Multi-frequency electromagnetic sounding of the Galilean moons

Induced magnetic fields provide the unique possibility to sound the conductive interior of planetary bodies. Such fields are caused by external time-variable magnetic fields. We investigate temporal variations of the jovian magnetospheric field at multiple frequencies at the positions of the Galilean moons and analyze possible responses due to electromagnetic induction within multi-layered interior models of all four satellites. At the jovian satellites the magnetic field varies with the synodic rotation period of Jupiter’s internal field (about 10 h), fractions of this period (e.g., 1/2 and 1/3) due to higher order harmonics of the internal field, the orbital periods of the satellites (∼40 h at Io to ∼400 h at Callisto) and the solar rotation period (about 640 h) and its harmonics due to variabilities of the magnetopause field. To analyze these field variations, we use a magnetospheric model that includes the jovian internal field, the current sheet field and fields due to the magnetopause boundary currents. With this model we calculate magnetic amplitude spectra for each satellite orbit. These spectra provide the strengths of the inducing signals at the different frequencies for all magnetic components. The magnetic fields induced in the interiors of the satellites are then determined from response functions computed for different multi-layer interior models including conductive cores and ocean layers of various conductivities and thicknesses. Based on these results we discuss what information about the ocean and core layers can be deduced from the analysis of induction signals at multiple frequencies. Even moderately thick and conductive oceans produce measurable signal strengths at several frequencies for all satellites. The conductive cores cause signals which will be hardly detectable. Our results show that mutual induction occurs between the core and the ocean. We briefly address this effect and its implications for the analysis of induced field data. We further note that close polar orbits are preferable for future Jupiter system missions to investigate the satellites interiors.     

On the implications of the shape of Mars

Determinations of the ellipticity of the surface of Mars reveal a large discrepancy with the dynamical value obtained from the precession of the orbits of the satellites. Because the dynamical value is in accordance with that expected on the basis of hydrostatic theory, assuming the density of Mars was almost uniform, the optical value was doubted. However, the discrepancy might be explained by solid state convection in the deep interior of Mars. Observations from occultations of the Mariner spacecraft have confirmed the existence of the discrepancy between the ellipticities; thus, the role of convection in the Martian interior may be a key to its evolution.     

Rotational distortion of stars of arbitrary structure to fourth-order sectorial harmonics

An explicit form of the radial part of the exterior gravitational potential of a planet or star, in hydrostatic equilibrium is expanded in terms of fourth-order sectorial harmonics. In developing the theory of figure one seeks to express the Clairaut equation for an equipotential spheroid (level surface) in an explicit form which is an integral form to quantities of fourth-order sectorial harmonics.     

Effects of zonal harmonics on the out-of-plane equilibrium points in the generalized Robe's circular restricted three-body problem

This article examines the effects of the zonal harmonics on the out-of-plane equilibrium points of Robe's circular restricted three-body problem when the hydrostatic equilibrium shape of the first primary is an oblate spheroid, the shape of the second primary is an oblate spheroid with oblateness coefficients up to the second zonal harmonic, and the full buoyancy of the fluid is considered. It is observed that the size of the oblateness and the zonal harmonics affect the positions of the out-of-plane equilibrium points L₆ and L₇. It is also observed that these points within the possible region of motion are unstable.     

Shapes of the saturnian icy satellites and their significance

The sizes and shapes of six icy saturnian satellites have been measured from Cassini Imaging Science Subsystem (ISS) data, employing limb coordinates and stereogrammetric control points. Mimas, Enceladus, Tethys, Dione and Rhea are well described by triaxial ellipsoids; Iapetus is best represented by an oblate spheroid. All satellites appear to have approached relaxed, equilibrium shapes at some point in their evolution, but all support at least 300 m of global-wavelength topography. The shape of Enceladus is most consistent with a homogeneous interior. If Enceladus is differentiated, its shape and apparent relaxation require either lateral inhomogeneities in an icy mantle and/or an irregularly shaped core. Iapetus supports a fossil bulge of over 30 km, and provides a benchmark for impact modification of shapes after global relaxation. Satellites such as Mimas that have smoother limbs than Iapetus, and are expected to have higher impact rates, must have relaxed after the shape of Iapetus was frozen.     

