Internal structures of the Galilean satellites

New models for the interiors of Io, Ganymede, and Callisto are proposed. The model of Io consists of a thin, high-rigidity outer layer separated from a solid interior by a thin, molten or partially molten shell. The modulus of rigidity of the outer layer must be at least 100 times larger than that of the underlying partially molten shell. These layers have thicknesses of order 100 km or less. The near-surface partially molten layer was most likely produced early in Io's history as a consequence of accretional heating; enhanced tidal heating in the outer rigid layer has kept the underlying region partially molten to the present day. The model of Ganymede consists of an ice outer layer, a shell of undifferentiated, primordial ice-silicate mixture, and a rock core. Accretional heating is responsible for melting the ice in the outer layers of Ganymede's initially homogeneous ice-silicate interior. Most of the rock in this outer layer accumulates in a shell on top of Ganymede's early cold and rigid central region; the water in the outer layer quickly refreezes. Heating of the undifferentiated region by the decay of radioactive elements in the silicate fraction would gradually warm it and reduce its viscosity. The rock layer would become gravitationally unstable and sink through the undifferentiated materials to form a rock core. Callisto's heavily cratered surface strongly suggests that relatively little, if any, ice-rock differentiation has occured in its interior.     

Thermal history of the Jovian satellite Io

The discovery of large volcanic eruptions on Io suggests that Io is one of the most geologically active planetary bodies. The energy source of this geologic activity is believed to be tidal heating induced by Jupiter. A number of thermal history calculations were done to investigate the effect of tidal heating on the thermal history of Io taking into account solid state convection and advective heat transfer. These simulations show that the total tidal heating energy in Io is almost equal to the advectively transferred heat, indicating that the observed heat flow from Io is nearly equal to the total tidal heating energy. Since total tidal heating energy is dependent on the radius of the liquid mantle and the internal dissipation factor (Q), the radius of the liquid mantle can be estimated for a given value of Q. Some reasonable thermal history models of Io were obtained using a model with Q ≈ 25–50 in which the magma source of Ionian volcanism is at a depth of 100–300 km. The models satisfy the heat flow data and the existence of a thick lithosphere. Using a model with Q = 25 and L = 300 km (thickness of the advective region) as the standard model (model II), we then studied the effect of convective heat transfer and the initial temperature distribution on the Ionian thermal history. In these calculations, the other parameters are the same as in the standard model (model II). These calculations show that although the temperature distribution in the central region reflects the difference in the efficiency of convective heat transfer and initial temperature distribution, the temperature distribution in the outer region does not changes appreciably.     

Turbulent viscosity and Jupiter's tidal Q

A recent estimate of tidal dissipation by turbulent viscosity in Jupiter's convective interior predicts that the current value of the planet's tidal Q ～ 5 × 10⁶. We point out a fundamental error in this calculation, and show that turbulent dissipation alone implies that at present Q ～ 5 × 10¹³. Our reduced estimat for the rate of tidal dissipation shows conclusively that tidal torques have produced only negligible modifications of the orbits of the Galilean satellites over the age of the solar system.     

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.     

Is Io's Mantle Really Molten?

The tremendous heating dissipated by jovian tides in Io's interior is essentially evacuated by an intense volcanic activity so that the heat is removed from the interior to the surface, much more by advection than by conduction through the lithosphere. The efficiency of this heat pipe cooling process is investigated through numerical models of convection performed in spherical geometry with a permeable top boundary. This new heat-transfer model provides a cooling twice as efficient as that obtained with an impermeable condition traditionally used in mantle convection modeling. The globally averaged temperature varies as Ra^−1/2, where Ra is the Rayleigh number, whereas the power law exponent is classically −1/4, so that the expected Ra would not be in excess of 10⁷. If the whole mantle of Io is involved in the convection process, the major portion could remain solid, while a possible molten zone could be confined to a 100-km-thick layer between the solid part and the core. This model predicts the existence of a strong lithosphere, which is required to support the observed topographic amplitude of the Io's relief.     

A post-Galileo view of Io's interior

We present a self-consistent model for the interior of Io, taking the recent Galileo data into account. In this model, Io has a completely molten core, substantially molten mantle, and a very cold lithosphere. Heat from magmatic activity can mobilize volatile compounds such as SO₂ in the lithosphere, and the movement of such cryogenic fluids may be important in the formation of surface features including sapping scarps and paterae.     

