Effects of low-viscous layers and a non-zero obliquity on surface stresses induced by diurnal tides and non-synchronous rotation: The case of Europa

In this study we present a semi-analytical Maxwell-viscoelastic model of the variable tidal stress field acting on Europa’s surface. In our analysis, we take into account surface stresses induced by the small eccentricity of Europa’s orbit, the non-zero obliquity of Europa’s spin axis - both acting on a diurnal 3.55-days timescale - and the reorientation of the ice shell as a result of non-synchronous rotation (NSR). We assume that Europa’s putative ocean is covered by an ice shell, which we subdivide in a low-viscous and warm lower ice layer (asthenosphere, viscosity 10¹²-10¹⁷ Pa s), and a high-viscous and cold upper ice layer (lithosphere, viscosity 10²¹ Pa s).Viscoelastic relaxation influences surface stresses in two ways: (1) through viscoelastic relaxation in the lithosphere and (2) through the viscoelastic tidal response of Europa’s interior. The amount of relaxation in the lithosphere is proportional to the ratio between the period of the forcing mechanism and the Maxwell time of the high-viscous lithosphere. As a result, this effect is only relevant to surface stresses caused by the slow NSR mechanism. On the other hand, the importance of the viscoelastic response on surface stresses is proportional to the ratio between the relaxation time (τ_j) of a given viscoelastic mode j and the period of the forcing function. On a diurnal timescale the fast relaxation of transient modes related to the low viscosity of the asthenosphere can alter the magnitude and phase shift of the diurnal stress field at Europa’s surface. The effects are largest, up to 20% in magnitude and 7° in phase for ice rigidities lower than 3.487 GPa, when the relaxation time of the aforementioned transient modes approaches the inverse of the average angular rate of Europa’s orbit. On timescales relevant for NSR (>10⁴ years) the magnitude and phase shift of NSR surface stresses can be affected by viscoelastic relaxation of the ocean-ice boundary. This effect, however, becomes only important when the behavior of the lithosphere w.r.t. NSR approaches the fluid limit, i.e. for strong relaxation in the lithosphere. The combination of NSR and diurnal stresses for different amounts of viscoelastic relaxation of NSR stresses in the lithosphere leads to a large variety of global stress fields that can explain the formation of the large diversity of lineament morphologies observed on Europa’s surface. Variation of the amount of relaxation in the lithosphere is likely due to changes in the spin rate of Europa and/or the rheological properties of the surface.In addition, we show that a small obliquity(<1°) can have a considerable effect on Europa’s diurnal stress field. A non-zero obliquity breaks the symmetric distribution of stress patterns with respect to the equator, thereby affecting the magnitude and orientation of the principal stresses at the surface. As expected, increasing the value of Europa’s obliquity leads to larger diurnal stresses at the surface, especially when Europa is located 90° away from the nodes formed by the intersection of its orbital and equatorial planes.     

The global shape of Europa: Constraints on lateral shell thickness variations

The global shape of Europa is controlled by tidal and rotational potentials and possibly by lateral variations in ice shell thickness. We use limb profiles from four Galileo images to determine the best-fit hydrostatic shape, yielding a mean radius of 1560.8±0.3 km and a radius difference a−c of 3.0±0.9 km, consistent with previous determinations and inferences from gravity observations. Adding long-wavelength topography due to proposed lateral variations in shell thickness results in poorer fits to the limb profiles. We conclude that lateral shell thickness variations and long-wavelength isostatically supported topographic variations do not exceed 7 and 0.7 km, respectively. For the range of rheologies investigated (basal viscosities from 10¹⁴ to ) the maximum permissible (conductive) shell thickness is 35 km. The relative uniformity of Europa's shell thickness is due to either a heat flux from the silicate interior, lateral ice flow at the base of the shell, or convection within the shell.     