Stagnant lid convection in the mid-sized icy satellites of Saturn

Thermal history models for the mid-sized saturnian satellites Mimas, Tethys, Dione, Iapetus, and Rhea have been calculated assuming stagnant lid convection in undifferentiated satellites and varying parameter values over broad ranges. Of all five satellites under consideration, only Dione, Rhea and Iapetus do show significant internal activities related to convective overturn for extended periods of time. The interiors of Mimas and Tethys do not convect or do so only for brief periods of time early in their thermal histories. Although we use lower densities than previous models, our calculations suggest higher interior temperatures but also thicker rigid shells above the convecting regions. Temperatures in the stagnant lid will allow melting of ammonia-dihydrate. Dione, Rhea and Iapetus may differentiate early and form early oceans, Iapetus only if ammonia is present. Mimas and Tethys with ammonia may differentiate if they accreted in an optically thick nebula with ambient temperatures around 250 K. Our models suggest that the outer shells of the satellites are largely primordial in composition even if the satellites differentiated. In these cases the deep interior may be layered with a pure ice shell underlain by an ammonia dihydrate layer and a rock core.     

The Martian Nonhydrostatic Components of Moment-of-Inertia

The author puts forward the proposal in this paper that all the terrestrial planets (Venus, the Earth, and Mars) as well as the Moon deviate from hydrostatic equilibrium to some degree. The Earth's level of deviation of these four celestial bodies is minimum, and that of Mars is maximum. Moreover, the author estimates Martian nonhydrostatic components of the principal moments-of-inertia using five models for the interior of Mars. Comparison with other terrestrial planets shows that setting the range of mean moment-of-inertia ratio, I/MR², in 0.345 ~ 0.355for Mars is reasonable. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Spherical harmonic representation of the gravity field from dynamic satellite data

Spherical harmonics are the natural parameters for the Earth's gravity field as sensed by orbiting satellites, but problems of resolution arise because the spectrum of effects is narrow and unique to each orbit. Comprehensive gravity models now contain many hundreds of thousands of observations from more than thirty different near-Earth artificial satellites. With refinements in tracking systems, newer data is capable of sensing the spherical harmonics of the field experienced by these satellites to very high degree and order. For example, altimeter, laser and satellite-tracking-satellite systems contain gravitational information well above present levels of satellite gravity field recovery (l = 20), but significant aliasing results because the orbital parameters are too restricted compared to the large number of spherical harmonics.It is shown however that the unique spectrum of information for each satellite contained within a comprehensive spherical harmonic model can be represented by simple gravitational constraint equations (lumped harmonics). All such constraints are harmonic in the argument of perigee (ω) with constants determinable directly from tracking data or reconstituted from the comprehensive solution:

$\text{(C}{}^1\text{, S}{}^1\text{) = (C}{}_{\mathrm{o}}\text{, S}{}_{\mathrm{o}}\text{) + Σ}{}_{\mathrm{i}}\text{= 1 (C}{}_{\mathrm{Ci}}\text{, S}{}_{\mathrm{Ci}}\text{) cos i ω + (C}{}_{\mathrm{Si}}\text{, S}{}_{\mathrm{Si}}\text{) sin i ω}$

. The constants are simple linear combinations of the geopotential harmonics. Through these lumped harmonics any satellite gravity field can be decomposed and then uniformly extended to any degree or tailored to a given orbit without reintegration of the trajectory and variational equations. They also make possible the inclusion of information into the field from special deep resonance passages, long arc zonal analyses, and satellites unique to other models. Numerous examples of the derivation, combination, extension and tailoring of the harmonics are presented. The importance of using data spanning an apsidal period is emphasized.     

Analytic solar structure

In this paper an attempt is made to derive an analytic solar model by assuming a one-parameter family of density distributions. The analytic representation of the solar interior is derived for hydrostatic equilibrium and energy conservation. The mathematics involved in equating the solar luminosity to the thermonuclear energy generation near the center is illustrated in terms of special functions.     

Effects of differential rotation on the gravitational figures of Jupiter and Saturn

It is assumed that observed zonal currents in the atmospheres of Jupiter and Saturn correspond to a state of permanent rotation, and that the angular velocity is constant on cylindrical surfaces parallel to the rotation axis. The equation of hydrostatic equilibrium for a rotating planet is solved under these restrictive assumptions, and the effect of the hypothesized rotation state on the planet's gravity harmonics and external shape is investigated. Spacecraft data on zonal currents are used to derive nearly model-independent corrections to the first four zonal gravity harmonic coefficients, which can be used to correct observed gravity harmonics to values appropriate for solid-body rotation. If the assumed rotation state is applicable, then zonal currents lead to measurable topography of isopycnic surfaces with respect to the reference fihure defined by the magnetospheric rotation period and the gravity harmonics. The amplitude of the topography is on the order of 5 km for Jupiter and 60 km for Saturn.     

