On the rotational dynamics of Prometheus and Pandora

Possible rotation states of two satellites of Saturn, Prometheus (S16) and Pandora (S17), are studied by means of numerical experiments. The attitude stability of all possible modes of synchronous rotation and the motion close to these modes is analyzed by means of computation of the Lyapunov spectra of the motion. The stability analysis confirms that the rotation of Prometheus and Pandora might be chaotic, though the possibility of regular behaviour is not excluded. For the both satellites, the attitude instability zones form series of concentric belts enclosing the main synchronous resonance center in the phase space sections. A hypothesis is put forward that these belts might form “barriers” for capturing the satellites in synchronous rotation. The satellites in chaotic rotation can mimic ordinary regular synchronous behaviour: they preserve preferred orientation for long periods of time, the largest axis of satellite’s figure being directed approximately towards Saturn.     

The chaotic rotation of Hyperion

A plot of spin rate versus orientation when Hyperion is at the pericenter of its orbit (surface of section) reveals a large chaotic zone surrounding the synchronous spin-orbit state of Hyperion, if the satellite is assumed to be rotating about a principal axis which is normal to its orbit plane. This means that Hyperion's rotation in this zone exhibits large, essentially random variations on a short time scale. The chaotic zone is so large that it surrounds the ½ and 2 states, and libration in the 3/2 state is not possible. Stability analysis shows that for libration in the synchronous and ½ states, the orientation of the spin axis normal to the orbit plane is unstable, whereas rotation in the 2 state is attitude stable. Rotation in the chaotic zone is also attitude unstable. A small deviation of the principal axis from the orbit normal leads to motion through all angles in both the chaotic zone and the attitude unstable libration regions. Measures of the exponential rate of separation of nearby trajectories in phase space (Lyapunov characteristic exponents) for these three-dimensional motions indicate the the tumbling is chaotic and not just a regular motion through large angles. As tidal dissipation drives Hyperion's spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone. At this point Hyperion becomes attitude unstable and begins to tumble. Capture from the chaotic state into the synchronous or ½ state is impossible since they are also attitude unstable. The 3/2 state does not exist. Capture into the stable 2 state is possible, but improbable. It is expected that Hyperion will be found tumbling chaotically.     

The rotation states predominant among the planetary satellites

On the basis of tidal despinning timescale arguments, Peale showed in 1977 that the majority of irregular satellites (with unknown rotation states) are expected to reside close to their initial (fast) rotation states. Here we investigate the problem of the current typical rotation states among all known satellites from a viewpoint of dynamical stability. We explore location of the known planetary satellites on the (ω₀, e) stability diagram, where ω₀ is an inertial parameter of a satellite and e is its orbital eccentricity. We show that most of the satellites with unknown rotation states cannot rotate synchronously, because no stable synchronous 1:1 spin-orbit state exists for them. They rotate either much faster than synchronously (those tidally unevolved) or, what is much less probable, chaotically (tidally-evolved objects or captured slow rotators).     

Synchronous spin-orbital resonance locking of large planetary satellites

A numerical investigation of the chaotic rotation of large planetary satellites before their synchronous spin-orbital resonance locking with regard to tidal friction is carried out. The rotational dynamics of seven large satellites greater than 1000 km in diameter and with known inertial parameters (Io, Europa, Ganymede, Callisto (J1–J4), Tethys (S3), Iapetus (S8), and Ariel (U1)) in the epoch of synchronous resonance locking is modeled. All of these satellites have a small dynamic asymmetry. The planar case is considered in which the satellite’s axis of rotation is orthogonal to the plane of orbit. The satellites possessing an initial rapid rotation pass through various resonant states during the tidal evolution. Here, the probability of their locking into these states exists. The numerical experiments presented in this paper have shown that, with a rather high arbitrariness in the choice of initial states, the satellites during the course of the tidal evolution of their rotational motion have passed without interruption through the regions of the 5: 2, 2: 1, and 3: 2 resonances in the phase space and are locked into the 1: 1 resonance. The estimate for the tidal deceleration time is obtained both theoretically and on the numerical experimental basis.     