Nonsynchronous Rotation Evidence and Fracture History in the Bright Plains Region, Europa

A geologic map for the Bright Plains in the Conamara Chaos region of Europa is presented and is used to unravel a detailed fracture sequence using cross-cutting relationships and fracture mechanics principles. Fracture orientations in the Bright Plains region rotated with time, consistently in a clockwise sense. This conclusion agrees with the observations of other researchers' northern Europan hemisphere investigations and points strongly toward the fracture sequence being controlled by the effect of nonsynchronous rotation, whereby the outer ice crust of Europa rotates slightly faster than the satellite's interior. This is convincing evidence that Europa's crust has been decoupled from the interior, possibly due to the presence of a liquid ocean beneath the crust.Tidal stresses induced in the ice crust by the combined effects of nonsynchronous rotation and diurnal tidal flexing can be calculated using the assumption that the crust behaves elastically over relatively short time scales (i.e., no viscous relaxation of stresses). The fracture orientations in the Bright Plains area were compared to a global scale tidal stress field to determine the longitudes at which each fracture set developed. The fracture sequence points strongly to the Bright Plains region of the crust having rotated at least 720° (and perhaps up to 900°) with respect to the satellite's interior during the visible fracture history. This amount exceeds previously published estimates of nonsynchronous rotation. The orientations of the most recent surface fractures are incompatible with the current state of stress in the Bright Plains region, implying a period of a few thousand years since the most recent fracturing events based on existing nonsynchronous rotation rate estimates.     

The librations, shape, and icy shell of Europa

Europa, the smallest of the Galilean satellites, has a young icy surface and most likely contains an internal ocean. The primary objective of possible future missions to Europa is the unambiguous detection and characterization of a subsurface ocean. The thickness of the overlying icy shell provides important information on the thermal evolution of the satellite and on the interaction between the ocean and the surface, the latter being fundamental for astrobiology. However, the thickness is not well known, and estimates range from several hundred of meters to some ten of kilometers. Here, we investigate the use of libration (rotation variation) observations to study the interior structure of Europa and in particular its icy shell. A dynamical libration model is developed, which includes gravitational coupling between the icy shell and the heavy solid interior. The amplitude of the main libration signal at 3.55 days (the orbital period) is shown to depend on Europa's shape and structure. Models of the interior structure of Europa are constructed and the equatorial flattening of the internal layers, which are key parameters for the libration, are calculated by assuming that Europa is in hydrostatic equilibrium. Europa's flattened shape is assumed to be due to rotation and permanent tides, and we extend the classical Radau equation for rotationally flattened bodies to include also tidal deformation. We show that the presence of an ocean increases the amplitude of libration by about 10%, depending mainly on the thickness of the icy shell. Therefore, libration observations offer possibility of detection of a subsurface ocean in Europa and estimation of the thickness of its overlying icy shell.     

Cycloid crack sequences on Europa: Relationship to stress history and constraints on growth mechanics based on cusp angles

The original model developed to explain cycloidal cracks on Europa interprets cycloids as tensile fractures that grow in a curved path in response to the constantly rotating diurnal tidal stress field. Cusps form when a new cycloid crack segment propagates at an angle to the first in response to a rotation of the principal tidal stress orientation during a period of no crack growth. A recent revised model states that a cycloid cusp forms through the creation of a secondary fracture called a tailcrack at the tip of an existing cycloid segment during shearing motion induced by the rotating tidal stress field. As the tailcrack propagates away from the cusp, it becomes the next cycloid segment in the chain. The qualitative tailcrack model uniquely accounts for the normal and shear stresses that mechanically must resolve onto the tip of an existing cycloid segment at the instant of cusp formation. In this work, we provide a quantitative framework and test of the hitherto purely conceptual tailcrack model. We first present a relative age sequence inferred from geologic mapping of multiply cross-cutting cycloids in Europa's trailing hemisphere and place this into the context of the global stress history. The age sequence requires a cumulative minimum of 630° of shell reorientation due to nonsynchronous rotation to account for the observed range of orientations of cycloids of different ages. We determined the back-rotated longitudes of formation of two cycloid chain examples and used mathematical modeling of europan tidal stresses to show that the tailcrack model for cusp formation is not only viable, but places constraints on the overall development of a cycloid chain by controlling the timing of cusp development within Europa's orbit. For all cusps analyzed, the exact ratio of resolved shear to normal stress required to form the cusp angles by a process of tailcracking, as governed by the principles of linear elastic fracture mechanics, is produced at the tip of a shearing cycloid segment during Europa's orbit. Cusp formation occurs after the point in the orbit at which the maximum tensile principal tidal stress occurs, implying that tensile tidal stresses are not directly responsible for cusp development. Instead, cusps develop when a tailcrack forms at the tip of a cycloid segment in response to the highly perturbed stress field induced during concomitant opening and shearing at the tip of the cycloid segment.     

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.     