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.     

Implications from Galileo Observations on the Interior Structure and Chemistry of the Galilean Satellites

Data from the recent gravity measurements by the Galileo mission are used to construct wide ranges of interior structure and composition models for the Galilean satellites of Jupiter. These models show that mantle densities of Io and Europa are consistent with an olivine-dominated mineralogy with the ratios of Mg to Fe components depending on mantle temperature for Io and on ice shell thickness for Europa. The mantle density and composition depend relatively little on core composition. The size of the core is largely determined by the core's composition with core radius increasing with the concentration of a light component such as sulfur. For Io, the range of possible core sizes is between 38 and 53% of the satellite's radius. For Europa, there is also a substantial effect of the thickness of the ice layer which is varied between 120 and 170 km on the core size. Core sizes are between 10 and 45% of Europa's radius. The core size of Ganymede ranges between one-quarter and one-third of the surface radius depending on its sulfur content and the thickness of the ice shell. A subset of the Ganymede models is consistent with an olivine-dominated mantle mineralogy. The thickness of the silicate mantle above the core varies between 900 and 1100 km. The outermost ice shell is about 900 km in thickness and is further subdivided by pressure-induced phase transitions into ice I, ice III, ice V, and ice VI layers. Callisto should be differentiated, albeit incompletely. It is proposed that this satellite was never molten at a large scale but differentiated through the convective gradual unmixing of the ice and the metal/rock component. Bulk iron-to-silicon ratios Fe/Si calculated for the inner pair of satellites, Io and Europa, are less than the CI carbonaceous chondrite value of 1.7±0.1, whereas ratios for the outer pair, Ganymede and Callisto, cover a broad range above the chondritic value. Although the ratios are uncertain, in particular for Ganymede and Callisto, the values are sufficiently distinct to suggest a difference in composition between these two pairs of satellites. This may indicate a difference in iron-silicon fractionation during the formation of both classes of satellites in the protojovian nebula.     

The equilibrium of a rotating body of arbitrary density

This paper extends Clairaut's theory of rotational equilibrium to third order terms in a small parameter and is meant to be a sequel to a 1962 publication by the author bearing on the same topic. It has been feasible to obtain the Clairaut equation, which governs the deformation of the equipotential surfaces within a rapidly rotating mass in hydrostatic equilibrium, as an ordinary differential equation. This has been achieved by eliminating the two integral terms which appeared in the original formulation. It is expected that the numerical integration of this newly obtained equation will contribute toward a more precise solution of certain geophysical problems — e.g., the determination of the geoid to an accuracy of ±1 m, and the correction to the travel-time of seismic waves; it should also assist in some planetary questions like the determination of the exterior shape for the rapidly rotating planets Jupiter and Saturn.     

Models, figures and gravitational moments of Jupiter’s satellite Io: Effects of the second order approximation

The effects of the equilibrium figure theory to within terms of the second order in a small parameter α on figure parameters and gravitational moments of the Galilean satellite Io have been considered. Integro-differential equations of the theory of figure to second order have been first solved numerically. Relations between the low-order coefficients of the gravitational field for satellites in hydrostatic equilibrium are generalized according to the second order theory. To show the effects of the second approximation, two three-layer trial models of Io are used. The considered models of the Io’s interiors differ by the size and density of the core, while having the same thickness and density of the crust, and the mantle density difference is only 20 kg/m³. The corrections of second order in smallness to the gravitational moments J₂ and C₂₂ decrease the third decimal digit of model gravitational moments by two units. As the effects of third and forth harmonics are determined mostly by outer layers of Io, to distinguish between model mantle density, the gravitational moments J₄, C₄₂ and C₄₄ should be determined to accuracy with three or four decimal digits. The second order corrections mostly effect the semi-axis a, and less the semi-axes b and c.     

