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.     

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.     

Rotational dynamics of planetary satellites: A survey of regular and chaotic behavior

The problem of observability of chaotic regimes in the rotation of planetary satellites is studied. The analysis is based on the inertial and orbital data available for all satellites discovered up to now. The Lyapunov spectra of the spatial chaotic rotation and the full range of variation of the spin rate are computed numerically by integrating the equations of the rotational motion; the initial data are taken inside the main chaotic layer near the separatrices of synchronous resonance in phase space. The model of a triaxial satellite in a fixed elliptic orbit is adopted. A short Lyapunov time along with a large range of variation of the spin rate are used as criteria for observability of the chaotic motion. Independently, analysis of stability of the synchronous state with respect to tilting the axis of rotation provides a test for the physical opportunity for a satellite to rotate chaotically. Finally, a calculation of the times of despinning due to tidal evolution shows whether a satellite's spin could evolve close to the synchronous state. Apart from Hyperion, already known to rotate chaotically, only Prometheus and Pandora, the 16th and 17th satellites of Saturn, pass all these four tests.     

Tidal locking of habitable exoplanets

Potentially habitable planets can orbit close enough to their host star that the differential gravity across their diameters can produce an elongated shape. Frictional forces inside the planet prevent the bulges from aligning perfectly with the host star and result in torques that alter the planet’s rotational angular momentum. Eventually the tidal torques fix the rotation rate at a specific frequency, a process called tidal locking. Tidally locked planets on circular orbits will rotate synchronously, but those on eccentric orbits will either librate or rotate super-synchronously. Although these features of tidal theory are well known, a systematic survey of the rotational evolution of potentially habitable exoplanets using classic equilibrium tide theories has not been undertaken. I calculate how habitable planets evolve under two commonly used models and find, for example, that one model predicts that the Earth’s rotation rate would have synchronized after 4.5 Gyr if its initial rotation period was 3 days, it had no satellites, and it always maintained the modern Earth’s tidal properties. Lower mass stellar hosts will induce stronger tidal effects on potentially habitable planets, and tidal locking is possible for most planets in the habitable zones of GKM dwarf stars. For fast-rotating planets, both models predict eccentricity growth and that circularization can only occur once the rotational frequency is similar to the orbital frequency. The orbits of potentially habitable planets of very late M dwarfs ( Open image in new window ) are very likely to be circularized within 1 Gyr, and hence, those planets will be synchronous rotators. Proxima b is almost assuredly tidally locked, but its orbit may not have circularized yet, so the planet could be rotating super-synchronously today. The evolution of the isolated and potentially habitable Kepler planet candidates is computed and about half could be tidally locked. Finally, projected TESS planets are simulated over a wide range of assumptions, and the vast majority of potentially habitable cases are found to tidally lock within 1 Gyr. These results suggest that the process of tidal locking is a major factor in the evolution of most of the potentially habitable exoplanets to be discovered in the near future.     

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).     

Photometric properties of outer planetary satellites

We present optical broadband photometry for the satellites J6, J7, J8, S7, S9, U3, U4, N1, and polarimetry for J6, obtained between 1970 and 1979. The outer Jovian satellites resemble C-type asteroids; J6 has a rotational lightcurve with period ～9.5 hr. The satellites beyond Jupiter also show C-like colors with the exception of S7 Hyperion. S9 Phoebe has a rotational lightcurve with period near either 11.25 or 21.1 hr. For U4 and N1 there is evidence for a lightcurve synchronous with the orbital revolution. The seven brighter Saturnian satellites show a regular relation between the ultraviolet dropoff and distance to the planet, probably related with differences in the rock component on their surfaces.     

Expression of Cassini's third law for Callisto, and theory of its rotation

The rotation of the main natural satellites of the Solar System is widely assumed to be synchronous, because this corresponds to an equilibrium state. In the case of the Moon, 3 laws have been formulated by Cassini, assuming a spin-orbit resonance and a 1:1 nodal resonance. The recent gravitational data collected by the spacecrafts Galileo (in the jovian system) and Cassini (in the saturnian system) allows us to study the rotation of other natural satellites, and to check the universality of Cassini's laws. This paper deals with the rotation of the Galilean satellites of Jupiter J-4 Callisto. In this study we use both analytical (like Lie transforms) and numerical methods (numerical detection of chaos, numerical integration, frequency analysis) to first check the reliability of Cassini Laws for Callisto, and then to give a first theory of its rotation, Callisto's being considered as a rigid body. We first show that the Third Cassini Law (i.e. the nodal resonance), is not satisfied in every reference frame, in particular in the most natural one (i.e. the J2000 jovian equator). The difference of the nodes presents a chaotic-like behavior, that we prove to be just a geometrical illusion. Moreover, we give a mathematical condition ruling the choice of an inertial reference frame in which the Third Cassini Law is fulfilled. Secondly, we give a theory of Callisto's rotation in the International Celestial Reference Frame (ICRF). We highlight a small motion (i.e. <200 m) of its rotation axis about its body figure, a 11.86-yr periodicity in Callisto's length-of-day, and the proximity of a resonance that forces 182-yr librations in Callisto's obliquity.     

