The escape of natural satellites from Mercury and Venus

It is suggested that the slow rotations of Mercury and Venus may be connected with the absence of natural satellites around them. If Mercury or Venus possessed a satellite at the time of formation, the tidal evolution would have caused the satellite to recede. At a sufficiently large distance from the planet, the Sun's gravitational influence makes the satellite orbit unstable. The natural satellites of Mercury and Venus might have escaped as a consequence of this instability.     

Obliquity evolution of extrasolar terrestrial planets

We have investigated the obliquity evolution of terrestrial planets in habitable zones (at ∼1 AU) in extrasolar planetary systems, due to tidal interactions with their satellite and host star with wide varieties of satellite-to-planet mass ratio (m/Mp) and initial obliquity (γ₀), through numerical calculations and analytical arguments. The obliquity, the angle between planetary spin axis and its orbit normal, of a terrestrial planet is one of the key factors in determining the planetary surface environments. A recent scenario of terrestrial planet accretion implies that giant impacts of Mars-sized or larger bodies determine the planetary spin and form satellites. Since the giant impacts would be isotropic, tilted spins (sinγ₀∼1) are more likely to be produced than straight ones (sinγ₀∼0). The ratio m/Mp is dependent on the impact parameters and impactors' mass. However, most of previous studies on tidal evolution of the planet-satellite systems have focused on a particular case of the Earth-Moon systems in which m/Mp?0.0125 and γ₀∼10° or the two-body planar problem in which γ₀=0° and stellar torque is neglected. We numerically integrated the evolution of planetary spin and a satellite orbit with various m/Mp (from 0.0025 to 0.05) and γ₀ (from 0° to 180°), taking into account the stellar torques and precessional motions of the spin and the orbit. We start with the spin axis that almost coincides with the satellite orbit normal, assuming that the spin and the satellite are formed by one dominant impact. With initially straight spins, the evolution is similar to that of the Earth-Moon system. The satellite monotonically recedes from the planet until synchronous state between the spin period and the satellite orbital period is realized. The obliquity gradually increases initially but it starts decreasing down to zero as approaching the synchronous state. However, we have found that the evolution with initially tiled spins is completely different. The satellite's orbit migrates outward with almost constant obliquity until the orbit reaches the critical radius ∼10-20 planetary radii, but then the migration is reversed to inward one. At the reversal, the obliquity starts oscillation with large amplitude. The oscillation gradually ceases and the obliquity is reduced to ∼0° during the inward migration. The satellite eventually falls onto the planetary surface or it is captured at the synchronous state at several planetary radii. We found that the character change of precession about total angular momentum vector into that about the planetary orbit normal is responsible for the oscillation with large amplitude and the reversal of migration. With the results of numerical integration and analytical arguments, we divided the m/Mp-γ₀ space into the regions of the qualitatively different evolution. The peculiar tidal evolution with initially tiled spins give deep insights into dynamics of extrasolar planet-satellite systems and discussions of surface environments of the planets.     

Lunar shape does not record a past eccentric orbit

The Moon has long been known to have an overall shape not consistent with expected past tidal forces. It has recently been suggested (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) that the present lunar moments of inertia indicate a past high-eccentricity orbit and, possibly, a past non-synchronous spin-orbit resonance. Here I show that the match between the lunar shape and the proposed orbital and spin states is much less conclusive than initially proposed. Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) spin and shape evolution scenarios also completely ignore the physics of the capture into such resonances, which require prior permanent deformation, as well as tidal despinning to the relevant resonance. If the early lunar orbit was eccentric, the Moon would have been rotating at an equilibrium non-synchronous rate determined by it eccentricity. This equilibrium supersynchronous rotation would be much too fast to allow a synchronous spin-orbit lock at e = 0.49, while the capture into the 3:2 resonance is possible only for a very constrained lunar eccentricity history and assuming some early permanent lunar tri-axiality. Here I show that large impacts in the early history of the Moon would have frequently disrupted this putative resonant rotation, making the rotation and eccentricity solutions of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) unstable. I conclude that the present lunar shape cannot be used to support the hypothesis of an early eccentric lunar orbit.     