Tidal heating and the long-term stability of a subsurface ocean on Enceladus

Tidal dissipation has been suggested as the heat source for the south polar thermal anomaly on Enceladus. We find that under present-day conditions and assuming Maxwellian behavior, tidal dissipation is negligible in the silicate core. Dissipation may be significant in the ice shell if the shell is decoupled from the silicate core by a subsurface ocean. We have run a series of self-consistent convection and conduction models in 2D axisymmetric and 3D spherical geometry in which we include the spatially-variable tidal heat production. We find that in all cases, the shell removes more heat from the interior than can be produced in the core by radioactive decay, resulting in cooling of the interior and the freezing of any ocean. Under likely conditions, a 40-km thick ocean made of pure water would freeze solid on a ∼30 Ma timescale. An ocean containing other chemical components will have a lower freezing point, but even a water-ammonia eutectic composition will only prolong the freezing, not prevent it. If the eccentricity of Enceladus were higher (e?0.015) in the past, the increased dissipation in the ice shell may have been sufficient to maintain a liquid layer. We cannot therefore rule out the presence of a transient ocean, as a relic of an earlier era of greater heating. If the eccentricity is periodically pumped up, the ocean may have thickened and thinned on a similar timescale as the orbital evolution, provided the ocean never froze completely. We conclude that the current heat flux of Enceladus and any possible subsurface ocean is not in steady-state, and is the remnant of an epoch of higher eccentricity and tidal dissipation.     

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.     

Geophysical implications of the Martian Gravity Field

The undulations of the Martian gravitational potential indicate lateral density variations in the Mars interior. A gravitating and solid Martian model deforms under the influence of these variations, producing stress differences of about 125 bars at a depth of about 200 km. Introduction of a partially molten core of 1300 km radius does not affect the stress distribution in the mantle significantly, whereas the assumption of a partially molten asthenosphere umderlying a solid lithosphere of about 300 km increases the stress differences appreciably. A strong linear correlation of the gravitational potential and the surface topography indicates that the extensive volcanism at the Tharsis region is a recent phenomenon. The high stresses associated with this region imply that there has been no extensive molten region within the upper 300 km since the volcanism.     

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.     

Modeling stresses on satellites due to nonsynchronous rotation and orbital eccentricity using gravitational potential theory

The tidal stress at the surface of a satellite is derived from the gravitational potential of the satellite's parent planet, assuming that the satellite is fully differentiated into a silicate core, a global subsurface ocean, and a decoupled, viscoelastic lithospheric shell. We consider two types of time variability for the tidal force acting on the shell: one caused by the satellite's eccentric orbit within the planet's gravitational field (diurnal tides), and one due to nonsynchronous rotation (NSR) of the shell relative to the satellite's core, which is presumed to be tidally locked. In calculating surface stresses, this method allows the Love numbers h and ?, describing the satellite's tidal response, to be specified independently; it allows the use of frequency-dependent viscoelastic rheologies (e.g. a Maxwell solid); and its mathematical form is amenable to the inclusion of stresses due to individual tides. The lithosphere can respond to NSR forcing either viscously or elastically depending on the value of the parameter , where μ and η are the shear modulus and viscosity of the shell respectively, and ω is the NSR forcing frequency. Δ is proportional to the ratio of the forcing period to the viscous relaxation time. When Δ?1 the response is nearly fluid; when Δ?1 it is nearly elastic. In the elastic case, tensile stresses due to NSR on Europa can be as large as ∼3.3 MPa, which dominate the ∼50 kPa stresses predicted to result from Europa's diurnal tides. The faster the viscous relaxation the smaller the NSR stresses, such that diurnal stresses dominate when Δ?100. Given the uncertainty in current estimates of the NSR period and of the viscosity of Europa's ice shell, it is unclear which tide should be dominant. For Europa, tidal stresses are relatively insensitive both to the rheological structure beneath the ice layer and to the thickness of the icy shell. The phase shift between the tidal potential and the resulting stresses increases with Δ. This shift can displace the NSR stresses longitudinally by as much as 45° in the direction opposite of the satellite's rotation.     