Convective-conductive transitions and sensitivity of a convecting ice shell to perturbations in heat flux and tidal-heating rate: Implications for Europa

We investigate the response of conductive and convective ice shells on Europa to variations of heat flux and interior tidal-heating rate. We present numerical simulations of convection in Europa's ice shell with Newtonian, temperature-dependent viscosity and tidal heating. Modest variations in the heat flux supplied to the base of a convective ice shell, ΔF, can cause large variations of the ice-shell thickness Δδ. In contrast, for a conductive ice shell, large ΔF involves relatively small Δδ. We demonstrate that, for a fluid with temperature-dependent viscosity, the heat flux undergoes a finite-amplitude jump at the critical Rayleigh number Racr. This jump implies that, for a range of heat fluxes relevant to Europa, two equilibrium states—corresponding to a thin, conductive shell and a thick, convective shell—exist for a given heat flux. We show that, as a result, modest variations in heat flux near the critical Rayleigh number can force the ice shell to switch between the thin, conductive and thick, convective configurations over a ∼10⁷-year interval, with thickness changes of up to ∼10-30 km. Depending on the orbital and thermal history, such switches might occur repeatedly. However, existing evolution models based on parameterized-convection schemes have to date not allowed these transitions to occur. Rapid thickening of the ice shell would cause radial expansion of Europa, which could produce extensional tectonic features such as fractures or bands. Furthermore, based on interpretations for how features such as chaos and ridges are formed, several authors have suggested that Europa's ice shell has recently undergone changes in thickness. Our model provides a mechanism for such changes to occur.     

Thermal-orbital evolution of Io and Europa

Coupled thermal-orbital evolution models of Europa and Io are presented. It is assumed that Io, Europa, and Ganymede evolve in the Laplace resonance and that tidal dissipation of orbital energy is an internal heat source for both Io and Europa. While dissipation in Io occurs in the mantle as in the mantle dissipation model of Segatz et al. (1988, Icarus 75, 187), two models for Europa are considered. In the first model dissipation occurs in the silicate mantle while in the second model dissipation occurs in the ice shell. In the latter model, ice shell melting and variations of the shell thickness above an ocean are explicitly included. The rheology of both the ice and the rock is cast in terms of a viscoelastic Maxwell rheology with viscosity and shear modulus depending on the average temperature of the dissipating layer. Heat transfer by convection is calculated using a parameterization for strongly temperature-dependent viscosity convection. Both models are consistent with the present orbital elements of Io, Europa, and Ganymede. It is shown that there may be phases of quasi-steady evolution with large or small dissipation rates (in comparison with radiogenic heating), phases with runaway heating or cooling and oscillatory phases during which the eccentricity and the tidal heating rate will oscillate. Europa's ice thickness varies between roughly 3 and 70 km (dissipation in the silicate layer) or 10 and 60 km (dissipation in the ice layer), suggesting that Europa's ocean existed for geological timescales. The variation in ice thickness, including both convective and purely conductive phases, may be reflected in the formation of different geological surface features on Europa. Both models suggest that at present Europa's ice thickness is several tens of km thick and is increasing, while the eccentricity decreases, implying that the satellites evolve out of resonance. Including lithospheric growth in the models makes it impossible to match the high heat flux constraint for Io. Other heat transfer processes than conduction through the lithosphere must be important for the present Io.     

The influence of obliquity on europan cycloid formation

Tectonic patterns on Europa are influenced by tidal stress. An important well-recognized component is associated with the orbital eccentricity, which produces a diurnally varying stress as Jupiter's apparent position in Europa's sky oscillates in longitude. Cycloidal lineaments seem to have formed as cracks propagated in this diurnally varying stress field. Maps of theoretical cycloid patterns capture many of the characteristics of the observed distribution on Europa. However, a few details of the observed cycloid distribution have not been reproduced by previous models. Recently, it has been shown that Europa has a finite forced obliquity, so Jupiter's apparent position in Europa's sky will also oscillate in latitude. We explore this new type of diurnal effect on cycloid formation. We find that stress from obliquity may be the key to explaining several characteristics of observed cycloids such as the shape of equator-crossing cycloids and the shift in the crack patterns in the Argadnel Regio region. All of these improvements of the fit between observation and theory seem to require Jupiter crossing Europa's equatorial plane 45° to 180° after perijove passage, suggestive of complex orbital dynamics that locks the direction of Europa's pericenter with the direction of the ascending node at the time these cycloids were formed.     