Reassessing the origin of Triton

The origin of Neptune’s large, circular but retrograde satellite Triton has remained largely unexplained. There is an apparent consensus that its origin lies in it being captured, but until recently no successful capture mechanism has been found. Agnor and Hamilton (Agnor, C.B., Hamilton, D.P. [2006]. Nature 441, 192-194) demonstrated that the disruption of a trans-neptunian binary object which had Triton as a member, and which underwent a very close encounter with Neptune, was an effective mechanism to capture Triton while its former partner continued on a hyperbolic orbit. The subsequent evolution of Triton’s post-capture orbit to its current one could have proceeded through gravitational tides (Correia, A.C.M. [2009]. Astrophys. J. 704, L1-L4), during which time Triton was most likely semi-molten (McKinnon, W.B. [1984]. Nature 311, 355-358). However, to date, no study has been performed that considered both the capture and the subsequent tidal evolution. Thus it is attempted here with the use of numerical simulations. The study by Agnor and Hamilton (Agnor, C.B., Hamilton, D.P. [2006]. Nature 441, 192-194) is repeated in the framework of the Nice model (Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F. [2005]. Nature 435, 459-461) to determine the post-capture orbit of Triton. After capture Triton is then subjected to tidal evolution using the model of Mignard (Mignard, F. [1979]. Moon Planets 20, 301-315; Mignard, F. [1980]. Moon Planets 23, 185-201). The perturbations from the Sun and the figure of Neptune are included. The perturbations from the Sun acting on Triton just after its capture cause it to spend a long time in its high-eccentricity phase, usually of the order of 10 Myr, while the typical time to circularise to its current orbit is some 200 Myr, consistent with earlier studies. The current orbit of Triton is consistent with an origin through binary capture and tidal evolution, even though the model prefers Triton to be closer to Neptune than it is today. The probability of capturing Triton in this manner is approximately 0.7%. Since the capture of Triton was at most a 50% event - since only Neptune has one, but Uranus does not - we deduce that in the primordial trans-neptunian disc there were some 100 binaries with at least one Triton-sized member. Morbidelli et al. (Morbidelli, A., Levison, H.F., Bottke, W.F., Dones, L., Nesvorný, D. [2009]. Icarus 202, 310-315) concludes there were some 1000 Triton-sized bodies in the trans-neptunian proto-planetary disc, so the primordial binary fraction with at least one Triton-sized member is 10%. This value is consistent with theoretical predictions, but at the low end. If Triton was captured at the same time as Neptune’s irregular satellites, the far majority of these, including Nereid, would be lost. This suggests either that Triton was captured on an orbit with a small semi-major axisa ? 50R_N (a rare event), or that it was captured before the dynamical instability of the Nice model, or that some other mechanism was at play. The issue of keeping the irregular satellites remains unresolved.     

The orbital-thermal evolution and global expansion of Ganymede

The tectonically and cryovolcanically resurfaced terrains of Ganymede attest to the satellite's turbulent geologic history. Yet, the ultimate cause of its geologic violence remains unknown. One plausible scenario suggests that the Galilean satellites passed through one or more Laplace-like resonances before evolving into the current Laplace resonance. Passage through such a resonance can excite Ganymede's eccentricity, leading to tidal dissipation within the ice shell. To evaluate the effects of resonance passage on Ganymede's thermal history we model the coupled orbital-thermal evolution of Ganymede both with and without passage through a Laplace-like resonance. In the absence of tidal dissipation, radiogenic heating alone is capable of creating large internal oceans within Ganymede if the ice grain size is 1 mm or greater. For larger grain sizes, oceans will exist into the present epoch. The inclusion of tidal dissipation significantly alters Ganymede's thermal history, and for some parameters (e.g. ice grain size, tidal Q of Jupiter) a thin ice shell (5 to 20 km) can be maintained throughout the period of resonance passage. The pulse of tidal heating that accompanies Laplace-like resonance capture can cause up to 2.5% volumetric expansion of the satellite and contemporaneous formation of near surface partial melt. The presence of a thin ice shell and high satellite orbital eccentricity would generate moderate diurnal tidal stresses in Ganymede's ice shell. Larger stresses result if the ice shell rotates non-synchronously. The combined effects of satellite expansion, its associated tensile stress, rapid formation of near surface partial melt, and tidal stress due to an eccentric orbit may be responsible for creating Ganymede's unique surface features.     