Adiabatic chaos in the Prometheus–Pandora system

The chaotic orbital motion of Prometheus and Pandora, the 16th and 17th satellites of Saturn, is studied. Chaos in their orbital motion, as found by Goldreich & Rappaport and Renner & Sicardy, is due to interaction of resonances in the resonance multiplet corresponding to the 121:118 commensurability of the mean motions of the satellites. It is shown rigorously that the system moves in adiabatic regime. The Lyapunov time (the 'time horizon of predictability' of the motion) is calculated analytically and compared to the available numerical–experimental estimates. For this purpose, a method of analytical estimation of the maximum Lyapunov exponent in the perturbed pendulum model of non-linear resonance is applied. The method is based on the separatrix map theory. An analytical estimate of the width of the chaotic layer is made as well, based on the same theory. The ranges of chaotic diffusion in the mean motion are shown to be almost twice as big compared to previous estimates for both satellites.     

Chaotic Dynamics of Satellite Systems

We consider the problem of calculating the Lyapunov time (the characteristic time of predictable dynamics) of chaotic motion in the vicinity of separatrices of orbital resonances in satellite systems. The primary objects of study are the chaotic regimes that have occurred in the history of the orbital dynamics of the second and fifth Uranian satellites (Umbriel and Miranda) and the first and third Saturnian satellites (Mimas and Tethys). We study the dynamics in the vicinity of separatrices of the resonance multiplets corresponding to the 3 : 1 commensurability of mean motions of Miranda and Umbriel and the multiplets corresponding to the 2 : 1 commensurability of mean motions of Mimas and Tethys. These chaotic regimes have most probably contributed much to the long-term orbital evolution of the two satellite systems. The equations of motion have been numerically integrated to estimate the Lyapunov time in models corresponding to various epochs of the system evolution. Analytical estimates of the Lyapunov time have been obtained by a method (Shevchenko, 2002) based on the separatrix map theory. The analytical estimates have been compared to estimates obtained by direct numerical integration.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 4, 2005, pp. 364–374.Original Russian Text Copyright © 2005 by Mel’nikov, Shevchenko.     

Generalized YORP evolution: Onset of tumbling and new asymptotic states

Asteroids have a wide range of rotation states. While the majority spin a few times to several times each day in principal axis rotation, a small number spin so slowly that they have somehow managed to enter into a tumbling rotation state. Here we investigate whether the Yarkovsky-Radzievskii-O'Keefe-Paddack (YORP) thermal radiation effect could have produced these unusual spin states. To do this, we developed a Lie-Poisson integrator of the orbital and rotational motion of a model asteroid. Solar torques, YORP, and internal energy dissipation were included in our model. Using this code, we found that YORP can no longer drive the spin rates of bodies toward values infinitely close to zero. Instead, bodies losing too much rotation angular momentum fall into chaotic tumbling rotation states where the spin axis wanders randomly for some interval of time. Eventually, our model asteroids reach rotation states that approach regular motion of the spin axis in the body frame. An analytical model designed to describe this behavior does a good job of predicting how and when the onset of tumbling motion should take place. The question of whether a given asteroid will fall into a tumbling rotation state depends on the efficiency of its internal energy dissipation and on the precise way YORP modifies the spin rates of small bodies.     

How fast do Galilean satellites spin?

Each of the Galilean satellites, as well as most other satellites whose initial rotations have been substantially altered by tidal dissipation, has been widely assumed to rotate synchronously with its orbital mean motion. Such rotation would require a small permanent asymmetry in the mass distribution in order to overcome the small mean tidal torque. Since Io and Europa may be substantially fluid, they may not have the strenght to support the required permanent asymmetry. Thus, each may rotate at the unknown but slightly nonsynchronous rate that corresponds to zero mean tidal torque. This behaviour may be observable by Galileo spacecraft imaging. It may help explain the longitudinal variation of volcanism on Io and the cracking of Europa's crust.     

The absence of satellites of asteroids

A CCD-imaging survey was made for satellites of minor planets at distances of about 0.1 to 7 arcmin from 1 Ceres, 2 Pallas, 4 Vesta, 6 Hebe, 7 Iris, 8 Flora, 15 Eunomia, 29 Amphitrite, 41 Daphne, and 44 Nysa, with cursory inspection of 192 Nausikaa. Satellites larger than 3 km were not found in this work, nor in previous photographic surveys. Not finding them appears to be consistent with theoretical studies of collisions in the asteroid belt by several authors. The satellites would have to be larger than at least 30 km to be collisionally stable. Tidal effects would lead to synchronous rotation and therefore long periods of rotation (several days), which are not generally observed. Taking tidal stability into account, we conclude that the only possible satellites for main-belt asteroids, with stability over eons, are near-contact binaries. The only other rare possibility for a satellite might be a piece of debris from a recent collision, and it would now be chaotic and collisionally unstable.     