Tidal dissipation within large icy satellites: Applications to Europa and Titan

This paper describes a new approach based on variational principles to calculate the radial distribution of tidal energy dissipation in any satellite. The advantage of the model with respect to classical solutions, is that it relates in a straightforward way the radial distribution of the time-averaged dissipation rate to its sensitivity to the corresponding distribution of viscoelastic parameters. This method is applied to Io-, Europa-, and Titan-like interiors, and it is tested against the results obtained by two classical methods by determining global dissipation as well as radial and lateral distributions within satellite interiors. By exploring systematically the different parameters defining the interior models, we demonstrate that the presence of a deep ocean below an outer ice layer strongly influences the tidal dissipation distribution in both the outer ice layer and in the innermost part of the satellite. On the one hand, the ocean by imposing a large radial displacement at the base of the outer ice I layer, controls the distribution of tidal strain rate within the outer layer, making the tidal strain rate field very weakly sensitive to the viscosity variations. Conversely, in the high-pressure ice layer below the ocean, both tidal strain rate and dissipation are very sensitive to any variation of the ice viscosity. On the other hand, for identical structures of the mantle and of the core, the presence of a subsurface ocean reduces the strength of dissipation in the silicate mantle. The existence of a liquid layer within Europa makes models of the silicate mantle less dissipative than the predictions for Io.     

Tidally Induced Volcanism

The dissipation of tidal energy causes the ongoing silicate volcanism on Jupiter's satellite, Io, and cryovolcanism almost certainly has resurfaced parts of Saturn's satellite, Enceladus, at various epochs distributed over the latter's history. The maintenance of tidal dissipation in Io and the occurrence of the same on Enceladus depends crucially on the maintenance of the respective orbital eccentricities by the existence of mean motion resonances with nearby satellites. A formation of the resonances among the Galilean satellites by differential expansion of the satellite orbits from tides raised on Jupiter by the satellites means the onset of the volcanism on Io could be relatively recent. If, on the other hand, the resonances formed by differential migration from resonant interactions of the satellites with the disk of gas and particles from which they formed, Io would have been at least intermittently volcanically active throughout its history. Either means of assembling the Galilean satellite resonances lead to the same constraint on the dissipation function of Jupiter Q _J 10⁶, where the currently high heat flux from Io seems to favor episodic heating as Io's eccentricity periodically increases and decreases. Either of the two models might account for sufficient tidal dissipation in the icy satellite Enceladus to cause at least occasional cryovolcanism over much of its history. However, both models are assumption-dependent and not secure, so uncertainty remains on how tidal dissipation resurfaced Enceladus.     

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.     

Solid tidal friction above a liquid water reservoir as the origin of the south pole hotspot on Enceladus

Earth, Jupiter's moon Io and Saturn's tiny moon Enceladus are the only solid objects in the Solar System to be sufficiently geologically active for their internal heat to be detected by remote sensing. Interestingly, the endogenic activity on Enceladus is only located on a specific region at the south pole, from which jets of water vapor and ice particles have been observed [Spencer, J.R., and 9 colleagues, 2006. Science 311, 1401-1405; Porco, C.C., and 24 colleagues, 2006. Science 311, 1393-1401]. The current polar location of the thermal anomaly can possibly be explained by diapir-induced reorientation of the satellite [Nimmo, F., Pappalardo, R.T., 2006. Nature 441, 614-616], but the thermal anomaly triggering and the heat power required to sustain it over geological timescales remain problematic. Using a three-dimensional viscoelastic numerical model simulating the response of Enceladus to tidal forcing, we explore the effect of a low-viscosity anomaly in the ice shell, localized to the south polar region, on the tidal dissipation patterns. We demonstrate that only interior models with a liquid water layer at depth can explain the observed magnitude of dissipation rate and its particular location at the south pole. Moreover, we show that tidally-induced heat must be generated over a relatively broad region in the southern hemisphere, and it is then transferred toward the south pole where it is episodically released during relatively short resurfacing events. As large tidal dissipation and internal melting cannot be induced in the south polar region in the absence of a pre-existing liquid decoupling layer, we propose that liquid water must have been present in the interior for a very long period of time, and possibly since the satellite formation. Owing to the orbital equilibrium requirement [Meyer, J., Wisdom, J., 2007. Icarus 188, 535-539], sustaining some liquid water at depth is impossible if heat is continuously emitted at a rate of 4-8 GW at the south pole. Based on that requirement, we propose that the current thermal emission is not in equilibrium with the heat production, and that the thermal emission rate is abnormally high at present time. Alternatively, continuous dissipation at a rate of 1-2 GW in the ice shell at the south pole should be sufficient to induce internal melting and it could sustain a layer of liquid water at depth over geologic timescales.     