Three-body capture of irregular satellites: Application to Jupiter

We investigate a new theory of the origin of the irregular satellites of the giant planets: capture of one member of a ∼100-km binary asteroid after tidal disruption. The energy loss from disruption is sufficient for capture, but it cannot deliver the bodies directly to the observed orbits of the irregular satellites. Instead, the long-lived capture orbits subsequently evolve inward due to interactions with a tenuous circumplanetary gas disk.We focus on the capture by Jupiter, which, due to its large mass, provides a stringent test of our model. We investigate the possible fates of disrupted bodies, the differences between prograde and retrograde captures, and the effects of Callisto on captured objects. We make an impulse approximation and discuss how it allows us to generalize capture results from equal-mass binaries to binaries with arbitrary mass ratios.We find that at Jupiter, binaries offer an increase of a factor of ∼10 in the capture rate of 100-km objects as compared to single bodies, for objects separated by tens of radii that approach the planet on relatively low-energy trajectories. These bodies are at risk of collision with Callisto, but may be preserved by gas drag if their pericenters are raised quickly enough. We conclude that our mechanism is as capable of producing large irregular satellites as previous suggestions, and it avoids several problems faced by alternative models.     

A co-accretional model of satellite formation

Numerical calculation of a simple accretion model including the effects of tidal friction indicate that coformation is tenable only if the planet's Q is less than about 10³. The parameter which most strongly affects the final mass ratio of the pair is the time at which the secondary embryo is introduced. Our model yields the proper Moon-Earth mass ratio if the Moon embryo is introduced when the Earth is only about $\text{1}\text{10}$ of its final mass. The lunar orbit remains at about 10 Earth radii throughout most of the growth.This model of satellite formation overcomes two difficulties of the “circumterrestrial cloud” model of Ruskol (1960, 1963, 1972): (1) The difficulty of accumulating a mass as great as the entire Moon before gravitational instability reduces the cloud to a small number of moonlets is removed. (2) The differences between terrestrial and outer planet satellite systems is easily understood in terms of the differences in Q between these planets. The high Q of the outer planets does not allow a satellite embryo to survive a significant portion of the accretion process, thus only small bodies which formed very late in the accumulation of the planet remain as satellites. The low Q of the terrestrial planets allows satellite embryos of these planets to survive during accretion, thus massive satellites such as the Earth's Moon are expected. The present lack of such satellites of the other terrestrial planets may be the result of tidal evolution, either infall following primary despinning (Burns, 1973) or escape due to increase in orbit eccentricity.     

Orbital resonances in the inner neptunian system: II. Resonant history of Proteus, Larissa, Galatea, and Despina

We investigate the orbital history of the small neptunian satellites discovered by Voyager 2. Over the age of the Solar System, tidal forces have caused the satellites to migrate radially, bringing them through mean-motion resonances with one another. In this paper, we extend our study of the largest satellites Proteus and Larissa [Zhang, K., Hamilton, D.P., 2007. Icarus 188, 386-399] by adding in mid-sized Galatea and Despina. We test the hypothesis that these moons all formed with zero inclinations, and that orbital resonances excited their tilts during tidal migration. We find that the current orbital inclinations of Proteus, Galatea, and Despina are consistent with resonant excitation if they have a common density . Larissa's inclination, however, is too large to have been caused by resonant kicks between these four satellites; we suggest that a prior resonant capture event involving either Naiad or Thalassa is responsible. Our solution requires at least three past resonances with Proteus, which helps constrain the tidal migration timescale and thus Neptune's tidal quality factor: 9000<QN<36,000. We also improve our determination of Qs for Proteus and Larissa, finding 36<QP<700 and 18<QL<200. Finally, we derive a more general resonant capture condition, and work out a resonant overlap criterion relevant to satellite orbital evolution around an oblate primary.     

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.     

Formation of the regular satellites of giant planets in an extended gaseous nebula II: satellite migration and survival

