, as well as to the Uranian tidal effective dissipation function. A determination is made of the resonances that could have been encountered in these histories, and of the orbital eccentricity constraints that would have led to certain capture. It is found that if the mass of Miranda were of the upper extreme value, it would have been locked in a long-term 2/1 orbital resonance with Ariel to form a Laplacelike, three-body resonance that encompassed Umbriel.

Data on the rotational dynamics of Mercury are examined together with possible events that could lead to the current state of rotation. It is shown that the dynamical evolution of Mercury's spin angular momentum controlled by the dissipative processes of tidal friction and relative motion between a solid mantle and a liquid core would lead naturally to the current state of rotation of the Mercury planet. To investigate the possibility that Mercury has a molten core, an experiment is designed for the measurement of the core properties of Mercury. It is shown that it is technically feasible to measure the four parameters necessary for the determination of the existence and the extent of a Mercurian molten core, including the amplitude of the physical vibration about the resonance spin rate, the obliquity, and the gravitational harmonic coefficients.

Most of the orbital resonances among the satellites of the major planets are thought to be assembled by differential tidal expansion of their orbits. Why this has not occurred for the Uranus system can be investigated by determining the resonances which would be encountered for various values of Uranus' tidal effective Q. The comparison and the possible escape of Ariel-Umbriel from the 5:3 resonance mean that it is not unreasonable that no orbital resonances are found among these satellites.

Six different categories of models for the formation of the moon within the context of the general theory of terrestial planet formation by the accumulation of protoplanets are discussed. These catagories are: (1) rotational fission; (2) precipitation fission; (3) intact capture; (4) disintegrative capture; (5) binary accretion; and (6) giant impact accretion. It appears that the only plausable mechanism proposed thus far involves the formation of the Moon following a giant impact that ejects portions of the differentiated Earth's mantle and parts of the impacting body into circumterrestrial orbit.

Several examples of satellite dynamics are presented where significant progress has been made in understanding a complex problem, where a long-standing problem has finally been solved, where newly discovered configurations have motivated novel descriptions or where an entirely new phenomenon has been revealed. The origin of orbital resonances in satellites and the solution of the problem of the origin and evolution of the three body Laplace resonance among the Galilean satellites are among the topics discussed.

For significant dissipation in Ariel, the past existence of an orbital resonance to force and maintain a substantial eccentricity in the orbit is hypothesized. The absolute maximum dissipation rate would be reduced both by the time necessary to establish the resonance in the first place and by a consequently larger minimum Q. The actual heating rate was much less than this maximum so even the existence of a 2/1 resonance may not be sufficient to account for Ariel's smooth sufrace.

The lack of commensurabilites among the mean motion of the satellites of Uranus is investigated by determining the probabilities of capture into those orbital resonances which might be encountered as the satellite orbits expand differentially from tidal torques. An alternate to this is investigated to evaluate the conditions necessary to allow sufficient tidal heating of Ariel for the observed resurfacing.

The 'braided' structure noted in Voyager images of the Saturn F ring is presently addressed by two models. In the first, the pattern is generated by a narrow and initially uniform ring's passing of a nearby satellite, followed by an embedded moonlet gravitational acceleration-induced doubling back so that trajectories of the ring particles traverse one end of the classic 'horseshoe' orbit. In the second model, the F ring is composed of two separated strands before the moon's passage, so that a long braided pattern can be generated by the subsequent drift in relative phase; the embedded moonlet is thereby obviated.

The origin of orbital resonances is shown in the demonstration of the evolution of a pair of planetary satellites through a commensurability of the mean motions by a sequence of diagrams of constant energy curves in a two-dimensional phase space; the closed curve corresponding to the motion in each successive diagram is identified by its adiabatically conserved area. It is found that two-body resonances serve as a basis in the solution of the problem of the origin and evolution of the three-body Laplace resonance among the Galilean satellites of Jupiter. The unusual rotation state of Saturn's satellite Hyperion which is expected to tumble chaotically for an indefinite amount of time is discussed.

The paper considers six different categories of models for the formation of the moon within the context of the general theory of terrestrial planet formation by accumulation of planetesimals. These categories are: (1) rotational fission, (2) precipitation fission, (3) intact capture, (4) disintegrative capture, (5) binary accretion, and (6) giant impact accretion. It appears that the only plausible mechanism proposed thus far involves the formation of the moon following a giant impact that ejects portions of the differentiated earth's mantle and parts of the impacting body into circumterrestrial orbit.