Origin and thermal evolution of icy satellites

O R I G I N A N D T H E R M A L E V O L U T I O N O F I C Y S A T E L L I T E S

ANGIOLETTA CORADINI 1AS (CNR ) - Planetologia, viale dell' Universitd 11, 00185 Roma (Italy)

COSTANZO FEDERICO Dipartimento di Scienze della Terra - Universitgt degli Studi, 06100 Perugia (Italy)

OLIVIER FORNI IAS - Bat 121, Universit~ Paris XI, 91405 Orsay Cedex (France)

and

GIANFRANCO MAGNI IAS (CNR) - Planetologia, viale dell'Universitd 11, 00185 Roma (Italy)

Abstract. The paper reviews the problem of formation and evolution of the so-called "regular satellites "of the giant planets, and it consists of two parts: the first describes the possible origin of the satellites, the second studies their evolution, attempting to stress the relations of the present status of the satellites with their evolutionary history.

The formation of regular satellite systems around giant planets is probably related to the formation of the central planet. Some characteristics of regular satellite systems are quite similar, and suggest a common origin in a disk present around the central body. This disk can originate through different mechanisms which we will describe, paying attention to the so-called "accretion disk" model, in which the satellite-forming material is captured. The disk phase links the formation of the primary body with the formation of satellites. The subsequent stages of the disk's evolution can lead first to the formation of intermediate size bodies, and through the collisional evolution of these bodies, to the birth of satellite "embryos' I able to gravitationally capture smaller bodies.

Given the scenario in which icy satellites may be formed by homogeneous accretion of planetesimals made of a mixtures of ice and silicates, if no melting occurs during accretion, the satellites have a homogeneous ice-rock composition. For the smaller satellites this homogeneous structure should not be substantially modified; only sporadic local events, such as large impacts, can modify the surface structure of the smaller satellites. For the larger satellites, if some degree of melting appears during accretion, a differentiation of the silicate part occurs, the amount of differentiation and hence the core size depending on the fraction of gravitational potential energy retained during the accumulation process. Melting and differentiation soon after the accretion, for the larger satellites, could also depend on the convective evolution in presence of phase transitions and generate an intermediate rock layer, considerably denser than the underlying, still homogeneous core, and unstable to overturning on a geologic time scale. Moreover the liquid water mantle could be a transient feature because the mantle would freeze over several hundred million years. For these large bodies the stable configuration is expected to be one consisting of a silicate core and a mantle of mixed rock and ice.

Key words: Satellites, Giant planets, Origin, Evolution

1. I n t r o d u c t i o n

The satell i tes of the outer Solar Sys tem are an e n s e m b l e of very different and

in t r igu ing objects, as has been revealed by the images gathered through the Voyager

mi s s ions (Smith et al . ,1979a; Smi th et al . , 1979b; Smi th et al. , 1981; Smi th et al. ,

Surveys in Geophysics 16: 533-591, 1995. ~) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

534 A. CORADINI ET AL.

TABLE I

Properties of the icy satellites

Satellite Distance Radius Mass Gravity Density Central 103 km km 1019 kg m s - 2 103 kg m -3 pressure

(Plan. Radii) Mpa

Io 422 (5.905) 1815 4- 5 8940 4- 20 1 . 8 0 3.55+0.03 5800

Europa 671 (9.937) 1569 4- 10 48004- 20 1.32 3.01 4- 0.06 3180

Ganymede 1070 (14.99) 2631 4- 10 14823 4- 5 1.44 1.93 4- 0.02 3610

Callisto 1880 (26.37) 2400 4- 10 10766 4- 5 1.25 1.83 4- 0.02 2700

Mimas 185 (3.075) 196 ± 3 4.55 ± 0.54 0.079 1.44 4- 0.18 10.9

Enceladus 238 (3.945) 250 4- 10 7.40 4- 3.6 0.079 1.13 4- 0.55 12.6

Tethys 295 (4.884) 530 4- 10 75.5 4- 9 0.18 1.21 ± 0.17 59.0

Dione 397 (6.256) 560 4- 5 105.2 -/- 3.3 0.22 1.43 4- 0.06 89.6

Rhea 527 (8.736) 765 4- 5 249 4- 15 0.28 1.33 4- 0.09 145

Titan 1222 (20.25) 2575 4- 2 13457 4- 3 1.35 1.881 4- 0.004 3280

Iapetus 3561 (59.03) 730 4- 10 188 4- 12 0.24 1.15 4-4- 0.08 100

Miranda 129.8 (4.95) 242 -4- 5 7.1 0.085 1.26 4- 0.39 13.0

Ariel 191.2 (7.30) 580 4- 5 140.4 0.267 1.65 ± 0.30 128

Umbriel 266.0 (10.15) 595 4- 10 118 0.239 1.44 4- 0.28 103

Titania 435.8 (16.64) 775 4- 10 287 0.327 1.50 ± 0.10 189

Oberon 582.6 (22.24) 800 4- 5 343 0.355 1.59 4- 0.09 226

Triton 354.8 (14.33) 1350 ± 5 2138 4- 14 2.075-4- 0.019 1100

Nereid 5513.4 (222.65) 170 4- 25

1982, Smith et al., 1986; Smith et al., 1989). More than fifty satellites have been discovered in the Solar System at the time of writing, ranging in size from collisional fragments (tens of kilometers) to thousands of kilometers. Satellites are generally classified as regular and irregular satellites, or collisional debris (Bums, 1986a).

The irregular satellites of the outer Solar System have orbits that are highly inclined to the planets' equatorial plane and having high eccentricities. Such characteristics suggest that they are captured objects, i.e., bodies formed within the Solar nebula and subsequently placed into orbits about their current planets. It has been suggested (Pollack and Bodenheimer, 1989) that the composition of the irregular satellites provides indications about the nature of the solid material that helped form the giant planets and that they can be used as standards to evaluate the additional processes undergone by the regular satellites.

ORIGIN AND THERMAL EVOLUTION OF ICY SATELLITES 535

The orbits of the regular satellites generally lie almost on the equatorial plane of their primary body and they have low eccentricities. Thus, the regular satellites were probably formed within disks of gas and dust that surrounded their parent planets during their early history. Among these satellites, some are apparently inactive bodies with ancient cratered surfaces and some are highly active ones. Recent endogenic processes have modified the surfaces of some satellites, while the surfaces of many other Jovian and Saturnian satellites have been modified by endogenic activity that took place soon after the heavy cratering episode early in the evolution of the Solar System. Information on the satellites of Uranus and Neptune was obtained from Voyager 2 when it.arrived at the two systems in January 1986 and in August 1989, respectively. Here we will briefly review the main characteristics of the satellite systems of Jupiter, Saturn and Uranus because they are the "observables" that we are trying to explain. We focus our attention on the so-called icy satellites that are characterized as "regular" from a dynamical point of view. Triton and Charon do not fit into this classification scheme. The genesis of the many distinctive surface features must relate in some way to the physical parameters of satellites and to the physical mechanisms through which they have been formed. Thus, in order to outline the degree of knowledge of the topic at present, we first describe the mechanisms of satellite formation inside an accretion disk, where it is now believed that regular satellites formed. Secondly, we look at the internal evolution, after having discussed the rheological properties of the materials of which the satellites are made. A better understanding of the rheological effects can in fact improve our knowledge of the internal evolution and of the cryovolcanic processes able to explain the unexpected observed diversities of the surfaces of the icy satellites.

1.1. JUPITER'S SATELLITES

There are 16 known satellites of Jupiter. Among them it is possible to identify two families that are clearly different both in size and in surface and bulk properties: the small satellites and the Galilean satellites. We will concentrate our effort in the study of the Galilean satellites, since they are probably the more representative in terms of the evolution history of the Jovian system and since, for their size and composition, they have experienced a long-lasting thermal history. However, for the sake of completeness we will briefly describe the entire satellite system.

The small satellites. Entering the Jovian system from the outside, the first four satellites are found, to have retrograde orbits and semimajor axes larger than twenty million kilometers. They are called, respectively, Sinope, Pasiphae, Carme and Ananke. The next group includes four satellites called Elara, Lysithea, Himalia and Leda. These satellites are characterized by highly inclined but direct orbits. These two groups are probably captured objects. The Galilean satellite system is found closer to Jupiter. The inner part of the system contains Amalthea, accompanied by three satellites discovered by Voyager: Thebe, located outside the Amalthea orbit,


and Adrastea and Metis, that are very close to the ring system and are problably related to its evolution.