For a satellite to survive in the disk the time scale of satellite migration must be longer than the time scale for gas dissipation. For large satellites (∼1000 km) migration is dominated by the gas tidal torque. We consider the possibility that the redistribution of gas in the disk due to the tidal torque of a satellite with mass larger than the inviscid critical mass causes the satellite to stall and open a gap (W.R. Ward, 1997, Icarus 26, 261-281). We adapt the inviscid critical mass criterion to include gas drag, and m-dependent nonlocal deposition of angular momentum. We find that such a model holds promise of explaining the survival of satellites in the subnebula, the mass versus distance relationship apparent in the saturnian and uranian satellite systems, the concentration of mass in Titan, and the observation that the satellites of Jupiter get rockier closer to the planet whereas those of Saturn become increasingly icy. It is also possible that either weak turbulence (close to the planet) or gap-opening satellite tidal torque removes gas on a similar time scale (10⁴-10⁵ years) as the orbital decay time of midsized (200-700 km) regular satellites forming in the inner disk (inside the centrifugal radius (I. Mosqueira and P.R. Estrada, 2003, Icarus, this issue)). We argue that Saturn’s satellite system bridges the gap between those of Jupiter and Uranus by combining the formation of a Galilean-sized satellite in a gas optically thick subnebula with a strong temperature gradient, and the formation of smaller satellites, closer to the planet, in a disk with gas optical depth ?1, and a weak temperature gradient.Using an optically thick inner disk (given gaseous opacity), and an extended, quiescent, optically thin outer disk, we show that there are regions of the disk of small net tidal torque (even zero) where satellites (Iapetus-sized or larger) may stall far from the planet. For our model these outer regions of small net tidal torque correspond roughly to the locations of Callisto and Iapetus. Though the precise location depends on the (unknown) size of the transition region between the inner and outer disks, the result that Saturn’s is found much farther out (at ∼3rcS, where rcS is Saturn’s centrifugal radius) than Jupiter’s (at ∼ 2rcJ, where rcJ is Jupiter’s centrifugal radius) is mostly due to Saturn’s less massive outer disk and larger Hill radius. However, despite the large separation between Ganymede and Callisto and Titan and Iapetus, the long formation and migration time scales for Callisto and Iapetus (I. Mosqueira and P.R. Estrada, 2003, Icarus, this issue) makes it possible (depending on the details of the damping of acoustic waves) that the tidal torque of Ganymede and Titan clears the gas disk out to their location, thus stranding Callisto and Iapetus far from the planet. Either way, our model provides an explanation for the presence of regular satellites outside the centrifugal radii of Jupiter and Saturn, and the absence of such a satellite for Uranus.     

Irregular satellite capture during planetary resonance passage

The passage of Jupiter and Saturn through mutual 1:2 mean-motion resonance has recently been put forward as explanation for their relatively high eccentricities [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461] and the origin of Jupiter's Trojans [Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R., 2005. Nature 435, 462-465]. Additional constraints on this event based on other small-body populations would be highly desirable. Since some outer satellite orbits are known to be strongly affected by the near-resonance of Jupiter and Saturn (“the Great Inequality”; ?uk, M., Burns, J.A., 2004b. Astron. J. 128, 2518-2541), the irregular satellites are natural candidates for such a connection. In order to explore this scenario, we have integrated 9200 test particles around both Jupiter and Saturn while they went through a resonance-crossing event similar to that described by Tsiganis et al. [Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F., 2005. Nature 435, 459-461]. The test particles were positioned on a grid in semimajor axes and inclinations, while their initial pericenters were put at just 0.01 AU from their parent planets. The goal of the experiment was to find out if short-lived bodies, spiraling into the planet due to gas drag (or alternatively on orbits crossing those of the regular satellites), could have their pericenters raised by the resonant perturbations. We found that about 3% of the particles had their pericenters raised above 0.03 AU (i.e. beyond Iapetus) at Saturn, but the same happened for only 0.1% of the particles at Jupiter. The distribution of surviving particles at Saturn has strong similarities to that of the known irregular satellites. If saturnian irregular satellites had their origin during the 1:2 resonance crossing, they present an excellent probe into the early Solar System's evolution. We also explore the applicability of this mechanism for Uranus, and find that only some of the uranian irregular satellites have orbits consistent with resonant pericenter lifting. In particular, the more distant and eccentric satellites like Sycorax could be stabilized by this process, while closer-in moons with lower eccentricity orbits like Caliban probably did not evolve by this process alone.     

Dynamical history of coplanar two-satellite systems

One of the possible early states of the Earth-Moon system was a system of several large satellites around the Earth. The dynamical evolution of coplanar three-body systems is studied; a planet (Earth) and two massive satellites (proto-moons) with geocentric orbits of slightly different radii. Such configurations may arise in multiple satellite systems receding from a planet due to tidal friction. The numerical integration of the equations of motion shows that initially circular Keplerian orbits are soon transformed into disturbed elliptic orbits which are intersecting. The life-time of such a coplanar system between two probable physical collisions of satellites is roughly from one day to one year for satellite systems with radii less than 20R⊕, and may reach 100 yr for three-dimensional systems. This time-scale is short in comparison with the duration of the removal of satellites due to tides raised on the planet, which is estimated as 10⁶–10⁸ yr for the same orbital dimensions. Therefore, the life-time of a system of several proto-moons is mainly determined by their tidal interactions with the Earth. For conditions which we have considered, the most probable result of the evolution was coalescence of satellites as the consequence of the collisions.     

On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.     