Empirical constraints on the salinity of the europan ocean and implications for a thin ice shell

Induced electrical currents within Europa inferred from Galileo spacecraft magnetometer instrument data have been interpreted as due to a salty europan ocean. Published compositional models for Europa's ocean, based on aqueous leaching of carbonaceous chondrites, range over five orders of magnitude in predicted magnesium sulfate concentrations. We combine the Galileo spacecraft magnetometer-derived oceanic conductivities and radio Doppler data-derived interior models with laboratory conductivity vs concentration data for both magnesium sulfate solutions and terrestrial seawater to determine empirically the range of salt concentrations permitted for Europa's ocean. Solutions for both a three-layer spherical model, and a five-layer half-space model, that satisfy current preferred best fits to magnetometer data imply high, near-saturation salt concentrations and require a europan ice shell of less than 15 km thick, with a best fit at 4 km ice thickness. Adding a conductive core and mantle has a negligible effect on the amplitude when ocean conductivities are greater than a few Siemens per meter. Similarly, we find that including a realistic ionosphere has a negligible effect. We examine the implications of these results for the subsurface habitability of Europa.     

Thermal Equilibrium States of Europa's Ice Shell: Implications for Internal Ocean Thickness and Surface Heat Flow

Ice-shell thickness and ocean depth are calculated for steady state models of tidal dissipation in Europa's ice shell using the present-day values of the orbital elements. The tidal dissipation rate is obtained using a viscoelastic Maxwell rheology for the ice, the viscosity of which has been varied over a wide range, and is found to strongly increase if an (inviscid) internal ocean is present. To determine steady state values, the tidal dissipation rate is equated to the heat-transfer rate through the ice shell calculated from a parameterized model of convective heat transfer or from a thermal conduction model, if the ice layer is found to be stable against convection. Although high dissipation rates and heat fluxes of up to 300 mWm⁻² are, in principle, possible for Europa, these values are unrealistic because the states for which they are obtained are thermodynamically unstable. Equilibrium models have surface heat flows around 20 mWm⁻² and ice-layer thicknesses around 30 km, which is significantly less than the total thickness of the H₂O-layer. These results support models of Europa with ice shells a few tens of kilometers thick and around 100-km-thick subsurface oceans.     

Constraints on dissipation in the deep interiors of Ganymede and Europa from tidal phase-lags

Jupiter’s satellites are subject to strong tidal forces which result in variations of the gravitational potential and deformations of the satellites’ surfaces on the diurnal tidal cycle. Such variations are described by the Love numbers \(k_2\) and \(h_2\) for the tide-induced potential variation due to internal mass redistribution and the radial surface displacement, respectively. The phase-lags \( \phi _{k_2}\) and \( \phi _{h_2}\) of these complex numbers contain information about the rheological and dissipative states of the satellites. Starting from interior structure models and assuming a Maxwell rheology to compute the tidal deformation, we calculate the phase-lags in application to Ganymede and Europa. For both satellites we assume a decoupling of the outer ice-shell from the deep interior by a liquid subsurface water ocean. We show that, in this case, the phase-lag difference \(\varDelta \phi = \phi _{k_2}- \phi _{h_2}\) can provide information on the rheological and thermal state of the deep interiors if the viscosities of the deeper layers are small. In case of Ganymede, phase-lag differences can reach values of a few degrees for high-pressure ice viscosities \({<}10^{14}\) Pa s and would indicate a highly dissipative state of the deep interior. In this case \(\varDelta \phi \) is dominated by dissipation in the high-pressure ice layer rather than dissipation within the ice-I shell. These phase lags would be detectable from spacecraft in orbit around the satellite. For Europa \(\varDelta \phi \) could reach values exceeding \(20^\circ \) and phase-lag measurements could help distinguish between (1) a hot dissipative silicate mantle which would in thermal equilibrium correspond to a very thin outer ice-I shell and (2) a cold deep interior implying that dissipation would mainly occur in a thick (several tens of km) outer ice-I shell. These measurements are highly relevant for ESA’s Jupiter Icy Moons Explorer (JUICE) and NASA’s Europa Multiple Flyby Mission, both targeted for the Jupiter system.     