Editorial

The Galilean satellites Io, Europa, and Ganymede interact through several stable orbital resonances where $\text{λ}{}_1\text{? 2λ}{}_2\text{+}\text{ω}{}_1\text{= 0, λ}{}_1\text{? 2λ}{}_2\text{+}\text{ω}{}_2\text{= 180°, λ}{}_2\text{? 2λ}{}_3\text{+}\text{ω}{}_2\text{= 0}$ and λ₁ ? 3λ₂ + 2λ₃ = 180°, with λ_i being the mean longitude of the ith satellite and $\text{ω}{}_{\mathrm{i}}$ the longitude of the pericenter. The last relation involving all three bodies is known as the Laplace relation. A theory of origin and subsequent evolution of these resonances outlined earlier (C. F. Yoder, 1979b, Nature279, 747–770) is described in detail. From an initially quasi-random distribution of the orbits the resonances are assembled through differential tidal expansion of the orbits. Io is driven out most rapidly and the first two resonance variables above are captured into libration about 0 and 180° respectively with unit probability. The orbits of Io and Europa expand together maintaining the 2:1 orbital commensurability and Europa's mean angular velocity approaches a value which is twice that of Ganymede. The third resonance variable and simultaneously the Laplace angle are captured into libration with probability ～0.9. The tidal dissipation in Io is vital for the rapid damping of the libration amplitudes and for the establishment of a quasi-stationary orbital configuration. Here the eccentricity of Io's orbit is determined by a balance between the effects of tidal dissipation in Io and that in Jupiter, and its measured value leads to the relation $\text{k}{}_1\text{?}{}_1\text{/Q}{}_1\text{≈ 900k}{}_{\mathrm{<mtext>J</mtext>}}\text{/Q}{}_{\mathrm{<mtext>J</mtext>}}$ with the k's being Love numbers, the Q's dissipation factors, and f a factor to account for a molten core in Io. This relation and an upper bound on Q₁ deduced from Io's observed thermal activity establishes the bounds 6 × 10⁴ < Q_J < 2 × 10⁶, where the lower bound follows from the limited expansion of the satellite orbits. The damping time for the Laplace libration and therefore a minimum lifetime of the resonance is 1600 Q_J years. Passage of the system through nearby three-body resonances excites free eccentricities. The remnant free eccentricity of Europa leads to the relation $\text{Q}{}_2\text{/?}{}_2\text{? 2 × 10}{}^{\mathrm{?4}}\text{Q}{}_{\mathrm{<mtext>J</mtext>}}$ for rigidity μ₂ = 5 × 10¹¹ dynes/cm². Probable capture into any of several stable 3:1 two-body resonances implies that the ratio of the orbital mean motions of any adjacent pair of satellites was never this large.A generalized Hamiltonian theory of the resonances in which third-order terms in eccentricity are retained is developed to evaluate the hypothesis that the resonances were of primordial origin. The Laplace relation is unstable for values of Io's eccentricity e₁ > 0.012 showing that the theory which retains only the linear terms in e₁ is not valid for values of e₁ larger than about twice the current value. Processes by which the resonances can be established at the time of satellite formation are undefined, but even if primordial formation is conjectured, the bounds established above for Q_J cannot be relaxed. Electromagnetic torques on Io are also not sufficient to relax the bounds on Q_J. Some ideas on processes for the dissipation of ideal energy in Jupiter yield values of Q_J within the dynamical bounds, but no theory has produced a Q_J small enough to be compatible with the measurements of heat flow from Io given the above relation between Q₁ and Q_J. Tentative observational bounds on the secular acceleration of Io's mean motion are also shown not to be consistent with such low values of Q_J. Io's heat flow may therefore be episodic. Q_J may actually be determined from improved analysis of 300 years of eclipse data.     

Tidal disruption of dissipative planetesimals

Tidal disruption is a potentially important process for the accumulation of the planets from planetesimals. The fact that stable equilibria do not exist for circular orbits inside the Roche limit has often been hypothesized to mean that any object that passes within the Roche limit is totally disrupted. We have disproven this hypothesis by solving the dynamic problem of the tidal disruption of a dissipative planetestimal during a close encounter with a protoplanet. The solution consists of a numerical integration of the three-dimensional, nonlinear equations of motion, including an approximate treatment of viscous dissipation in the solid regions of the planetesimal. The numerical methods have been extensively tested on a series of one-, two- (Jeans), and three-(Roche) dimensional test problems involving the equilibrium of a body subjected to tidal forces. The results may be scaled to planetesimals of arbitrary size, providing that the scaled equation of state applied. The calculations show that a strongly dissipative planetesimal which passes by the Earth on a parabolic orbit with a perigee within the Roche limit (≈3R_Earth) is not tidally disrupted (even for grazing incidence), and loses no more than a few percent of its mass. This result applies to bodies of radius R which have a kinematic viscosity ν ? 10¹²(R/1000km)² cm²sec^?1. Less dissipative planetesimals (ν ≈ 10¹³(R/1000 km)² cm²sec^?1) may lose up to about 20% of their mass. There are two coupled reasons why this result differs from previous hypotheses: (1) in a dynamic encounter, there is insufficient time to disrupt the planetesimal, and (2) even in circular orbit, the small velocities in the solid region imply that many orbital periods are necessary to completely disrupt the planetesimal. Hence solid and partially molten planetesimals will not experience substantial tidal disruption; completely molten bodies may be sufficiently inviscid to undergo tidal disruption.     