Analytical description of physical librations of saturnian coorbital satellites Janus and Epimetheus

Janus and Epimetheus are famously known for their distinctive horseshoe-shaped orbits resulting from a 1:1 orbital resonance. Every 4 years these two satellites swap their orbits by a few tens of kilometers as a result of their close encounter. Recently Tiscareno et al. (Tiscareno, M.S., Thomas, P.C., Burns, J.A. [2009]. Icarus 204, 254-261) have proposed a model of rotation based on images from the Cassini orbiter. These authors inferred the amplitude of rotational librational motion in longitude at the orbital period by fitting a shape model to Cassini ISS images. By a quasi-periodic approximation of the orbital motion, we describe how the orbital swap impacts the rotation of the satellites. To that purpose, we have developed a formalism based on quasi-periodic series with long- and short-period librations. In this framework, the amplitude of the libration at the orbital period is found proportional to a term accounting for the orbital swap. We checked the analytical quasi-periodic development by performing a numerical simulation and find both results in good agreement. To complete this study, the results obtained for the short-period librations are studied with the help of an adiabatic-like approach.     

Secular ring instability in the protoplanetary accretion disk

The basic parameters describing the angular momentum distribution within the Uranus system and of its tidal evolution have been estimated. The nine satellites orbiting under the synchronous zone of Uranus is the maximum number in the solar system and it makes the Uranus system different compared with any other in the Solar system, however the satellites in question are relatively small and their contribution of the tidal dynamics of the system is small compared with that due to UI and UV. The time for existence of the nine satellites as integrated bodies can be estimated as 1.4 × 10⁹ y (UVI) and more. The total tidal decrease in the Uranus angular velocity of rotation is estimated as 7 × 10^–9s^–1.     

Dynamics of Enceladus and Dione inside the 2:1 mean-motion resonance under tidal dissipation

In a previous work (Callegari and Yokoyama, Celest. Mech. Dyn. Astr. 98:5–30, 2007), the main features of the motion of the pair Enceladus–Dione were analyzed in the frozen regime, i.e., without considering the tidal evolution. Here, the results of a great deal of numerical simulations of a pair of satellites similar to Enceladus and Dione crossing the 2:1 mean-motion resonance are shown. The resonance crossing is modeled with a linear tidal theory, considering a two-degrees-of-freedom model written in the framework of the general three-body planar problem. The main regimes of motion of the system during the passage through resonance are studied in detail. We discuss our results comparing them with classical scenarios of tidal evolution of the system. We show new scenarios of evolution of the Enceladus–Dione system through resonance not shown in previous approaches of the problem.     

Surfaces of section in the Miranda-Umbriel 3:1 inclination problem

The recent numerical simulations of Tittemore and Wisdom (1988, 1989, 1990) and Dermottet al. (1988), Malhotra and Dermott (1990) concerning the tidal evolution through resonances of some pairs of Uranian satellites have revealed interesting dynamical phenomena related to the interactions between close-by resonances. These interactions produce chaotic layers and strong secondary resonances. The slow evolution of the satellite orbits in this dynamical landscape is responsible for temporary capture into resonance, enhancement of eccentricity or inclination and subsequent escape from resonance. The present contribution aims at developing analytical tools for predicting the location and size of chaotic layers and secondary resonances. The problem of the 3:1 inclination resonance between Miranda and Umbriel is analysed.     

Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited

This report is a review of Darwin’s classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.     

Triaxiality of satellites and small bodies in the Solar System

The triaxial figures are very common shape of most of planetary satellites as well as of smaell bodies as asteroids. There are 21 satellites in the Solar System triaxial figures of which were detected in situ evidently (Davies et al., 1995). However, the total number of triaxiaxial satellites in the Solar System should be in fact larger. In this paper the general theory of triaxiality due to tidal forces is discussed in regard to the very recent numerical data. Since they orbit synchronously, as a rule: their orbital periods are equal to the rotational periods, the tidal forces may be responsible for their triaxial figures. On the other hand the origin of triaxiality of asteroids due to another process and the of their figures cannot be axplained by the tidal effects.     