Coupling of thermal evolution and despinning of early Iapetus

The Cassini mission revealed two spectacular characteristics of Iapetus: (1) a geologically old and high equatorial ridge, which is unique in the Solar System and (2) a large flattening of 35 km consistent with the equilibrium figure for a hydrostatic body rotating with a period of 16 h, whereas the current spin period is 79.33 days. This study describes three-dimensional simulations of solid-state convection within an undifferentiated Iapetus. It investigates the implications for the evolution of the interior thermal structure and its spin rate and global shape using radially layered viscoelastic models. The role of the concentration in the short-lived radiogenic element [²⁶Al], just after accretion is completed, is specifically addressed. The first result is to show that whatever the [²⁶Al] value, convection occurs. As suggested by Castillo-Rogez et al. [Castillo-Rogez, J., Matson, D., Sotin, C., Johnson, T., Lunine, J., Thomas, P. [2007] Icarus, 190, 179-202], convection reduces the warming of the interior compared to the conductive evolution and therefore limits the conditions for despinning. In our calculations, two conceptual linear viscoelastic models are used. When considering a Maxwell rheology, the interior temperature (viscosity) never reaches a value high (low) enough to induce despinning. In order to promote dissipation at low temperature, a Burgers rheology, which includes an additional dissipation peak, is introduced. For favorable parameter values, this latter rheology leads to despinning. However, only models associated with large amounts of short-lived radiogenic elements lead to the observed flattening. This suggests that the accretion process needs to be completed shortly after the formation of CAIs (Calcium-Aluminum-rich Inclusions) (?4 Myr). For [²⁶Al] varying between 72 and 46 ppb, the observed flattening is obtained only for a limited range of initial spin period, between 9.5 and 10.2 h. For [²⁶Al] ranging between 30 and 15 ppb, initial spin rates smaller than 8.5 h are required. For smaller values of [²⁶Al], the body is too cold and viscous to acquire a significant flattening even if a rotation period close to the body disruption limit is considered. Even with a thin lithosphere during the early stage, our simulations show that Iapetus never reaches the equilibrium figure for a hydrostatic body due to the non-zero rigidity of the lithosphere. The 35 km value of the flattening is the result of the partial relaxation of an ancient larger flattening ranging between 45 and 80 km, depending on the evolution of the lithosphere thickness mainly controlled by the radiogenic content. A thin lithosphere is consistent with an early building of the equatorial ridge. The lithosphere thickening due to interior cooling can explain the preservation of the ridge throughout the remaining evolution of Iapetus.     

The equilibrium figures of Phobos and other small bodies

The shape of a close planetary satellite is distorted from a self-gravitating sphere into a triaxial ellipsoid maintained by tidal and centrifugal forces. Using the family of Roche ellipsoids calculated by Chandrasekhar, it should be possible in some cases to determine the density of an inner satellite by an accurate measurement of its shape alone. The equilibrium figure of Phobos is expected to be the most extreme of any satellite. The shape of Phobos as observed by Mariner 9 approaches but appears not to be a Roche ellipsoid, although the uncertainties of measurement remain too large to exclude the possibility. In any case, Phobos is so small that even the low mechanical strength of an impact-compressed regolith is sufficient to maintain substantial departures from the equipotential figure. If larger close satellites, particularly Amalthea, are found to be Roche ellipsoids, their densities can be estimated immediately from the data presented.Asteroids of size comparable to Phobos and Deimos appear to have more irregular shapes than the Martian satellites. This may reflect the absence of a deep regolith on those asteroids due to the low effective escape velocity for impact ejecta. For Phobos and Deimos, on the other hand, ejecta will tend to remain in orbit about Mars until swept up again by the satellite, contributing to a deeper equilibrium layer of debris.     

The effect of gravitational and pressure torques on Titan's length-of-day variations

Cassini radar observations show that Titan's spin is slightly faster than synchronous spin. Angular momentum exchange between Titan's surface and the atmosphere over seasonal time scales corresponding to Saturn's orbital period of 29.5 year is the most likely cause of the observed non-synchronous rotation. We study the effect of Saturn's gravitational torque and torques between internal layers on the length-of-day (LOD) variations driven by the atmosphere. Because static tides deform Titan into an ellipsoid with the long axis approximately in the direction to Saturn, non-zero gravitational and pressure torques exist that can change the rotation rate of Titan. For the torque calculation, we estimate the flattening of Titan and its interior layers under the assumption of hydrostatic equilibrium. The gravitational forcing by Saturn, due to misalignment of the long axis of Titan with the line joining the mass centers of Titan and Saturn, reduces the LOD variations with respect to those for a spherical Titan by an order of magnitude. Internal gravitational and pressure coupling between the ice shell and the interior beneath a putative ocean tends to reduce any differential rotation between shell and interior and reduces further the LOD variations by a few times. For the current estimate of the atmospheric torque, we obtain LOD variations of a hydrostatic Titan that are more than 100 times smaller than the observations indicate when Titan has no ocean as well as when a subsurface ocean exists. Moreover, Saturn's torque causes the rotation to be slower than synchronous in contrast to the Cassini observations. The calculated LOD variations could be increased if the atmospheric torque is larger than predicted and or if fast viscous relaxation of the ice shell could reduce the gravitational coupling, but it remains to be studied if a two order of magnitude increase is possible and if these effects can explain the phase difference of the predicted rotation variations. Alternatively, the large differences with the observations may suggest that non-hydrostatic effects in Titan are important. In particular, we show that the amplitude and phase of the calculated rotation variations are similar to the observed values if non-hydrostatic effects could strongly reduce the equatorial flattening of the ice shell above an internal ocean.     