Subsurface oceans and deep interiors of medium-sized outer planet satellites and large trans-neptunian objects

The detection of induced magnetic fields in the vicinity of the jovian satellites Europa, Ganymede, and Callisto is one of the most surprising findings of the Galileo mission to Jupiter. The observed magnetic signature cannot be generated in solid ice or in silicate rock. It rather suggests the existence of electrically conducting reservoirs of liquid water beneath the satellites' outermost icy shells that may contain even more water than all terrestrial oceans combined. The maintenance of liquid water layers is closely related to the internal structure, composition, and thermal state of the corresponding satellite interior. In this study we investigate the possibility of subsurface oceans in the medium-sized icy satellites and the largest trans-neptunian objects (TNO's). Controlling parameters for subsurface ocean formation are the radiogenic heating rate of the silicate component and the effectiveness of the heat transfer to the surface. Furthermore, the melting temperature of ice will be significantly reduced by small amounts of salts and/or incorporated volatiles such as methane and ammonia that are highly abundant in the outer Solar System. Based on the assumption that the satellites are differentiated and using an equilibrium condition between the heat production rate in the rocky cores and the heat loss through the ice shell, we find that subsurface oceans are possible on Rhea, Titania, Oberon, Triton, and Pluto and on the largest TNO's 2003 UB₃₁₃, Sedna, and 2004 DW. Subsurface oceans can even exist if only small amounts of ammonia are available. The liquid subsurface reservoirs are located deeply underneath an ice-I shell of more than 100 km thickness. However, they may be indirectly detectable by their interaction with the surrounding magnetic fields and charged particles and by the magnitude of a satellite's response to tides exerted by the primary. The latter is strongly dependent on the occurrence of a subsurface ocean which provides greater flexibility to a satellite's rigid outer ice shell.     

Surface-bounded atmosphere of Europa

A 1-D collisional Monte Carlo model of Europa's atmosphere is described in which the sublimation and sputtering sources of H₂O molecules and their molecular fragments are accounted for as well as the radiolytically produced O₂. Dissociation and ionization of H₂O and O₂ by magnetospheric electron, solar UV-photon and photo-electron impact, and collisional ejection from the atmosphere by the low-energy plasma are taken into account. Reactions with the surface are discussed, but only adsorption and atomic oxygen recombination are included in this model. The size of the surface-bounded oxygen atmosphere of Europa is primarily determined by a balance between atmospheric sources from irradiation of the satellite's icy surface by the high-energy magnetospheric charged particles and atmospheric losses from collisional ejection by the low-energy plasma, photo- and electron-impact dissociation, and ionization and pick-up from the surface-bounded atmosphere. A range of sources rates for O₂ to H₂O are used with a larger oxygen-to-water ratio than suggested by laboratory measurements in order to account for differences in adsorption onto grains in the regolith. These calculations show that the atmospheric composition is determined by both the water and oxygen photochemistry in the near-surface region, escape of suprathermal oxygen and water into the jovian system, and the exchange of radiolytic water products with the porous regolith. For the electron impact ionization rates used, pick-up ionization is the dominant oxygen loss process, whereas photo-dissociation and atmospheric sputtering are the dominant sources of neutral oxygen for Europa's neutral torus. Including desorption and loss of water enhances the supply of oxygen species to the neutral torus, but hydrogen produced by radiolysis is the dominant source of neutrals for Europa's torus in these models.     

Equilibrium Convection on a Tidally Heated and Stressed Icy Shell of Europa for a Composite Water Ice Rheology

Water ice I rheology is a key factor for understanding the thermal and mechanical state of the outer shell of the icy satellites. Ice flow involves several deformation mechanisms (both Newtonian and non-Newtonian), which contribute to different extents depending on the temperature, grain size, and applied stress. In this work I analyze tidally heated and stressed equilibrium convection in the ice shell of Europa by considering a composite viscosity law which includes diffusion creep, basal slip, grain boundary sliding and dislocation creep, and. The calculations take into account the effect of tidal stresses on ice flow and use grain sizes between 0.1 and 100 mm. An Arrhenius-type relation (useful for parameterized convective models) is found then by fitting the calculated viscosity between 170 and 273 K to an exponential regression, which can be expressed in terms of pre-exponential constant and effective activation energy. I obtain convective heat flows between ~40 and ~60 mW m^?2, values lower than those usually deduced (~100 mW m^?2) from geological indicators of lithospheric thermal state, probably indicating heterogeneous tidal heating. On the other hand, for grain sizes larger than ~0.3 mm the thicknesses of the ice shell and convective sublayer are ~20–30 km and ~5–20 km respectively, values in good agreement with the available information for Europa. So, some fundamental geophysical characteristics of the ice shell of Europa could be arising from the properties of the composite water ice rheology.     