The Galilean satellites. Io, the innermost Galilean satellite is the densest of the group. It is characterized by an orange-yellowish surface, completely devoid of craters, but instead characterized by a strong volcanic activity. Voyager 1 observed eight strong eruptions, the most spectacular of which was that of the volcano Pele. During the second Voyager passage, four months later, Pele was not erupting, and instead extended plumes were observed to escape from the Loki volcano. Europa is the smallest Galilean satellite and has a lower density than Io. In the Voyager high-resolution pictures, Europa displays a surface covered by an extensive pattern of long fractures, but completely devoid of impact craters. The interpretation is that a relatively young icy layer covers the satellite, as can also be inferred by the high albedo (0.6). The observed lineaments are supposed to be the results of tension cracks that have fractured the ice lithosphere, allowing rotation and translation of intact blocks of ice (Golombeck and Banerdt, 1990). Ganymede is the largest and most massive Galilean satellite. Its surface is heterogeneous in colour and morphology: dark, highly cratered regions are present together with brighter, faulted regions. In the dark regions, the density of craters, their size distribution and their shape suggest that we are looking at an ancient surface, probably the original crust. These regions are also characterized by the presence of long folds, sometimes cut by younger craters: these were probably created by an ancient bombardment by large bodies. The large impacts were probably cancelled by surface relaxation, typical of a non rigid, volatile-rich crust. The bright regions are characterized by grooves one kilometer deep and thousands of kilometers long. These faults are possibly related to the tectonic evolution of the satellite that might have undergone a strong differentiation. These features are generally believed to have originated through brittle failure and graben formation associated with large scale extensional tectonics in regions of endogenic resurfacing (Squyres and Croft, 1986). In a second phase, the fractures would have been filled up by bright icy material. Callisto: existing images convey an impression of surface uniformity. The crater distribution is typical of the old surfaces of the Solar System. Only one large basin has been found, Valhalla, characterized by a diameter larger than 500 km. Valhalla is surrounded by series of concentric ridges, which extend out to about 1500 km from its center.

The dichotomy between Ganymede and Callisto, similar in size and bulk composition and moving on adjacent orbits, requires an explanation in the framework of comparative planetology.

1.2. SATURN'S SATELLITES

The system of Saturn consists of 21 satellites, the largest of which were discovered prior to the space era. The smaller twelve were discovered by Voyager. In the case of Saturn, two groups of bodies can also be recognized: the large icy satellites


and the small, probably captured, objects. In spite of a superficial resemblance, the satellite system of Saturn is quite distinct from that of Jupiter, and is in fact unique in the Solar System.

The small satellites and rings. Fourteen satellites belong to this group: Phoebe and Hyperion are found in entering the Saturn system: these two objects are completely different from one another, Phoebe being quasi-spherical and dark while Hyperion is irregularly shaped and heavily cratered, at least in the areas imaged by Voyager. Phoebe could be an undifferentiated asteroid captured by Saturn, while Hyperion may be the leftover of a collisional destruction of a larger body, possibly formed locally, as suggested by the icy composition. The other twelve satellites are close to Saturn and some of them occupy Lagrangian positions with respect to other satellites of Saturn. Dione and Tethys both have two Lagrangian satellites tens of kilometers in size.

Satellite-ring interactions. A complete description of the complex structure of the Saturn rings is beyond the scope of our paper (for an extensive description see for example (Cuzzi et al., 1984): however some attention should be payed to the complex interplay between rings and small satellites. The system of the small satellites of Saturn and their interactions with the rings constitutes an important chapter in the evolution of the Solar System, because phenomena that were important in determining the evolution of the primordial protosolar nebula may still be active there. In fact, although the rings are well inside the Roche limit, it is believed that some accretion processes are taking place: particles remain gravitationally bound when they reside on the surface of larger particles (Weidenschilling et al., 1984). Furthermore, the net tidal stress within the small particles is generally small: for that reason particle collisions can produce accretion instead of destruction, partic- ularly inside the A-Ring (Weidenschilling et al., 1984), and meter-sized particles can be formed. The observation of such particles in the Saturn rings is evidence that collisional evolution of particles can lead to the formation of meter-sized bodies. The presence of a strong tidal interaction probably does not allow the survival of larger bodies. The shape of the rings is also affected by the collisional evolution of its constituents. Generally speaking, collision conserves angular momentum and dissipates energy; instead, as a consequence of differential rotation, the angular momentum is transfered outward and the ring spreads out (Borderies et al., 1984). Therefore the overall radial width of the rings can be explained by viscous diffusion. Instead, the large range of structures (Cuzzi et al., 1984) can be interpreted in terms of viscous instabilities and satellite perturbations. The viscous instability has been used to explain the formation of fine multiringlet structures in the B ring; the torque exerted on the ring material by satellites near resonance has been used to explain the formation of gaps in the rings, like, for example, the Encke division. Moreover, the torque exerted by a pair of small satellites on a narrow ring has been considered to be a confinement mechanism. The interaction between rings and satellites can also be responsible for the orbital evolution of satellites, because an external satellite extracts angular momentum from the ring, and its orbit con-

5 3 8 A. CORADINI ET AL.

qequently expands. The problem of orbital evolution, however, is far from being solved (Borderies et al., 1984). We have mentioned these issues because these mechanisms were probably important for the evolution of the satellite systems.

The icy satellites. After Titan, the six largest satellites in the Saturn system are: Mimas, Enceladus, Tethys, Dione, Rhea and Iapetus. The first five can be considered as regular satellites, having circular and prograde orbits lying very close to the equatorial plane of Saturn with semimajor axes smaller than 10 Saturnian radii. The sixth, Iapetus, moves at a greater distance (about 60 Satumian radii) on an orbit inclined at 14.7 ° . The density of these satellites is only slightly larger than that of water (Table I). This fact, combined with the images of the surface and with spectroscopic data, indicates that their composition is ice. Mimas has a heavily cratered surface. The largest crater is 130 km in diameter, about 1/3 of the diameter of Mimas. This impact crater is characterized by a central peak 6 km high, located at the center of a depression about 10 km deep. Traces of fractures that were produced during this impact are still present on the satellite surface. Indications of endogenic activity are completely absent. Enceladus is the brigthest of the Solar System bodies and consequently one of the coolest because it reflects about 90% of the incident light. This may indicate that the surface is completely made up of clean ice, relatively "fresh", i.e. not exposed for a long time to the bombardment of cosmic ray particles. The surface is characterized by almost cratefless, large areas, where flows of icy material can be identified. In the cratered areas, the crater size distribution indicates a relatively young crust. The average diameter is small: no craters larger than 35 km in diameter were found. The overall appearance of the surface seems to indicate that several episodes of resurfacing have taken place and that long-lasting endogenic activity has characterized the life of Enceladus. Moreover the young terrains are also characterized by striation cracks and faults. The characteristics of the surface can be interpreted easily if it is accepted that magmatic activity has taken place. Tethys has a diameter twice that of Enceladus. At the resolution of Voyager images (about 5 km) two types of regions were identified: cratered terrains, areas that are almost saturated, and rejuvenated areas, where some kind of endogenic activity cancelled the older craters. Tethys also shows evidence of a heavy bombardment: a large impact produced a crater 400 km in diameter. The crater is shallowed and appears degraded. The floor rebound typical of cratering in icy bodies was probably also accompanied by ice flows that filled the inner part of the crater. Dione has a surface characterized by different structures, valleys and craks forming a complex network. Moreover, significant differences in surface albedo are present. This can be interpreted in terms of different chemistry. Darker regions could be richer in silicaceous material or made up of older ice. In any case the satellite has a density value implying a large silicate/ice content. Rhea is the largest of the inner satellites, with a diameter of 1500 km. Despite its size, this satellite does not exhibit a clear evidence of internal activity. However, differences in albedo and the presence on the surface of "clear" areas can be used as an indication of recent deposition of icy material


on the surface. In any case, the original surface seems to be preserved because it is almost covered by craters. Two crater distributions were identified, one interpreted as due to the original bombardement and the other probably due to a local source. Iapetus is the most mysterious of the satellites of Saturn, having one hemisphere completely dark and the other very bright. The dark material has been generally interpreted as a deposit on a bright surface. The origin of this deposit can be either endogenic (volcanic activity) or by accretion from an external source.

The Saturn system is dominated by Titan. Titan has a thick atmosphere composed primarily of nitrogen and methane. The presence of the atmosphere precludes the direct observation of the surface of the satellite. After the Voyager encounter a wide debate on the nature of the Titan surface started. The pre-Voyager models of Titan were essentially based on cosmochemistry and on the previous observation of atmospheric methane. Therefore it was assumed that it was possible to have a methane ocean under the clouds, able to supply the atmosphere, continuously depleted of methane via photolysis. Post-Voyager analysis of the radio occulta- tion profile revealed that the methane is subsaturated in the lower parts of the atmosphere. This fact, coupled with the large rate of depletion of the methane via photolysis, ruled out the pure methane hypothesis.

Our knowledge of the interior is very limited, too. The body is probably made largely of water, ice and rock, but the most important questions concern its volatile reservoirs: following Stevenson (1991), Titan could be essentially like Ganymede with the addition of a veneer of volatiles CH4 or C2H6 and N2, or it has an interior which is rich in methane and nitrogen. The question of internal volatile content is not directly addressed by the current observations: in fact only the average density is known and it comes to 1880 kg/m 3. This intermediate value is between those of Ganymede and Callisto, and, unfortunately is coherent with both hypotheses. Volatile-poor and volatile-rich models have been proposed and Stevenson (1991) favours the second hypothesis, the history of volatiles being intimately interconnected with that of the satellite and of its atmosphere. In fact, the presence of the atmosphere, whatever its origin, strongly suggests a thermal evolution of the satellite, greatly affected by the presence of volatiles. New data are needed, mainly on the nature of Titan's surface, and we hope that the Cassini mission will provide some of the data needed.