The equilibrium of rubble-pile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates

Many new small moons of the giant planets have been discovered recently. In parallel, satellites of several asteroids, e.g., Ida, have been found. Strikingly, a majority of these new-found planetary moons are estimated to have very low densities, which, along with their hypothesized accretionary origins, suggests a rubble internal structure. This, coupled to the fact that many asteroids are also thought to be particle aggregates held together principally by self-gravity, motivates the present investigation into the possible ellipsoidal shapes that a rubble-pile satellite may achieve as it orbits an aspherical primary. Conversely, knowledge of the shape will constrain the granular aggregate's orbit—the closer it gets to a primary, both primary's tidal effect and the satellite's spin are greater. We will assume that the primary body is sufficiently massive so as not to be influenced by the satellite. However, we will incorporate the primary's possible ellipsoidal shape, e.g., flattening at its poles in the case of a planet, and the proloidal shape of asteroids. In this, the present investigation is an extension of the first classical Darwin problem to granular aggregates. General equations defining an ellipsoidal rubble pile's equilibrium about an ellipsoidal primary are developed. They are then utilized to scrutinize the possible granular nature of small inner moons of the giant planets. It is found that most satellites satisfy constraints necessary to exist as equilibrated granular aggregates. Objects like Naiad, Metis and Adrastea appear to violate these limits, but in doing so, provide clues to their internal density and/or structure. We also recover the Roche limit for a granular satellite of a spherical primary, and employ it to study the martian satellites, Phobos and Deimos, as well as to make contact with earlier work of Davidsson [Davidsson, B., 2001. Icarus 149, 375–383]. The satellite's interior will be modeled as a rigid-plastic, cohesion-less material with a Drucker–Prager yield criterion. This rheology is a reasonable first model for rubble piles. We will employ an approximate volume-averaging procedure that is based on the classical method of moments, and is an extension of the virial method [Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium. Yale Univ. Press, New Haven] to granular solid bodies.     

A gas-poor planetesimal capture model for the formation of giant planet satellite systems

Assuming that an unknown mechanism (e.g., gas turbulence) removes most of the subnebula gas disk in a timescale shorter than that for satellite formation, we develop a model for the formation of regular (and possibly at least some of the irregular) satellites around giant planets in a gas-poor environment. In this model, which follows along the lines of the work of Safronov et al. [1986. Satellites. Univ. of Arizona Press, Tucson, pp. 89-116], heliocentric planetesimals collide within the planet's Hill sphere and generate a circumplanetary disk of prograde and retrograde satellitesimals extending as far out as ∼RH/2. At first, the net angular momentum of this proto-satellite swarm is small, and collisions among satellitesimals leads to loss of mass from the outer disk, and delivers mass to the inner disk (where regular satellites form) in a timescale ?¹⁰⁵ years. This mass loss may be offset by continued collisional capture of sufficiently small <1 km interlopers resulting from the disruption of planetesimals in the feeding zone of the giant planet. As the planet's feeding zone is cleared in a timescale ?¹⁰⁵ years, enough angular momentum may be delivered to the proto-satellite swarm to account for the angular momentum of the regular satellites of Jupiter and Saturn. This feeding timescale is also roughly consistent with the independent constraint that the Galilean satellites formed in a timescale of ¹⁰⁵-¹⁰⁶ years, which may be long enough to accommodate Callisto's partially differentiated state [Anderson et al., 1998. Science 280, 1573; Anderson et al., 2001. Icarus 153, 157-161]. In turn, this formation timescale can be used to provide plausible constraints on the surface density of solids in the satellitesimal disk (excluding satellite embryos for satellitesimals of size ∼1 km), which yields a total disk mass smaller than the mass of the regular satellites, and means that the satellites must form in several ∼10 collisional cycles. However, much more work will need to be conducted concerning the collisional evolution both of the circumplanetary satellitesimals and of the heliocentric planetesimals following giant planet formation before one can assess the significance of this agreement. Furthermore, for enough mass to be delivered to form the regular satellites in the required timescale one may need to rely on (unproven) mechanisms to replenish the feeding zone of the giant planet. We compare this model to the solids-enhanced minimum mass (SEMM) model of Mosqueira and Estrada [2003a. Icarus 163, 198-231; 2003b. Icarus 163, 232-255], and discuss its main consequences for Cassini observations of the saturnian satellite system.     

When and where were the satellites of uranus formed?