Fracture penetration in planetary ice shells

The proposed past eruption of liquid water on Europa and ongoing eruption of water vapor and ice on Enceladus have led to discussion about the feasibility of cracking a planetary ice shell. We use a boundary element method to model crack penetration in an ice shell subjected to tension and hydrostatic compression. We consider the presence of a region at the base of the ice shell in which the far-field extensional stresses vanish due to viscoelastic relaxation, impeding the penetration of fractures towards a subsurface ocean. The maximum extent of fracture penetration can be limited by hydrostatic pressure or by the presence of the unstressed basal layer, depending on its thickness. Our results indicate that Europa's ice shell is likely to be cracked under 1-3 MPa tension only if it is ?2.5 km thick. Enceladus' ice shell may be completely cracked if it is capable of supporting ∼1-3 MPa tension and is less than 25 km thick.     

Elastic ice shells of synchronous moons: Implications for cracks on Europa and non-synchronous rotation of Titan

A number of synchronous moons are thought to harbor water oceans beneath their outer ice shells. A subsurface ocean frictionally decouples the shell from the interior. This has led to proposals that a weak tidal or atmospheric torque might cause the shell to rotate differentially with respect to the synchronously rotating interior. Applications along these lines have been made to Europa and Titan. However, the shell is coupled to the ocean by an elastic torque. As a result of centrifugal and tidal forces, the ocean would assume an ellipsoidal shape with its long axis aligned toward the parent planet. Any displacement of the shell away from its equilibrium position would induce strains thereby increasing its elastic energy and giving rise to an elastic restoring torque. In the investigation reported on here, the elastic torque is compared with the tidal torque acting on Europa and the atmospheric torque acting on Titan.Regarding Europa, it is shown that the tidal torque is far too weak to produce stresses that could fracture the ice shell, thus refuting an idea that has been widely advocated. Instead, it is suggested that the cracks arise from time-dependent stresses due to non-hydrostatic gravity anomalies from tidally driven, episodic convection in the satellite’s interior.Two years of Cassini RADAR observations of Titan’s surface have been interpreted as implying an angular displacement of ∼0.24° relative to synchronous rotation. Compatibility of the amplitude and phase of the observed non-synchronous rotation with estimates of the atmospheric torque requires that Titan’s shell be decoupled from its interior. We find that the elastic torque balances the seasonal atmospheric torque at an angular displacement ?0.05°, effectively coupling the shell to the interior. Moreover, if Titan’s surface were spinning faster than synchronous, the tidal torque tending to restore synchronous rotation would almost certainly be larger than the atmospheric torque. There must either be a problem with the interpretation of the radar observations, or with our basic understanding of Titan’s atmosphere and/or interior.     

Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

The fate of dwarf galaxies in clusters and the origin of intracluster stars

The main goal of this paper is to compare the relative importance of destruction by tides vs. destruction by mergers, in order to assess if tidal destruction of galaxies in clusters is a viable scenario for explaining the origin of intracluster stars. We have designed a simple algorithm for simulating the evolution of isolated clusters. The distribution of galaxies in the cluster is evolved using a direct gravitational N-body algorithm combined with a subgrid treatment of physical processes such as mergers, tidal disruption, and galaxy harassment. Using this algorithm, we have performed a total of 148 simulations. Our main results are:

The spatial morphology of Europa's near-surface O2 atmosphere

Results from a three-dimensional ballistic model of Europa's O₂ atmosphere are presented. Hubble Space Telescope (HST) ultraviolet observations show spatially non-uniform O₂ airglow from Europa. One explanation for this is that the O₂ atmosphere is spatially non-uniform. We show that non-uniform ejection of O₂ alone cannot reproduce the required morphology, but that a non-uniform distribution of reactive species in Europa's porous regolith can result in a non-uniform O₂ atmosphere. By allowing O₂ molecules to react with Europa's visibly dark surface material, we produced a spatially non-uniform atmosphere which, assuming uniform electron excitation of O₂ over the trailing hemisphere, compares favorably with the morphology suggested by the HST observations. This model, which requires a larger source of O₂ than has previously been estimated, can in principal be tested by the New Horizons observations of Europa's O₂ atmosphere.