1.3. URANUS' SATELLITES

The main satellites of Uranus are, in order of distance from the primary: Miranda, Ariel, Umbriel, Titania and Oberon. Their densities are slightly larger than those of the Saturnian satellites, ranging from 1500 to 1700 kg]m 3. They all have a low albedo (12% in the case of Umbriel) and a surface probably made of a mixture of water-ice and carbon compounds. Several types of volcanic units have been recognized on the icy Uranian satellites indicating that after their formation, the satellites generated enough internal energy to erupt fluid onto their surface. Titania


TABLE 11

Properties of the small satellites of Neptune

Designation Name Radius Semimajor Inclination Eccentricity Mass

(kin) axis (RN) (g)

1989N1 Proteus 208 4-8 4.75 0.04 0.0004 4.0 x 1022

1989N2 Larissa 96 4-7 2.97 0.20 0.0014 4.3 x 1021

1989N4 Galatea 79 4-12 2.50 0.05 0.0001 3.7 x 1021

1989N3 Despina 744-10 2.12 0.07 0.0001 2.1 x 1021

1989N5 Thalassa 404-8 2.02 0.21 0.0002 3.2 X 10 21

1989N6 Naiad 294-6 1.94 4.73 0.0003 9.9 x 1019

and Ariel have long depressions and valleys; Ariel has smooth flooded surfaces and Oberon is characterized by the presence of a very high mountain. Several geologic structures have been recognized on Miranda, including cliffs, faults and valleys. Voyager 2 has also discovered a few other small satellites inside the system of rings, while bodies close enough to the ring to be considered responsible for its confinement have been not detected.

1.4. NEPTUNE'S SATELLITES

Three of the giant planets possess regular satellite systems; Neptune does not. Instead, it has one massive satellite, Triton that follows a very inclined, retrograde orbit; a second satellite, Nereid, with a very eccentric and inclined orbit; and a system of small but regular satellites. Triton, the largest satellite of Neptune, shows evidence of a remarkable geologic history, with active geyser-like eruptions. However, the relatively high density and its retrograde orbit offer strong evidence that Triton, as its companion Nereid, which follows a highly eccentric orbit, are not original members of the Neptune family, but are captured objects (Farinella et al., 1980; McKinnon, 1984; Goldreich et al., 1989). Nereid has an orbit with eccentricity 0.75, therefore its distance from Neptune varies from 1.39 to 9.73 million kilometers. The motion of Nereid is direct and its orbital period is of 359 days. It is observable with difficulty, having magnitude 19, and its physical properties are almost unknown. The small satellites are six in number: excluding 1989N6 Naiad, with an inclination of 4.7 °, their orbits lie on the equatorial plane (Table II).

ORIGIN AND THERMAL EVOLUTION OF ICY SATELLITES

2. F o r m a t i o n o f the Sa te l l i t e s

541

In this section we address the origin of regular satellites, more strictly related to the origin of the central body. Some characteristics of regular satellite systems are similar, and suggest a common origin in an accretion disk around the central body. If this disk phase is present, it links the formation of the primary body to that of satellites. It has been suggested that the satellite disks were generated either from the outer envelopes of the central body or from material collected by the protoplanet from the protosolar nebula. In the first case the composition of the satellites would provide useful constraints on the composition of their planets' atmosphere and in the second case on conditions in the early solar system (Pollack and Bodenheimer, 1989). A completely different approach assumes that the satellites were formed from debris produced by a catastrophic event.

Further, satellitesimals can be formed from the agglomeration of dust in the circumplanetary disk (Coradini et al., 198 l b) or debris of varying sizes can be created by large impacts, or planetesimals can be captured by the disk. These different mechanisms imply different bulk compositions for planetary satellites, allowing us, in principle, to discriminate between them (Lissauer, 1987). However, the further evolutionary history - that for the larger satellites implies differentiation and formation of the core - may have canceled or obliterated the original differences. An understanding of the formation of the regular satellite systems cannot be obtained without paying some attention to the formation and evolution of the central body. It is therefore useful to give here a brief summary of the present state of knowledge on this topic.

The formation and evolution of the giant planets has been modelled in the framework of two different scenarios. In the first one, giant protoplanets are formed by gravitational instabilities in a massive solar nebula (Cameron, 1978; Bodenheimer et al., 1980), and they are characterized by solar chemical compositions and masses probably larger than the present masses of the giant planets. The solid core is formed through sedimentation of solid material to the center of the structure, or by capture of solid planetesimals (Pollack et al., 1979).

In the second scenario, the cores are formed first, through an accumulation mechanism similar to the mechanism generally accepted for the formation of the terrestrial planets (Safronov, 1969; Safronov and Ruskol, 1982). As the core grows larger, more and more nebular gas is collected in its sphere of influence, until a large and massive envelope is formed. At this time a rapid contraction phase begins. At present the core accretion model is favoured for two main reasons: first, the massive protoplanets cannot survive due to rotational instabilities; second, the Voyager missions have discovered that all the giant planets have a solid core. The masses of these cores seem to vary only slightly with the distance from the Sun, despite the large differences in total mass that characterize the planets themselves. However, this model does present some difficulties, the chief one being the rather long timescales required to build up the core, according to the so-called "minimum


mass" models of the solar nebula. Minimum-mass models assume that the mass of the nebula was the minimum necessary to build up the planets, augmented with gas in order to reach solar composition. These long timescales conflict with expectations about the lifetime of the solar nebula, based, for example, on what is known about the length of the T Tauri phase in young stars. However, giant planets may form much more rapidly in more massive nebulae. Besides Lissauer (1987) and, recently, Greenzweig and Lissauer (1990) have shown that accretion rates at moderately low random velocities are enhanced by a factor of ,,~2, when taking account of the 3-body approximation; it has also been shown (Greenzweig and Lissauer, 1990) that gravitational enhancements due to runaway accretion in cross sections are greater for planets occupying a smaller fraction of their Hill spheres. This effect, as was predicted by Lissauer (1987), allows more rapid accretion of the cores of the outer planets. Another way to attain a rapid accretion of Uranus and Neptune cores is the so-called "macro-accretion" (Ip, 1989). This model consists in the simulation of the random effects of gravitational scattering and accretion on a system of Earth-sized objects. The results are the formation of Uranus-size planets in a few 108 years. Furthermore, the core accretion model does not predict any particular value for the envelope mass, so that a further mechanism is required to stop envelope accretion and to determine what its angular momentum is.

The physical nature and the development of the "core instability", was investigated, using a spherically symmetric protoplanetary model; consisting of a growing rigid core and a gaseous envelope of solar composition by Wuchterl (1991 a, 1991 b, 1991 c). In the framework of this theory the critical mass of the core can be defined as the mass beyond which the core very rapidly collects the surrounding gas. It has been shown (Wuchterl, 1991a) that along a time sequence of growing cores there is a limiting time beyond which no more static models exist. The assumption made is that the "core" is surrounded by a thick nebula (with a density p of 1.5 x 10 -7 kg/m 3) that completely fills up the Hill sphere. In these conditions the spherical symmetry is appropriate. Wuchterl (1991a), comparing his results with those of Perri and Cameron (1974), Mizuno et al. (1978), Mizuno (1980) and Bodenheimer and Pollack (1986), who made similar assumptions, concludes that the differences are due to the fact that the planetary envelopes, at the critical mass, have different structures. Moreover Wuchterl (1991b)studied the local linear stability of the critical mass model, dropping the assumption of the perturbation adiabaticity. He showed that a large portion of the envelope is located in regions of vibrational instability. Therefore a hydrodynamical calculation is needed to investigate the evolution of a protoplanet beyond the critical mass. In fact Wuchterl (1991 a), using his critical model as initial condition, studied a hydrodynamical evolution and the subsequent phases of the core instability. In this model, a very short quasi-hydrostatic contraction phase immediately follows the attainment of the critical mass, then non-linear hydrodynamical waves are excited and an outflow is obtained. After the ejection of a large part of the envelope mass, the activity declines and a new quasi-static state of the protoplanet appears. This remnant object has a pulsating


envelope of 1.4 MEarth around a core of 13.1 MEarth thus giving rise in the Jupiter zone to an object similar in mass to Uranus and Neptune. No possibility to reach the present mass of Jupiter was found in this numerical simulation and it seems that hydrostatic and thermal equilibrium cannot be reached if the dynamical evolution beyond the critical mass is followed. This result is in contrast with the result of Mizuno (1980).

The implications of the work of Wuchterl for the formation of the giant planets must be investigated and the role of the different assumptions under which the numerical simulations were performed must be clarified. As an example the effects of the rotation and of the accretion of the solid material onto the core in damping the oscillations has been neglected.

From the previous discussion we can argue that refinements are needed in modeling the final phases of giant planet formation, i.e. a hydrodynamical 3D simulation of the central body accretion must be performed. This is essential if the gas accretion is studied starting from a "low-gas" nebula. In this case, the mass present in the Hill lobe is small, due to the smaller gas density, and mainly concentrated on the central plane of the disk. Therefore the gas accretion on the central body passes through the replenishment of the sphere of action and through the formation of a disk-shaped envelope. The description of this process requires a 3D hydrodynamical simulation, and does not necessarily imply a rapid collapse phase. The formation of a disk surrounding the protoplanet can be important to link the formation of the regular satellites to that of the central body.

2.1. THE DISK PHASE

There is a general consensus on the formation of the regular satellites inside a disk of gas and dust that surrounded the parent planet. Four possible models for the formation of a disk in which satellites originate can be proposed (Pollack and Bodenheimer, 1989). Briefly, they are:

1. the accretion disk model in which the satellite-forming disk is derived directly from solar nebula gas entering the region around the planet (Coradini et al., 1989, Magni et al., 1990);

2. the spin-out disk model, in which the disk was formed from the outer parts of the planets envelope being left behind as the planet contracted (Pollack et al., 1986; Korycansky et al., 1991);

3. the blow-out disk model, in which an impact by a large (~ MEarth) planetesimal ejected material to form a disk (Singer, 1975; Slattery et al., 1992);

4. the co-accretion model, in which solar nebula planetesimals collide within the planet's gravitational sphere of influence to form a planetesimal disk (Safronov and Ruskol, 1982; Safronov et al., 1986).