Structure of ices on satellites

Ices on satellites in the solar system undergo changes produced by meteoritic bombardment, pressure, and thermal effects. The effect of the meteoritic bombardment on (porous) ices is some densification, but mainly the formation of crystalline H₂O polymorphs and the establishment of a rough equilibrium ratio between hexagonal and amorphous forms below 150°K. As a result of the low temperatures, the pressure densification of porous ices is significant only at depths of at least hundreds of meters for large satellites. The densification process is controlled by creep, that is, by slow plastic deformation of the solid matrix for medium porosities and by diffusion for low porosities. The isothermal effect on porous ice is an extremely slow densification process caused by surface and volume diffusion. A thermal gradient leads to migration of pores toward the warmer end and, since the velocity of the pores is proportional to their size, to their clustering. As a result, smaller pores become eliminated and the pore size distribution changes. Quantitative analysis of these effects has been made for ices including the integrodifferential coagulation equation which gives the new pore size distribution and the steepening of the gradient of porosity. For CO₂-ice the rates of these effects can be estimated to be several orders of magnitude higher than for H₂O-ice. Various physical properties are significantly affected and, in particular, it is concluded that, on a time scale of 10⁸ to 10⁹ years, in satellites with a cold interior the outer icy layers may have become densified while the opposite is true when satellites such as Europa and perhaps Enceladus have an internal source of heat.     

Nonhydrostatic effects and the determination of icy satellites’ moment of inertia

We compare the moment of inertia (MOI) of a simple hydrostatic, two layer body as determined by the Radau–Darwin Approximation (RDA) to its exact hydrostatic MOI calculated to first order in the parameter q = Ω²R³/GM, where Ω, R, and M are the spin angular velocity, radius, and mass of the body, and G is the gravitational constant. We show that the RDA is in error by less than 1% for many configurations of core sizes and layer densities congruent with those of solid bodies in the Solar System. We then determine the error in the MOI of icy satellites calculated with the RDA due to nonhydrostatic effects by using a simple model in which the core and outer shell have slight degree 2 distortions away from their expected hydrostatic shapes. Since the hydrostatic shape has an associated stress of order ρΩ²R² (where ρ is density) it follows that the importance of nonhydrostatic effects scales with the dimensionless number σ/ρΩ²R², where σ is the nonhydrostatic stress. This highlights the likely importance of this error for slowly rotating bodies (e.g., Titan and Callisto) and small bodies (e.g., Saturn moons other than Titan). We apply this model to Titan, Callisto, and Enceladus and find that the RDA-derived MOI can be 10% greater than the actual MOI for nonhydrostatic stresses as small as ∼0.1 bars at the surface or ∼1 bar at the core–mantle boundary, but the actual nonhydrostatic stresses for a given shape change depends on the specifics of the interior model. When we apply this model to Ganymede we find that the stresses necessary to produce the same MOI errors as on Titan, Callisto, and Enceladus are an order of magnitude greater due to its faster rotation, so Ganymede may be the only instance where RDA is reliable. We argue that if satellites can reorient to the lowest energy state then RDA will always give an overestimate of the true MOI. Observations have shown that small nonhydrostatic gravity anomalies exist on Ganymede and Titan, at least at degree 3 and presumably higher. If these anomalies are indicative of the nonhydrostatic anomalies at degree 2 then these imply only a small correction to the MOI, even for Titan, but it is possible that the physical origin of nonhydrostatic degree 2 effects is different from the higher order terms. We conclude that nonhydrostatic effects could be present to an extent that allows Callisto and Titan to be fully differentiated.