Spin states and climates of eccentric exoplanets

The known extrasolar planets exhibit a wide range of orbital eccentricities e. This has a profound influence on their rotations and climates. Because of tides in their interiors, mostly solid exoplanets are expected eventually to despin to a state of spin-orbit resonance, where the orbital period is some integer or half-integer times the rotation period. The most important of these resonances is the synchronous state, where the planet's spin period exactly equals its orbital period (like Earth's Moon, and indeed most of the regular satellites in the Solar System). Such planets seem doomed to roast on one side and freeze on the other. However, synchronous planets rock back and forth by an angle of ∼2Arcsine with respect to the sub-stellar point. For e=0.055 (as for the Moon), this optical libration amounts to only ∼6°; but for a synchronous planet with e=0.50, for example, it would rise to ∼59°. This greatly expands the temperate “twilight zone” near the terminator and considerably improves the planet's prospects for habitability. For e?0.72389, the optical libration exceeds 90°; for such planets, the sector of permanent night vanishes, while the sunniest region splits in two. Furthermore, the synchronous state is not the only possible spin resonance. For example, Mercury (with e≈0.206) has an orbital period exactly 1.5 times its rotation period. A terrestrial exoplanet with e=0.40, say, is liable to have an orbital period of 2.0, 2.5, or 3.0 times its spin period. The corresponding insolation patterns are generally complicated, and all different from the synchronous state. Yet these non-synchronous resonances also protect certain longitudes from the worst extremes of temperature and solar radiation, and improve the planet's habitability, compared to non-resonant rotation. These results also have implications for the direct detectability of extrasolar planets, and the interpretation of their thermal emissions.     

Rotational modeling of Hyperion

Saturn’s moon, Hyperion, is subject to strongly-varying solid body torques from its primary and lacks a stable spin state resonant with its orbital frequency. In fact, its rotation is chaotic, with a Lyapunov timescale on the order of 100 days. In 2005, Cassini made three close passes of Hyperion at intervals of 40 and 67 days, when the moon was imaged extensively and the spin state could be measured. Curiously, the spin axis was observed at the same location within the body, within errors, during all three fly-bys—~ 30° from the long axis of the moon and rotating between 4.2 and 4.5 times faster than the synchronous rate. Our dynamical modeling predicts that the rotation axis should be precessing within the body, with a period of ~ 16 days. If the spin axis retains its orientation during all three fly-bys, then this puts a strong constraint on the in-body precessional period, and thus the moments of inertia. However, the location of the principal axes in our model are derived from the shape model of Hyperion, assuming a uniform composition. This may not be a valid assumption, as Hyperion has significant void space, as shown by its density of 544± 50  kg m⁻³ (Thomas et al. in Nature 448:50, 2007). This paper will examine both a rotation model with principal axes fixed by the shape model, and one with offsets from the shape model. We favor the latter interpretation, which produces a best-fit with principal axes offset of ~ 30° from the shape model, placing the A axis at the spin axis in 2005, but returns a lower reduced χ ² than the best-fit fixed-axes model.     

Nebular drag and capture into spin-orbit resonance

The majority of planetary satellites whose spin period is known are observed to be in synchronous spin-orbit resonance. The commonly accepted explanation for this observation is that it is due to the effects of tidal evolution. However, cosmogonic theories state that the formation of planetary and satellite systems occurs within a primordial solar nebula and circumplanetary nebulae, respectively. In this paper the influence of nebular drag on the capture into spin-orbit resonance is analysed. The results show that the torques generated are important for these resonances in a wide range of cases. Using the protojovian nebula model by Lunine and Stevenson (1982), conservative estimates of the despinning time scales for the Galilean satellites are computed. In comparison the despinning time scale from tidal effects are several orders of magnitude larger.on leave of absence from Departamento de Matemática, Faculdade de Engenharia de Guaratinguetá, UNESP, CP 205, 12500-000, Guaratinguetá, SP, Brazil     

Rotation of Synchronous Satellites Application to the Galilean Satellites

An analytical theory of the rotation of a synchronous satellite is developed for the application to the rotation of the Galilean satellites. The theory is developed in the framework of Hamiltonian mechanics, using Andoyer variables. Special attention is given to the frequencies of libration as functions of the moments of inertia of the satellite. This revised version was published online in July 2006 with corrections to the Cover Date.     