It should be noted that the blow-out model has been proposed to explain the origin of the Earth-Moon system and that of the Uranian syste m, as it will be seen briefly


TABLE 17I Dynamical characteristics of the regular satellites. The semimajor axes are given in planetary radii and the masses in planetary masses. Note the similarity in relative mass and size of the three systems

Planet Satelli te Semimajor Eccentricity Inclination Relative Total satellite axis (forced) (forced) mass mass (solid)

free free

Jupiter Io 5.905 (0.0041) 0.040 4.704 x 10 -5 2.0 X 10 - 4

10-5

Europa 9.397 (0.0101) 0.470 2.526 x 10 -5 10 -4

Ganymede 14.99 (0.0006) 0.195 7.803 x 10 -5 0.0015

Callisto 26.37 0.007 0.281 5.677 x 10 -5

Saturn Mimas 3.075 0.0202 (1.53) 6.6 x 10 -8 2.4 X 10 - 4

Enceladus 3.945 (0.0045) 0.02 1.5 x 10 -s Tethys 4.884 0.0000 (1.09) 1.03 × 10 - 6

Dione 6.256 (0.0022) 0.02 1.08 X 10 - 6

Rhea 8.736 (0.0010) 0.35 4.4 X 10 - 6

0.0003 Titan 20.25 0.0292 0.33 2.36 X 10 - 4

Uranus Miranda 4.95 0.0027 4.22 7.94 X 10 - 7 1.0 x 10 - 4

Ariel 7.30 0.0034 0.31 1.45 x 10 -5 Umbriel 10.15 0.0050 0.36 1.54 x 10 -5 Titania 16.64 0.0022 0.14 4.00 x 10 -5 Oberon 22.24 0.0008 0.10 3.49 x 10 -5

later on, while the co-accretion model has been invoked only to explain the origin

of the Moon.

Our discussion will be limited to the regular satellite systems of Jupiter, Saturn

and Uranus (Table III). As far as the Neptune system is concerned we will try to

show that also in this case, a "second generation" regular satellite system can be formed, and that, possibly a "first generation" one was present once. The Uranus

system has been included and modelled in a similar way as the others, notwith-

standing its peculiar dynamical characteristics. In fact, the equator of Uranus is

inclined by 98 ° with respect to the plane of its orbit. However, its five satellites

are remarkably regular, having near zero eccentricities and inclinations, with orbit

planes tilted by about 98 ° with respect to the plane of the ecliptic.

It is generally assumed that the inclination of Uranus has been produced by

the impact of a body with a mass of the order of 10% of the central body. This


would imply either the direct impact of a large body with blow out of material, or tidal capture followed by accretion onto the primary. In both cases the final angular momentum will be the same (Singer, 1975). Two scenarios, discussed in Magni et al. (1990), can be proposed for the Uranian satellite formation: either satellites were formed before the tilting took place, or after it. In both cases the chemistry and dynamics are rather complex, but, due to the presence of the extra incoming mass, the amount of solid material relative to the gas, present in the Uranian disk should be larger than in the case of Jupiter and Saturn. It has been shown (Slattery et al. , 1992) that a giant impact on the primitive Uranus can produce a disk of icy material swirling about Uranus. The impact of different Earth-sized (from 1 to 3 MEarth) bodies onto Uranus was simulated, using smooth particle hydrodynamics and different values of angular momentum of the bodies. In most cases ices were left in orbit and the present period of rotation and inclination of Uranus obtained. Therefore the regular satellites of Uranus can be formed in a disk produced by a giant impact, with mechanisms similar to those invoked for the formation of the satellites of Jupiter and Saturn.

The Neptune system has been briefly described in the previous section. It is interesting to note that the regular satellite system has a much smaller total mass than that of the other systems, when expressed in terms of primary mass. It has been suggested that the present appearance of the Neptune system is related to its dynamical history and that a " first generation" regular satellite was originally formed (Goldreich et al., 1989). If Triton is indeed a captured object, that was once in heliocentric orbit, its insertion in a Neptune-centric orbit should have deeply perturbed the system of the regular satellites. Goldreich et al. (1989) suggested that the immediate post-capture orbit should be highly eccentric with a semimajor axis of 103 RNeptun e and an oscillating periapse, but with a minimum value of 5 RNeptun e. The tidal dissipation caused the evolution of the orbit to its present status. The same study suggested that during its orbital evolution, Triton destroyed the regular satellites and perturbed the orbit of Nereid. Moreover predictions were made on what would be observed in the Neptune system by the Voyager spacecraft: regular satellites with orbit well inside 5 RNeptun e could survive, but they should be on highly inclined orbits, due to the chaotic perturbations forced by Triton. Instead the observed regular satellite system is lying on the equatorial plane, with the exclusion of Naiad. It has been suggested (Banfield and Murray, 1992) that Triton's capture has increased the regular satellite eccentricity and inclination to such a large extent as to favour mutual collision among already formed satellites. These original satellites have been destroyed, leaving a large amount of debris on the equatorial plane. Neptune's inner satellites have been rebuilt on the equatorial plane of Neptune, after the circularization of Triton's orbit. This scenario implies that the formation of regular satellites is a very efficient process and it can take place in a relatively short time, even in difficult dynamical conditions. From the previous discussion it is apparent that the formation of first or second generation


regular satellites is not difficult, provided a disk shaped structure is formed. In what follows we will briefly examine the existing disk models.

Quantitative analysis of the accretion model (Coradini et al., 1989) and of the spin out model (Korycansky et al., 1991) have been performed. The strongest argument in favor of the accretion model is the similarities of the satellite systems of the giant planets among themselves (Table III) and with the Solar System, as has already been mentioned.

The accretion disk structure can be described by numerical models, taking account of energy transport, time evolution of the accretion, viscous shear and turbulence. The accretion problem requires hydrodynamical and time-dependent techniques: in fact, mass infall generates supersonic flows and shock waves, and the gravitational force of the Sun is also time dependent. Two alternative approaches can be used, for which a satisfactory approximate treatment can be developed: (1) a "large scale" description, where, distances are comparable to the Hill radius, and a polytropic gas law and hydrodynamical scheme can be used; (2) a "small scale" approximation, where the region containing the regular satellites is described. Here, the equilibrium between mass infall to the planet and mass motions in the disk can be assumed, and mass infall rate depends on the gas viscosity. Models can thus be computed using the general scheme for the treatment of accretion disks, developed in Shakura and Sunyaev (1973) and Lynden-Bell and Pringle (1974). Our models include a detailed treatment of the vertical turbulent convection and radiative transfer, taking account of both gas and grain opacities in determining self-consistent values of turbulent viscosity. The strength of turbulent viscosity is computed, taking account of the local values of physical parameters such as density, pressure and temperature, and is directly related to the growth factor of the convective instability (Canuto et al., 1984; Cabot et al., 1987a; Cabot et al., 1987b).

Time dependent "large scale" model. The time evolution of the accretion process has been studied using Eulerian two-dimensional hydrodynamics. Two different, but internally consistent 2D hydrodynamical schemes have been developed. In the first (flat) model, isotropy in the vertical dimension is assumed; the force field contains gravitational interactions with the Sun and the planet, and centrifugal and Coriolis forces. This model can give a satisfactory description of the situation in the midplane of the disk, but no information on its vertical structure can be supplied. In the second scheme (axisymmetric model), the vertical structure is described in detail, but the force field and the other physical parameters in the midplane are averaged over the azimuthal angle.

Since the protoplanet growth was not followed in detail, the mass distribution around the central body at the starting point of the evolution is unknown. The accretional evolution of the central body in a solar nebula already depleted in solid material accumulated in the solid core, can be followed for several free fall times, in order to allow the matter in the Hill lobe to relax, until a slowly varying, quasi- equilibrium state is reached. The results obtained in the case of the flat model


of Jupiter, Saturn and Uranus, starting when the protoplanet has already gained 95% of its present mass, show that at an early stage the accretion is asymmetrical and a disk structure can be identified. This structure extends from the planet to a distance of ,,~ 0.1RHm and it is denser for Jupiter than for Saturn and Uranus, as a consequence of the larger Jovian gravity strength, and of the higher density of the protosolar nebula at the distance of Jupiter. In a second stage, after several free-fall times, two regions can be identified in the disk: an inner region, in which the gas has a prograde rotation, and an outer one that is only gravitationally bound to the planet, not having enough energy to escape from the Hill lobe. Thus a region in the disk can be isolated, where the angular momentum has the same orientation as the regular satellite systems, and where the gas is gravitationally bound to the protoplanet. This region can be called the satellite accretion disk. This model is useful, but from the comparison between the two geometries used, namely the "polar one" and the "planar one" (Coradini et al., 1989) it is apparent that the two situations are quite different, mainly if the central body accretion rate should be evaluated. Thus we developed a hydrodynamical 3D computer program to estimate also the time evolution of the accretion process. This program in fact allows us to follow the accretion of the central body, taking into account the anisotropy in the gravitational field due to the presence of the Sun. This program however has been developed with some assumptions, that will be given later on.