Uranus exhibits an unusually large obliquity compared to other planets of the solar system; its equator is inclined by 98° to the plane of its orbit. However its five satellites are remarkably regular, with eccentricities and inclinations very nearly zero, but of course with orbit planes that are tilted by ～98° to the plane of the ecliptic. This circumstance is used here to relate the formation of satellites to planet formation. Six different cases are discussed, of which two can be ruled out and two others are highly improbable. In the analysis, use is made of the fact that satellites in near-equatorial orbits could not follow a rapid (“non-adiabatic”) change of the planet's obliquity. We assume, also, that the observed obliquity is the result of the last stages of planet accumulation. We can therefore exclude contemporaneous formation of planet and satellites, and conclude instead that the satellites were formed or acquired after the planet's axis had been tilted. A plausible scenario involves the tidal capture of a body having 5% to 10% of the planet's mass—sufficient to account for the tilt—followed by its accretion. However, tidal forces break up the body into chunks, slow the accretion, and allow ～1% of the chunks to form the satellites through interaction with a temporary dense atmosphere. The same reasoning may apply also for Saturn and Jupiter. It should be noted that the synchronous orbit it well within the Roche limit for all three planets.     

The fate of ejecta from Hyperion

Ejecta from Saturn's moon Hyperion are subject to powerful perturbations from nearby Titan, which control their ultimate fate. We have performed numerical integrations to simulate a simplified system consisting of Saturn (including optical flattening as well as dynamical oblateness), its main ring system (treated as a massless flat annulus), the moons Tethys, Dione, Titan, Hyperion, and Iapetus, and the Sun (treated simply as a massive satellite). At several different points in Hyperion's orbit, 1050 massless particles, more or less evenly distributed over latitude and longitude, were ejected radially outward from 1 km above Hyperion's mean radius at speeds 10% faster than escape speed from Hyperion. Most of these particles were removed within the first few thousand years, but ∼3% of them survived the entire 100,000-year duration of the simulations. Ejecta from Hyperion are much more widely scattered than previously thought, and can cross the orbits of all of Saturn's satellites. About 9% of all the particles escaped from the saturnian system, but Titan accreted ∼78% of the total, while Hyperion reaccreted only ∼5%. This low efficiency of reaccretion may help to account for Hyperion's small size and rugged shape. Only ∼1% of all the particles hit other satellites, and another ∼1% impacted Saturn itself, while ∼3% of them struck its main rings. The high proportion of impacts into Saturn's rings is surprising; these collisions show a broad decline in impact speed with time, suggesting that Hyperion ejecta gradually spread inwards. Additional simulations were used to investigate the dependence of ejecta evolution on launch speed, the mass of Hyperion, and the presence of the Sun. In general, the wide distribution of ejecta from Hyperion suggests that it does contribute to “Population II” craters on the inner satellites of Saturn. Ejecta which escape from a satellite into temporary orbit about its planet, but later reimpact into the same moon or another one produce “poltorary” impacts, intermediate in character between primary and secondary impacts. It may be possible to distinguish poltorary craters from primary and secondary craters on the basis of morphology.     

Ridge formation and de-spinning of Iapetus via an impact-generated satellite

We present a scenario for building the equatorial ridge and de-spinning Iapetus through an impact-generated disk and satellite. This impact puts debris into orbit, forming a ring inside the Roche limit and a satellite outside. This satellite rapidly pushes the ring material down to the surface of Iapetus, and then itself tidally evolves outward, thereby helping to de-spin Iapetus. This scenario can de-spin Iapetus an order of magnitude faster than when tides due to Saturn act alone, almost independently of its interior geophysical evolution. Eventually, the satellite is stripped from its orbit by Saturn. The range of satellite and impactor masses required is compatible with the estimated impact history of Iapetus.     

Tidal interactions and disruptions of giant planets on highly eccentric orbits

We calculate the evolution of planets undergoing a strong tidal encounter using smoothed particle hydrodynamics (SPH), for a range of periastron separations. We find that outside the Roche limit, the evolution of the planet is well-described by the standard model of linear, non-radial, adiabatic oscillations. If the planet passes within the Roche limit at periastron, however, mass can be stripped from it, but in no case do we find enough energy transferred to the planet to lead to complete disruption. In light of the three new extrasolar planets discovered with periods shorter than two days, we argue that the shortest-period cases observed in the period-mass relation may be explained by a model whereby planets undergo strong tidal encounters with stars, after either being scattered by dynamical interactions into highly eccentric orbits, or tidally captured from nearly parabolic orbits. Although this scenario does provide a natural explanation for the edge found for planets at twice the Roche limit, it does not explain how such planets will survive the inevitable expansion that results from energy injection during tidal circularization.     

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.