The evolution of the moon

The thermal evolution of the Moon as it can be defined by the available data and theoretical calculations is discussed. A wide assortment of geological, geochemical and geophysical data constrain both the present-day temperatures and the thermal history of the lunar interior. On the basis of these data, the Moon is characterized as a differentiated body with a crust, a 1000-km-thick solid mantle (lithosphere) and an interior region (core) which may be partially molten. The presence of a crust indicates extensive melting and differentiation early in the lunar history. The ages of lunar samples define the chronology of igneous activity on the lunar surface. This covers a time span of about 1.5 billion yr, from the origin to about 3.16 billion yr ago. Most theoretical models require extensive melting early in the lunar history, and the outward differentiation of radioactive heat sources.Thermal history calculations, whether based on conductive or convective computation codes define relatively narrow bounds for the present day temperatures in the lunar mantle. In the inner region of the 700 km radius, the temperature limits are wider and are between about 100 and 1600°C at the center of the Moon. This central region could have a partially or totally molten core.The lunar heat flow values (about 30 ergs/cm²s) restrict the present day average uranium abundance to 60 ± 15 ppb (averaged for the whole Moon) with typical ratios of K/U = 2000 and Th/U = 3.5. This is consistent with an achondritic bulk composition for the Moon.The Moon, because of its smaller size, evolved rapidly as compared to the Earth and Mars. The lunar interior is cooling everywhere at the present and the Moon is tectonically inactive while Mars could be and the Earth is definitely active.     

The production of Ganymede's magnetic field

One of the great discoveries of NASA's Galileo mission was the presence of an intrinsically produced magnetic field at Ganymede. Generation of the relatively strong (750 nT) field likely requires dynamo action in Ganymede's metallic core, but how such a dynamo has been maintained into the present epoch remains uncertain. Using a one-dimensional, three layer thermal model of Ganymede, we find that magnetic field generation can only occur if the sulfur mass fraction in Ganymede's core is very low (?3%) or very high (?21%), and the silicate mantle can cool rapidly (i.e. it has a viscosity like wet olivine). However, these requirements are not necessarily compatible with cosmochemical and physical models of the satellite. We therefore investigate an alternative scenario for producing Ganymede's magnetic field in which passage through an eccentricity pumping Laplace-like resonance in Ganymede's past enables present day dynamo action in the metallic core. If sufficient tidal dissipation occurs in Ganymede's silicate mantle during resonance passage, silicate temperatures can undergo a runaway which prevents the core from cooling until the resonance passage ends. The rapid silicate and core cooling that follows resonance escape triggers dynamo action via thermal and/or compositional convection. To test the feasibility of this mechanism we couple our thermal model with an orbital evolution model to examine the effects of resonance passage on Ganymede's silicate mantle and metallic core. We find that, contrary to expectations, there are no physically plausible scenarios in which tidal heating in the silicates is sufficient to cause the thermal runaway necessary to prevent core cooling. These findings are robust to variations in the silicate rheology, tidal dissipation factor of Jupiter (QJ), structure of the ice shell, and the inclusion of partial melting in the silicate mantle. Resonance passage therefore appears unlikely to explain Ganymede's magnetic field and we must appeal to the special conditions described above to explain the presence of the field.     

Tidal dissipation in rotating fluid bodies: a simplified model

We study the tidal forcing, propagation and dissipation of linear inertial waves in a rotating fluid body. The intentionally simplified model involves a perfectly rigid core surrounded by a deep ocean consisting of a homogeneous incompressible fluid. Centrifugal effects are neglected, but the Coriolis force is considered in full, and dissipation occurs through viscous or frictional forces. The dissipation rate exhibits a complicated dependence on the tidal frequency and generally increases with the size of the core. In certain intervals of frequency, efficient dissipation is found to occur even for very small values of the coefficient of viscosity or friction. We discuss the results with reference to wave attractors, critical latitudes and other features of the propagation of inertial waves within the fluid, and comment on their relevance for tidal dissipation in planets and stars.