The Eulerian equations of hydrodynamics, written in spherical coordinates (r, 0, ~b), are:

Op 0~ + V . (pv) = 0, (1)

O(pv~)

Ot r

o(pvo) l, orb op (vT o + v . (p ov) = - ; ( o N + + _ + v~cot0)~ (3)

0(pA) Orb Op + V . (pAv) = -(p-0-~ ÷ ~-~). (4)

0~T--

Here the first equation is the continuity equation and the remaining three describe momentum transfer. In the equations p is the mass density, v - (vr , vo, r e ) is the fluid velocity and A = r sin 0 v~ is the specific angular momentum. The gravitational potential rb is determined taking into account the mass of the central body Me, the mass of the Sun M o, and the mass MH of the gas inside the Hill lobe, as follows:

rb(r) = % ( r ) + rbc(r) + rbH(r) (5)

where

rb®(r) - - G M e , (6) f


10

5

- 5

. . . - " - ' ~ T i " v - . . . • " . . . . . - - . . . ' . . : ; . . . . . . , . : . . i . ~ : . . ' : : . . . " . .

..... \ , , . . . . / ....

....... ~ . . ""% • , . , : . . , . . ' ~0 .......

. . . . . . . . . . . . . . . . . . . . . . . mulll iiiiiiiii . . . . . . . . . . . . . . . . . . . . . .

.....~ ......... . . . .~.. . . . . " : : ' . . ..... . .

/ \

- 1 0 , , i , l , , , , I , , , i t l l l l l

- 1 0 - 5 0 5 10 Fig. 1. The grid used in the hydrotiynamical program for Jupiter. The grid is characterized by N~ × No x N , = 41 x 71 × 11 contained in an annular section surrounding the growing protoplanet. The vertical section of the mesh is shown at the center of the figure.

G M ~ • Sc(r) = - - - - , (7)

rc

and, if r < rH we assume,

G M H [3 1 r~

while, if r > ~'H

(8)

G M H CH(r) -- , (9)

rH

here rc = to(r, O, 05) is the distance from the protoplanet, and the last term in the gravitational potential ~/4 is computed considering only the gas contained in the Hill lobe, assumed to-be uniformly distributed inside the lobe.

The program is written for a grid made ofNT x No × N~ = 41 × 71 x 11 elements contained in an annular section of width of 4 A.U. as the one depicted in Figure 1. The grid becomes more refined towards the accrefing core, in order to have a better description of the accretion process and of the structure of the disk. The method used to numerically integrate the system is called the "donor cells"


method and it has been described in Boss (1980). This method ensures that, when the equations of motion are written in conservative form, a rigorous conservation of the mass and momentum is obtained throughout the numerical grid. Furthermore, with the donor cell method the density can be prevented from becoming negative in regions of severe mass depletion by taking a moderately small time step. The evolution is carried on under the following assumptions:

1. The grid is rotating with the central body, therefore apparent forces are present, and due to the discrete structure of the hydrodynamical equations, angular momentum is not exactly conserved and there is a slow and small variation of the total angular momentum, even when it is seen from an inertial reference frame.

2. The accretion of the core is simulated assuming that the material entering the central cell is deposited onto the core after one typical "free fall time" of the central cell.

3. The gas self-gravitation is approximately evaluated only inside the Hill lobe.

4. The gas has no viscosity and its equation of state is a polytropic law with exponent 1.43, typical of a mixture of helium and molecular hydrogen.

5. Initially the gas is in a stationary state, i.e. pressure gradients exactly compen- sate the gravitational and apparent force gradients and each gas element moves on circular orbits.

6. Due to the great amount of computer time required to integrate the equations, the time evolution has been exactly computed in the first 80 years, in order to obtain a sufficient relaxation of the structure of the nebula around the accreting core. Then integration intervals of 15 years were used, in order to calculate the mean accretion rate of the core: it was thus possible to extrapolate for the time corresponding to a fixed amount (as an example 5%) of variation of both the core mass and the total mass of the feeding zone.

7. The density distribution of the feeding zone, at each extrapolation, is simply scaled by a constant depletion factor, depending on the accretion rate of the core. The velocities were instead left unchanged. A typical extrapolation time was considered of a few tens of years.

8. The accreting core, while it is capturing mass, exchanges also angular momentum with the surrounding medium; therefore it moves inside the feeding zone. This drift does not strongly influence the accretion being less than 0.15 A.U. during the whole accretion process for Jupiter and Saturn.

9. The program was tested by changing the free parameters one by one in order to control the sensitivity of the model to the different parameters and to assumptions. The most important parameter was found to be the dimension of the cells aroun the core: anyway, a typical cell volume of (0.06RH) 3 determines a nearly asymptotic value of the total accretion time.

The accretion process of Jupiter and Saturn was simulated: the time evolution of the mass of the core for both planets is shown in Figure 2. The mass of the protosolar


0.8

0.6

0.4

0.2

.... jSaurn

O.O t t r , I a ~ , ~ l , , , , I , , ~ , I ~ , , , I , , ,

0 5.0x10 5 1.0x10 4 1.5x10 4 2.0x10 4 2.5x10 4 3.OxlO 4 Time (years)

Fig. 2. Time evolution of the mass of Jupiter and Saturn

nebula in the calculations is of 0.015Msun and its density on the central plane ,~ expressed by the following law (Coradini et al., 1981a):

Pneb(kg/m 3) 1.3 - ,~ -6 ~ 2 72 = lu ra. d (10)

and the temperature is taken as:

T ( K ) = 600 r~. 1. (11)

In Figure 2 the accretion time scale is of the order of 2.7 x 104 and 1.7 x 10 4

years respectively. This time is significantly longer than the one evaluated before (Coradini et al., 1989) using a bidimensional flat model. The difference is due to the refinement of the model: first of all, the third dimension was introduced in Coradini et al. (1989) in an approximate way, combining the "fiat model" results with the "polar model" ones; the accretion from the pole was overevaluated because in this geometry the 'barrier' of the angular momentum is negligible, while in the correct 3D geometry it is not; moreover, in the 3D model it is possible to put the boundary conditions far enough from the accreting body, while in the 2D geometry only a small region around the protoplanet can be followed.

Finally, the grid was refined, in the 3D model, more than one order of magnitude near the accreting core, with respect to the 2D model. The mass and radius of the disk as function of time for Jupiter (upper) and Saturn (lower) are reported in Figure


d

0 .50

0 .40

0 .30

0 .20

0 .10

0 .00

0

I ' ' ' ' I ' ' ' ' I ' ' ' '

.H-+

. . . . . . . . _H_++t+FH'+;;I::~+++++

l l ; t n l u l p ~ l l F J ,

s l U l n ~ r l l l

_H+4_+ +++H+

~ , , ~ ' , e ~ ......... • M a s s

V , , t , I , , , , I , , , r I , , , , I , ,

5 . 0 x l O 3 1 .Ox lO 4 1 . 5 x 1 0 4 2 . 0 x l O 4

Time (yea rs )

I ' ' ' ' I ' ' ' '

E

0

3"--"

O n - "

, , I , , , , 1

2.5x10 4 5.0x10 4

d

0 . 0 5 0

0 . 0 4 0

0 . 0 3 0

0 . 0 2 0

0 . 0 1 0

+

(, + + +

-H-+

I I I - ~

+ +

+ + +

+ Radius

* Mass

Y ~

+ +

+

4.0

3.5

2.5

E o

O

3.0 v

-O o

re"

0 . 0 0 0 ~ , , , I . . . . I , , , , I . . . . 2 .0 0 5 . 0 x 10 ̀ 5 1 .0x 104 1 .5x 104 2 . 0 x 104

Time ( yea rs )

Fig. 3. Mass and radius of the disk as function of time for Jupiter (upper) and Saturn (lower): the gaps in the plot represent the extrapolated intervals.

552 A. C O R A D I N I ET AL.

3: the gaps in the plot represent the extrapolated intervals. Here we define the disk as the ensemble of the cells that:

- are inside the potential well due to gravitational and centrifugal forces; - have total energy less than the maximum (average) value of the boundary of

the well.

The total mass in the disk is about 0.2 MEart h = 6 x 10-4MJupiter for Jupiter and about 0.4 MEarth = 5 X 10 -4 Msaturn for Saturn; the two disks almost replenish the Hill lobe. It has to be outlined that, after several free fall times, the disk region is very stable, and grows in mass and radius, with the central core. Besides, it coincides with the region that has prograde rotation with respect to the core. Finally, the mass of the region reaches a value of the same order of magnitude of the total mass of the regular satellite (Table III).

This region can be considered as the satellite accretion disk (Figures 4, 5). It should be stressed that the attainment of the limit mass does not imply that

the accretion stops. In fact the accretion rate ~ / i s still positive, for the following reasons: the amount of gas present in the feeding zone is larger than what is needed to form the giant planets, and no mechanisms to deplete it were taken into account, such as the combined effect of the accretion of different planets accompanied by the mass depletion of the protosolar nebula.