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     

Mercury's capture into the 3/2 spin-orbit resonance including the effect of core-mantle friction

The rotation of Mercury is presently captured in a 3/2 spin-orbit resonance with the orbital mean motion. The capture mechanism is well understood as the result of tidal interactions with the Sun combined with planetary perturbations [Goldreich, P., Peale, S., 1966. Astron. J. 71, 425-438; Correia, A.C.M., Laskar, J., 2004. Nature 429, 848-850]. However, it is now almost certain that Mercury has a liquid core [Margot, J.L., Peale, S.J., Jurgens, R.F., Slade, M.A., Holin, I.V., 2007. Science 316, 710-714] which should induce a contribution of viscous friction at the core-mantle boundary to the spin evolution. According to Peale and Boss [Peale, S.J., Boss, A.P., 1977. J. Geophys. Res. 82, 743-749] this last effect greatly increases the chances of capture in all spin-orbit resonances, being 100% for the 2/1 resonance, and thus preventing the planet from evolving to the presently observed configuration. Here we show that for a given resonance, as the chaotic evolution of Mercury's orbit can drive its eccentricity to very low values during the planet's history, any previous capture can be destabilized whenever the eccentricity becomes lower than a critical value. In our numerical integrations of 1000 orbits of Mercury over 4 Gyr, the spin ends 99.8% of the time captured in a spin-orbit resonance, in particular in one of the following three configurations: 5/2 (22%), 2/1 (32%) and 3/2 (26%). Although the present 3/2 spin-orbit resonance is not the most probable outcome, we also show that the capture probability in this resonance can be increased up to 55% or 73%, if the eccentricity of Mercury in the past has descended below the critical values 0.025 or 0.005, respectively.     

Theory of the rotation of Janus and Epimetheus

The saturnian coorbital satellites Janus and Epimetheus present a unique dynamical configuration in the Solar System, because of high-amplitude horseshoe orbits, due to a mass ratio of order unity. As a consequence, they swap their orbits every 4 years, while their orbital periods is about 0.695 days. Recently, Tiscareno et al. (Tiscareno, M.S., Thomas, P.C., Burns, J.A. [2009]. Icarus 204, 254-261) got observational informations on the shapes and the rotational states of these satellites. In particular, they detected an offset in the expected equilibrium position of Janus, and a large libration of Epimetheus.We here propose to give a three-dimensional theory of the rotation of these satellites in using these observed data, and to compare it to the observed rotations. We consider the two satellites as triaxial rigid bodies, and we perform numerical integrations of the system in assuming the free librations as damped.The periods of the three free librations we get, associated with the three dimensions, are respectively 1.267, 2.179 and 2.098 days for Janus, and 0.747, 1.804 and 5.542 days for Epimetheus. The proximity of 0.747 days to the orbital period causes a high sensitivity of the librations of Epimetheus to the moments of inertia. Our theory explains the amplitude of the librations of Janus and the error bars of the librations of Epimetheus, but not an observed offset in the orientation of Janus.     

Angular momentum and tidal evolution

Five satellites of Neptune orbit under the synchronous zone. In this sense the Neptune's system is similar to that of Uranus (nine satellites) and differs from Jupiter (two) and Saturn (zero). The basic parameters describing the angular momentum within the Neptune's system and of its tidal evolution are estimated. The main character of the tidal dynamics is due to the retrograde Triton. The total tidal decrease in the spin angular momentum of Neptune is compared with those of Uranus, Jupiter and Saturn.     

Long-term evolution of the spin of Mercury: I. Effect of the obliquity and core-mantle friction

The present obliquity of Mercury is very low (less than 0.1°), which led previous studies to always adopt a nearly zero obliquity during the planet’s past evolution. However, the initial orientation of Mercury’s rotation axis is unknown and probably much different than today. As a consequence, we believe that the obliquity could have been significant when the rotation rate of the planet first encountered spin-orbit resonances. In order to compute the capture probabilities in resonance for any evolutionary scenario, we present in full detail the dynamical equations governing the long-term evolution of the spin, including the obliquity contribution.The secular spin evolution of Mercury results from tidal interactions with the Sun, but also from viscous friction at the core-mantle boundary. Here, this effect is also regarded with particular attention. Previous studies show that a liquid core enhances drastically the chances of capture in spin-orbit resonances. We confirm these results for null obliquity, but we find that the capture probability generally decreases as the obliquity increases. We finally show that, when core-mantle friction is combined with obliquity evolution, the spin can evolve into some unexpected configurations as the synchronous or the 1/2 spin-orbit resonance.     

Orbital resonances in the inner neptunian system: I. The 2:1 Proteus-Larissa mean-motion resonance

We investigate the orbital resonant history of Proteus and Larissa, the two largest inner neptunian satellites discovered by Voyager 2. Due to tidal migration, these two satellites probably passed through their 2:1 mean-motion resonance a few hundred million years ago. We explore this resonance passage as a method to excite orbital eccentricities and inclinations, and find interesting constraints on the satellites' mean density () and their tidal dissipation parameters (Qs>10). Through numerical study of this mean-motion resonance passage, we identify a new type of three-body resonance between the satellite pair and Triton. These new resonances occur near the traditional two-body resonances between the small satellites and, surprisingly, are much stronger than their two-body counterparts due to Triton's large mass and orbital inclination. We determine the relevant resonant arguments and derive a mathematical framework for analyzing resonances in this special system.