The shapes and rotational dynamics of minor planetary satellites

Statistical analysis of the available data on the sizes and inertial parameters for all hitherto known satellites of the Solar system’s planets is performed. Analytical approximations are derived for the size distribution of satellites. Empirical relations are obtained to approximately estimate the inertial parameters of a satellite from its size. These relations can be used in statistical studies of the possibility of manifestations of various nonstandard rotational modes of planetary satellites. In particular, the probability of the “Amalthea effect” (the presence of two centers of synchronous resonance in the phase space of rotational motion) is shown to be much higher for minor (with diameters smaller than 100 km) satellites moving in close-to-circular orbits than for other satellites.     

Time-varying transformations for Hill–Clohessy–Wiltshire solutions in elliptic orbits

The relative motion of chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be approximated by linearized time-periodic equations of motion. The linear time-invariant Hill–Clohessy–Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Two Lyapunov–Floquet transformations and an integral-preserving transformation are here presented which relate the linearized time-varying equations of relative motion to the Hill–Clohessy–Wiltshire equations in a one-to-one manner through time-varying coordinate transformations. These transformations allow the Hill–Clohessy–Wiltshire equations to describe the linearized relative motion for elliptic chief satellites.     

Observations and Theoretical Analysis of Lightcurves of Natural Satellites of Planets

We present the results of photometric observations of Saturn's seventh satellite Hyperion and four other planetary satellites: Saturn's moon Phoebe and three Jovian satellites Himalia, Elara, and Pasiphae. The observations have been conducted from September, 1999 to March, 2000, and during September–October, 2000. Analysis of periodic variations in Hyperion's lightcurve was performed. The lightcurve was modeled using the software package developed for calculating the rotational dynamics of a satellite. Our data generally indicate that over the period of observations Hyperion was in the chaotic mode of rotation.     

Some unsolved problems in evolutionary dynamics in the solar system

Several unsolved problems in the evolutionary histories leading to current dynamical configurations of the planets and their systems of satellites are discussed. These include the possibilities of rather tight constraints on the primordial rotation states of Mercury and Venus and the stabilizing mechanism for the latter's retrograde spin, a brief mention of the problem of origin of the moons of Earth and Mars, the excessive heat flow from Jupiter's satellite lo which is not compatible with an otherwise self-consistent model of origin of the Laplace three-body libration, the mechanism for the long history of resurfacing of Saturn's satellite Enceladus and the possibly short lifetime of the A ring and the mechanisms for resurfacing the satellites of Uranus, especially Ariel, if the high stability of the mean motion orbital resonances at the 2/1 commensurability involving Ariel and Umbriel precludes a long term occupancy of the resonance. Finally, excessive times of accumulation of the outer planets in current models may possibly be reducible from the effects of nebular gas drag.     

Unusual rotation modes of minor planetary satellites

An analysis of the character of the possible dynamics of all hitherto known planetary satellites shows two satellites—Amalthea (J5) and Prometheus (S16)—to have the most unusual structure of the phase space of possible rotational motion. These are the only satellites whose phase space of planar rotation may host synchronous resonances of three different kinds: the α resonance, the β resonance, and a mode corresponding to the period doubling bifurcation of the α resonance. We analyze the stability of these states against the tilt of the rotational axis.     

Orbital variation of the synchronous satellite and its calculation

This paper investigates the orbital variation of the 24-hour synchronous satellite and gives a method for calculating the variation. The results provide a theoretical basis for the design and calculation of the orbit of the communication satellites.     

Evolutionary characteristics of the orbits of outer Saturnian,Uranian, and Neptunian satellites

We present the results of our systematic study of the long-period orbital evolution of all of the outer Saturnian, Uranian, and Neptunian satellites known to date. The plots of the orbital elements against time give a clear idea of the pattern of the orbital evolution of each satellite. The tabular data allow us to estimate the basic parameters of the evolving orbits, including the ranges of variation in the semimajor axes, eccentricities, and ecliptical inclinations as well as the variation periods and mean motions of the arguments of pericenters and the longitudes of the nodes. We compare the results obtained by numerically integrating the rigorous equations of the perturbed motion of the satellites with the analytical and numerical-analytical results. The satellite orbits with a librational pattern of variation in the arguments of pericenters are set apart.