A more detailed description of the inner regions of satellite accretion disks is obtained, once the mass infall rate into the inner regions has been determined, and stationary accretion has been reached. In an "optically thin disk" the viscous stress acting between different radial sections of the disk can be determined globally (Lynden-Bell and Pringle, 1974). The strength of this viscosity depends on the presence of turbulence. Turbulent viscosity is several orders of magnitude larger than laminar viscosity. Generally, in the astrophysical approximation, the turbulent viscosity is expressed in terms of the speed of sound and of the characteristic scale height of the model, scaled by an ad hoc parameter a (Shakura and Sunyaev, 1973). Following Canuto et al. (1984), a can be expressed in terms of the growth rate of the unstable modes of the physical mechanisms generating turbulence. For optically thick satellite disks, convection has been found to be the dominant instability mechanism in the inner regions (Coradini and Magni, 1984). In the outer radiative regions of the disk, the Kelvin-Helmoltz instability, as feeding mechanism for turbulence, has been introduced (Coradini and Magni, 1984). Although in satellite accretion disks, the ratio of thermal to gravitational energies is at times substantial, hydrostatic equilibrium can be assumed. Thus the pressure gradient is balanced by the vertical component of the gravitational attraction, most of the disk mass being confined in regions where the disk geometry is not too distorted. A satellite accretion disk can be modelled assuming stationary accretion and axisymmetry.

For stationary disks, in the thin disk approximation, the vertical structure can be determined once the mass infall rate, the turbulent viscosity and the energy transfer mechanisms are known. Assuming for Jupiter and Saturn, an accretion time of 250 years and a mass infall rate per unit surface ~/c ~ r-°'5, the values aAl.

ORIGIN AND THERMAL EVOLUTION OF ICY SATELLITES 5 5 3

1

O

g -I

Z5

- 2

4 5 7 Distonce from Sun (ou)

2FV ' ' b ' ' ' ~ . . . . ~ , ' ' ' ( ' ~ ~¢ '~ ' ' ) > ~ ' ' ' ' ~ ' ' ' - | \ \ / /

- 2 : , ,

8 9 10 11 Distcnce from Sun (uu)

Fig. 4. Planar view of the logarithmic isodensity contour plot for Jupiter (upper) and Saturn (lower) when the core mass has reached the mass of the planet. The density is normalized to the reference values of 1.4 × 10 .8 kg /m 3 for Jupiter and of 2.8 × 10 . 9 kg /m 3 for Saturn.

obtained for the total mass of the disk are 4.4 × 10 25 kg for Jupiter and 9.2 × 1025

kg for Saturn. These masses are larger than the effective satellite masses, equal to 2.8 × 10 25 kg and 1.1 × 10 25 kg, respectively. In this modeling a large fraction of the disk mass resides in the outer part, requiring an inward drift of the solid component to form the satellites where they are today, while the excess of gas can be removed by tidal effects. In both Jupiter and Saturn disks, the convective zone


• , , , • i • • • b - • • , • • • , • • - i - • , , •

O. 60 .... 7.5c

" - - - - - - - - - - - _ L . _ . - - - - - - - - - - - - - - 1 . 3 0 - - " 3 ~ . 0 .50 ~

0 .40 =~ ~ - -0 .90 _

cn 0 .50 - o . 7 ~ ~ - -

O.OC _n 35

4.6 4.8 5.0 5.2 5.4 5.6 5.8 Dis tance f rom Sun (au)

• ~ . ~ , - ~ . 7 ~ . . ~ _ ~ . . - - -~ ~ . . . . . . o . o ~ _ _ ~ _ ~

0 6 0 ~ o . 6 s ~ ~ ~ • F - ~ ~ ~ ~ - o . ~ - - - - _

._. 0.50 ~ o ~ ~ ~ _~ ~- -%. ~ _ ~ ~ , ~ _

9.0 9.2 9.4 9.6 9.8 10.0 Distonce from Sun (au)

Fig. 5. Vertical view of the logarithmic isodensity contour plot for Jupiter (upper) and Saturn (lower) when the core mass has reached the mass of the planet. The density is normalized to the reference values of 1.4 × 10 .8 kg/m 3 for Jupiter and of 2.8 x 10 .9 kg/m ~ for Saturn.

overlaps the area where the regular satellites are present: for Jupiter this zone is located at a distance of 5.7 RJupite r and at a height of 4.1 RJupite r above the central plane; while for Saturn it is located at a distance of 0.5 RSaturn and at a height of 0.6 RSatum above the central plane. Moving from the planet outwards, the iron grains start to condense, first in the upper part of the disk where the temperature is lower, and later in the central plane. The icy particles appear only at very large distances from the protoplanet, and simultaneously within the whole column. In this model the convection develops vertically, due to the large vertical gradients of temperature that are present in the disk, when the condensation of grains increases the gas opacity.

As we have already noted, the modeled surface density distribution differs from that observed in present satellite systems. Furthermore, the temperature in the internal region of the disks is too high to allow condensation of ices, while the satellites of Jupiter, Saturn and Uranus are largely composed of ices. Thus, it can be argued that the most important conditions for satellite evolution appear when


the mass infall stops. Simulation of this further stage has not yet been done. So the accretion disk model predicts the amount of mass in the disks quite well, but not its radial distribution.

Let us now concentrate on the spin-out model. A requirement of this model is that a sufficient amount of angular momentum be transported outward during planetary formation so that a disk of sufficiently large size and mass can form from high angular momentum gas. Korycansky et al. (1991) state that the amount of specific angular momentum required to make Such a disk i.e. the specifLc angular momentum of the regular satellites, is approximately 20-50 times that of the average specific angular momentum of the systems as a whole: planet plus satellites. Thus, in order to form the required disk, there must be a partition of the system's angular momentum at the end of accretion. There must be a sufficiently large amount of angular momentum transported into the outermost envelope for disk formation. A test of the feasibility of the spin-out model will be whether or not such a partition of angular momentum of the gaseous envelopes of the protoplanet can be achieved. An essential part of the calculations is the modeling of the transport of angular momentum. The prescription of Korycansky et al. (1991) for angular momentum transport is based on expectations of the action of turbulent convection, and on numerical hydrodynamical evolution of rotational instabilities. These results, under the assumptions of unidimensionality, quasi-sphericity and hydrostatic equilibrium, show that angular momentum transport mechanisms are quite efficient in moving angular momentum in the outer parts of the gaseous envelopes of the protoplanets as they form. Thus, a small fraction of the mass of the protoplanet wind up with a large fraction of the total angular momentum.

Attempts have been made to calculate models with mass and angular momentum corresponding to that of Saturn at the present time. Korycansky et al. (1991) found that disk formation during contraction, after the end of the accretion, was so efficient in removing angular momentum that the resulting model planets had only ~ 20% of the angular momentum present before contraction. The resulting disks had approximately the same mass and were somewhat larger than required to explain the regular satellite systems, indicating that the scenario in which the protosatellite disks are formed by the spin-out of material from the contracting protoplanets is possible. There was a serious discrepancy in that the model planets had too little angular momentum, compared to Saturn itself. The limitation imposed by the unidimensional schematization of the calculations probably prevents the reaching of realistic values of angular momentum, so that 3D calculations are needed.

2.2. FORMATION AND EVOLUTION OF SATELLITES

The physical characteristics of the satellite disks, obtained by means of numerical modeling, are different from those observed in the present system of Jupiter, Saturn and Uranus (Coradini et al., 1989; Magni et al., 1990; and Korycansky et al., 1991). Thus, as discussed above, the most important conditions for satellite evolution are


determined when mass infall stops. As Stevenson et a/.(1986) have pointed out, when the mass infall stops the temperature in the disk begins to decrease. During the cooling phase, nucleation of condensates from the gaseous material occurs. This nucleation can be either heterogeneous (on the surface of pre-existing refractory seed nuclei) or homogeneous. Growth of these condensation centers to micro-sized grains is achieved through diffusion and rapid coagulation in a time interval shorter than 103 years (Stevenson et al., 1986). Subsequent growth can be slower, but it is aided by the relative motion between gas and grains, or by the possible sticking of grains during collisions (Coradini et al., 1980; Volk et al., 1980; Nagakawa et al., 1981; Nagakawa et al., 1986; Weidenschilling et al., 1984). Condensation is likely to begin near the disk photosphere, where the temperature is lowest, and causes the formation of a cloud of micro-sized grains. Unlike the gas, which is supported by pressure gradients, the dust grains settle toward the midplane of the disk. The density towards the central plane increases until the configuration becomes gravitationally unstable, forming rings that can break up in fragments distributed in circular orbits. The fragments can collapse, reaching an equilibrium configuration, i.e. attaining solid body density or becoming virialized structures.

The formation ofplanetesimals through gravitational instabilities has been studied both in a gas-free scenario (Safronov, 1969; Goldreich and Ward, 1973) and in a two-fluid system, i.e. gas and solid particles, coupled by viscosity (Coradini et al., 198 la; Coradini et al., 198 lb; Sekiya, 1983). The mass obtained for planetesimals (hereafter we will call "satellitesimals" the object formed in circumplanetary orbit and "planetesimals" those formed in circumsolar orbits) in a satellite disk is about 1012-1014 kg (Coradini et al., 1981b).

The scenario previously described implies high values of particles density on the central plane, consequently a high quiescence of the disk. Weidenschilling (1984) suggests that satellitesimals can be formed through binary collisions among grains not settled to the disk midplane due to the turbulence. Harris and Kaula (1975) and Harris (1978) suggested that satellites have been formed from the same circumsolar planetesimals responsible for the growth of the cores of the giant planets, and captured in the satellite accretion disk during the final phases of the formation of the central body. This model differs from the model of Ruskol (1960, 1963, 1972) in terms of the inclusion of the tidal effects. Safronov et al.(1986) present a different scenario for the formation of the satellites. In a preliminary phase, prior to the beginning of gas accretion, a swarm of solid material began to form, due to inelastic collisions and to subsequent captures of particles by the gravitational field of the planet. The presence of circumplanetary gas decreases the particle relative velocities, increasing the probability of constructive collisions, but it increases also the timescale of the formation of the satellitesimals. Smaller heliocentric planetesimals can be captured by gas drag interactions with the disk, while larger bodies can be captured by collisions. In this stage gas drag is still present leading to contraction of the orbits and losses of the inner part of the swarm


to the central body. This complicated scenario is probably closer to reality, but detailed disk models are needed to investigate the possibilities that it suggests.

The further evolution of the satellitesimals is characterized by collisional accretion and orbital decay of their planetocentric orbit due to dissipative effects such as gas drag and mutual collisions. The collisional accretion of the planetesimals goes on until an embryo is formed, that prevails and captures mass from its own feeding zone (Safronov, 1969). Numerical simulations of the circumplanetary swarm have been done by Harris and Kaula (1975), who also took account of also the effects of tidal friction. They have inferred that tidal friction is insufficient to prevent embryo satellites from spiralling into the central planet during accretion. Thus the regular satellite systems should be formed very late during the process of planet formation. It has been shown (Coradini et al., 198 l b) that in a gaseous satellite disk the collisional growth of planetesimals is fast enough to allow the formation of embryos in spite of orbital decay, at least in the region of regular satellites. Only planetesimals lying beyond 100 planet radii can spiral inward without any appreciable growth. In any case, the simple application of the Safronov accumulation theory (Safronov, 1969) does not yield masses in agreement with the present mass of the satellites, unless a value less than 5 is attributed to the so-called "Safronov number" 0. Small values of 0 would imply high relative velocities of the planetesimals, wihich is not consistent with the assumption of a dense circumplanetary gas disk. In particular, Jupiter's satellites always have masses smaller than the present masses of the satellites, while for Saturn's satellites the masses are always larger, with the exception of Titan.

To improve the numerical simulation of the satellite accretion, new effects must be included; for example the enhancement of the collisional cross-section when the three-body approximation is taken into account (Greenzweig and Lissauer, 1990), or the simultaneous accretion in many interacting distant zones, as suggested by Spaute et al. (1991 ) for the planetesimal swarm around the Sun. In such a simulation it is necessary to know, as starting conditions, the total mass and the surface mass distribution of the satellite disk, as well as the chemical composition: in particular the ratio of rock to gas and rock to water that characterizes the disk. For these reasons the modeling of the disk phase is crucial.

The chemical evolution of the satellite disk is also important to match the present chemical composition of the satellites. Except for Io and Europa, the regular satellites are made primarily of water ice and rock. Interior models of these satellites, constrained to match their mean densities, provide estimates of the relative proportion of rock and water ice contained within them. A mixture of 55% rock and 45% ice by mass characterizes all three regular satellite systems, based on interior models of Ganymede and Callisto for the Jovian system, Titan for the Saturnian system, and Titania and Oberon for the Uranian system. This composition seems to indicate the formation of the satellites within a CO-depleted satellite disk. In fact the CO can sequester two-thirds of the available oxygen, greatly reducing the amount of available water if the amount of O is fixed. Thus the more water


that is present, the more the disk should be depleted in carbon monoxide. On the other hand, Pollack and Bodenheimer (1989) assume that the satellite disk cannot be depleted in CO more than the protosolar nebula and therefore the satellite disk must be replenished with water by dissolution of the icy planetesimals in the planet envelope left behind to form a disk, as the planet contracted. Lunine (1990), discussing the formation and the characteristics of Titan, notes that the satellite density falls between those for rock-ice bodies formed or in a methane-rich disk, or in a carbon-monoxide- rich disk. Since it is possible for Titan to lose water by vaporization during accretion, its density suggests that Titan formed in a methane rich disk and lost water during accretion or formed in a disk with roughly the same amount of methane and carbon monoxide.

3. Internal Evolution

The studies of Lewis (1971 a, 1971 b) were the first to consider the chemical composition, structure and evolution of icy bodies; later investigations by Consolmagno and Lewis (1976, 1977,1978 ) developed more detailed models of internal evolution for a range of chemical models. Both equilibrium and disequilibrium condensations were considered as well as homogeneous and inhomogeneous accretion.

The evolution of any planetary body is governed by its internal sources of heat and by the mechanisms by which heat is transferred from the interior to the surface. The Consolmagno and Lewis models included radioactivity due to chondritic abundances of U, Th and K in the silicate component and, for homogeneous accretion, the heat resulting from differentiation of a core. Because of uncertainties in the viscosity or strength of solid icy materials, the models did not consider heat transfer due to solid-state convection. Radioactive heating in a body of the size of Ganymede or Callisto is sufficient to cause large amounts of melting of the icy component and complete ice-silicate differentiation. The models predicted that, in the absence of solid-state convection, the icy mantles of Ganymede and Callisto would largely be at present in the liquid state. Subsequent studies have shown that solid-state deformation could significantly affect the internal evolution of ice- silicate bodies. Reynolds and Cassen (1979)showed that a solid ice lithosphere, overlying an internal liquid water layer would be unstable to thermal convection. Thus it is necessary to briefly review the present knowledge of possible energy sources and of the rheology of the ices and of the icy mixtures in order to determine if convection can affect the evolution of the icy moons.

3.1. ENERGY SOURCES

The earliest source of energy is likely to have been accretion (Schubert et al., 1981; Coradini et al., 1982; Lunine and Stevenson, 1982; Stevenson et al., 1986; Squyres et al., 1988; Coradini et al., 1989). In an impact, a fraction of the impactor kinetic


energy is irreversibly trapped as heat in the target. For reasonable values of the trapped energy and for satellites with radii smaller than 800 km, the accumulation temperatures are not expected to attain the melting temperature of H20 ice, while accumulation temperatures great enough for eutectic melting of ammonia hydrates may have been attained (Ellsworth and Schubert, 1983; Federico and Lanciano, 1983; Squyres et al., 1988). Ganymede, Callisto, Titan are large enough that accumulation temperatures in the outer layers may reach the melting temperatures of both H20 ice and NH3. As a consequence large icy satellites may have a differentiated internal structure, while small bodies may be a homogeneous mixture of ices and silicates after a process of homogeneous accumulation.

A different degree of initial melting and slightly different chemical compositions have been proposed as the cause of the difference between Ganymede and Callisto on one hand and Titan on the other: the presence on Titan of a dense atmosphere, that can be generated through the outgassing process (Atreya, 1986; Lunine and Stevenson, 1987; Coradini et al., 1989). Recently, Zahnle et al. (1992) have proposed instead that the presence or not of an atmosphere is a consequence of the escape of atmospheric gases due to hypervelocity impacts. Because the impact velocity of stray bodies striking Titan is lower than those striking Ganymede and Callisto, the biggest satellite of Saturn would have retained a higher fraction of incoming atmophiles and suffers less atmospheric erosion. Thus Titan's atmosphere may be an expression of a late accreting, volatile rich veneer.

During accretion some energy becomes available for heating if the satellite differentiates into a non-homogeneous internal structure. The energy dissipated during differentiation is a substantial fraction, roughly 15 %, of the total accretional energy for the large icy satellites and this percentage diminishes to roughly 7% if the silicates are hydrates (Schubert et al., 1986). Model calculations for the larger satellites (Coradini et al., 1982) suggest early global differentiation more or less contemporaneous with the completion of accumulation.

After accretion, thermal energy is generated by the decay of radioactive U, Th and K. The strength of this source is connected with the silicate mass fraction, that is also a crucial parameter because it determines the characteristics of the internal structure of the differentiated satellites (Schubert et al., 1986). The knowledge of the total mass and of the global density do not allow us to fix the internal discontinuities in a unique way. Another uncertainty affecting the silicate mass fraction is the percentage of the anhydrous vs. hydrate silicates present in the structure and the occurence of phase transitions in the water ice component for satellites larger than Rhea.

For some satellites tidal heating is a significant source of energy, resulting from periodic distortion of planet and satellite shape. This distortion is inelastic and causes energy to be dissipated as heat in the planet and in the satellite itself. To conserve angular momentum, energy is simultaneously added to the orbit, causing it to expand. Tides within the satellite's interior also cause orbital evolution. For synchronism to be mantained through the orbital expansion, the satellite's spin must


slow down in pace with the satellite's decreased orbital rate. This is accomplished by the long axis of the deformed satellite being slightly offset from the line of centers, which allows the planet to produce the appropriate decelerating torque.

Tides on a satellite moving in an elliptical orbit have two additional features. First, radial tides in which the size of the tidal bulge varies as the distance of the satellite to the planet. In the satellite, energy is dissipated through this continual flexing. Second, the instantaneous orbital angular velocity, which varies according to Kepler's second law, oscillates around its mean value, which is the satellite's spin rate for a synchronous satellite. The tidal bulge is generally misaligned with the planet-satellite line and again energy is lost as the tidal bulge oscillates back and forth across the satellite's face. These effects dissipate energy and circularize orbits competing with planetary tides that usually tend to increase orbital eccentricities (Bums, 1986b). Tidal energy is dissipated in a synchronously rotating viscoelastic satellite at the rate (Segatz et al., 1988)

5 /)Tidal -- 21 RSatellit e z~5e2~(k2) (12)

2 G

where n is the orbital angular velocity, e is the eccentricity and/t~Satellit e the radius of the satellite. The ,~(k2) is the imaginary part of the tidal potential Love number of second degree and quantifies the effects of the satellite structure and rheology. It is equivalent to f (k2/Q)Sat , Q being the dissipation function for the satellite and f the structural factor used by Cassen et al. (1982). These last two parameters are extremely uncertain because they depend on the interior structure and on the composition of the satellite. The potential Love number describes the radial surface deformation for a normalized external disturbance potantial. Peale et al.(1979) applied their elastic theory to modeling Io and succesfully predicted that tidal heating is the energy source for Io's volcanism. The orbital resonance of Io with Europa and Ganymede, Laplace resonance, forces the orbit of Io to be slightly eccentric. The eccentricity causes the tide raising potential of Jupiter on Io to oscillate and if Io is sufficiently deformable, a significant part of the tidal energy can be dissipated as heat. The time history of the Laplace resonance, and in general the time variation of the forced eccentricity are poorly known and difficult to reconstruct, being also connected with the dissipation factor of the planet QPlanet. Moreover, the internal structure of the satellite strongly influences through (/¢2/Q)Satellite the strength of the tidal source and is crucial in determining its efficiency. For example, the presence of an "ocean" or the presence of a partially melted "astenosphere" can affect tidal dissipation. On the Earth, tidal dissipation occurs primarily in the oceans. How the internal structure will affect tidal dissipation will be discussed later on, but the problem of the orbital evolution of resonant satellites will not be discussed here (see e.g. Peale et al., 1979; Yoder, 1979; Yoder and Peale, 1981; Greenberg, 1981; Peale, 1986 ). This problem remains unsettled because orbital evolution depends o n QPlanet, (k2/Q)Satellite and other parameters, all of which are

OR/GIN AND THERMAL EVOLUTION OF ICY SATELLITES 561

not determined with certainty. In what follows values of QSatellite in the range 10 to 100 have been used.

3.2. RHEOLOGY

The rheological properties of ices play an important role in determining the tectonic evolution of the icy satellites. The strength of ice in the plastic flow regime determines the rate of topographic relaxation of crater wall and influences the characteristics of the sub-solidus convection. In fact, if heat can be transported to the surface by solid-state convection, melting will not occur; if, on the other hand, ice is resistant to deformation, dissipation of heat will not be efficient and the rise of the internal temperatures will produce melting and differentiation.

Rheological laws of pure water ice and mixtures are becoming better under- stood as a result of recent experimental work. These experiments, however, have been performed on pure ice and mixtures that do not completely coincide with the "natural" ones, and laboratory strain rates are at least five orders of magnitude faster than solid-state convection in the interiors of a planetary body. Hydrostatic pressures in these experiments can reach 600 MPa, a value that differs from the central pressure of the large icy satellites for a factor from 2 to 6 (Table I), while laboratory temperatures can be as low as 77 K. Viscosity models, based on theoretical and experimental results, have been reviewed by Schubert et al. (1986). It is important to note that the fitting of the experimental data gives a law of the type:

H* = Ao-~e RT (13)

where ~ is the strain rate, cr the effective stress, A and m material parameters, H* the activation enthalpy and R the gas constant. The viscosity is given by :

H* ~r 1 crl_,~ e~-~ (14) ~/(o-, T) - 2~ - 2 A

If diffusion creep is dominant, m = 1 and the viscosity is Newtonian, independent of stress; if dislocation creep is dominant, m ranges from 3 to 5 and the viscosity is non-Newtonian and proportional to crl-'~. Experimental data (Durham et al., 1983; Kirby et al., 1985; Sotin et al., 1985) on the deformation ofpolycrystalline water ice. at high pressure and low temperature are collected in Table IV. It should be noted that for Ice VI the pressures are in the range of 1.0 to 1.7 GPa.

In a homogeneously mixed satellite, the presence of silicate in the ice would increase the viscosity by an order of magnitude. For example, Friedson and Steven- son (1983) have theoretically found that a 50% by volume ice-rock mixture has a viscosity 6 to 16 times greater than that of pure ice, depending of the size distribution of the silicate particles. This theoretical result seems to be confirmed by experimental results, recently obtained by Durham et al. (1992). For mixture of solid particles plus ice, ice being a material that is markedly non-Newtonian, the


TABLE IV

Steady state flow parameters for ices

log A m H* (MPa m/s) (kJ/mole)

ICE In

243 < T < 268 11.8-t-0.4 4.0±0.6 91±2

195 < T < 243 5.1±0.03 4.0+0.1 61±2

158 < T < 195 -3.1 4.0 31

ICE II 2.7±0.4 5.0±0.3 57±3

ICE III 63 ± 3 5.5±0.5 325±25

ICE VI

253 < T < 293 -10 .6+0 .2 1.93±0.02 28.5 + 8 10-3p

viscosity is larger than expected even for Newton±an fluids, and the mixture turns out to be roughly 2 orders of magnitude more viscous than pure ice.

To date the measured strength of water ice shows that all ices get stronger with decreasing temperature and that a mixture of ice plus silicate particles is stronger than ordinary ice. Ice II is stronger than ice I, ice V is weaker than ice I and ice III is a great deal weaker.

Non-Newton±an behaviour seems to characterize ices in the system NH3- H20 and methane clathrate, with an approximate formula CH4- 6H20. At temperatures less than 200 K, the presence of ammonia weakens water ice, especially around T = 176 K, where partial melting occurs. More recently, Yarger et al. (1993) have experimentally shown a far more complex behaviour of ammonia-water compounds. Initiation of melting is possible through self-heating at temperatures well below 176 K.

The so-called methane clathrate (CH 4. H20) has about the same strength as water ice, but in all these experiments the confining pressure is 50 MPa and the strain rate is equal to 3.5 10-6/s. From a theoretical and experimental point of view the rheology of ices and of ice mixtures of planetological interest seems to be non-Newton±an and two competing effects act on the viscosity: the behaviour of the material is weaker owing to the presence of ammonia and harder owing to the presence of silicaceous grains. However, using the experimental values of the parameters and extrapolating the values of the viscosity to conditions relevant to the interior of the icy satellites, it can be asserted that solid-state convection can take place in the interior of the icy moons. In fact the rheology of the ice- silicate mixture is dominated by the creep behaviour of ices rather than by that of silicates. On the contrary, Lewis, (1971b), Consolmagno and Lewis, (1978) and Parmentier and Head (1979a, 1979b) in their modelling assumed that the viscosity


of the homogeneous ice-silicate mixture was much larger than the viscosity of pure ices.

The knowledge of rheological properties for ammonia-water-methanol liquids, representing possible cryovolcanic substances seen on the surface of the icy satellites, is fundamental in explaining the nature of the resurfacing processes. Recently, Kargel et al. (1991) have shown that these aqueous mixtures have viscosities ranging from 10 -3 Pa s for pure water to 104 Pa s for the ammonia-water-methanol peritectic. With supercooling and/or partial crystallization, these substances may attain viscosities several orders of magnitude greater than those given above. Kargel et al.(1991) also showed that, as the water content of ammonia-water slurries increases from the peritectic, their rheology departs from Newtonian with a power-law index ra _< 2.4.

3.3. SUB-SOLIDUS CONVECTION

The thermomechanical equations (Navier-Stokes equation plus energy equation) that govern natural convection are complicated, especially when the dependence on temperature and pressure of the involved thermodynamical parameters is taken into account. Thus a number of simplifying assumptions are necessary and the validity of the obtained solutions must be tested. Owing to these difficulties the parameterization of heat flow in terms of Nusselt number-Rayleigh number power law has been an important tool in modelling thermal histories of planetary objects. This approach, commonly referred to as parameterized convection, has been used in a variety of forms by different authors, but it is always based on the simple dependence of the Nusselt number N u , on the Rayleigh number Ra (see for example Sharpe and Peltier, 1979; Schubert, 1979: Schubert et al., 1979; Turcotte et al., 1979 )

N u = a R a n (15)

where a and fl are constants that depend on the type of heating and on the boundary conditions; a can also depend on the geometry. For a fluid layer of width d the Rayleigh number results

e~g p A ) Ra - (16)

where c~ is the thermal expansion coefficient, g the gravitational acceleration, AT the superadiabatic temperature difference across the layer, k the thermal diffusivity and r] is the dynamic viscosity and p the reference density. From linear stability analysis, convection sets in a fluid layer when:

R a ~ Racr (17)


If the fluid layer is uniformly heated from within, R a can still be defined using Equation (16), except that in this case the temperature difference across the layer is the temperature difference due to the conductive steady state and it results that:

Ra c~ g p2 H d 5 - ( 1 8 )

K k ~

where H is the heat generation rate per unit mass and K is the thermal conductivity. The value of the critical Rayleigh number depends on the boundary conditions, on the aspect ratio and on the type of heating. On the basis of theoretical results a value ranging from 1000 to 3000 has been normally used in the numerical simulations. After the onset of convection the flow adjusts itself to distribute the heat that is not transported by conduction. A large number of laboratory experiments and theoretical and numerical investigations have shown that in a convective system the flow can be given by

qconv = NUqcond (19)

The above relation holds for a fluid layer heated from below and from within. For constant viscosity convection at Rayleigh numbers well above the critical one, the exponent /3 in the order of 1/4 to 1/3 is well documented (Schubert, 1979; Jarvis and Peltier, 1982). When the viscosity depends on temperature and pressure and the fluid is not Newtonian, numerical investigations by Christensen (1984a, 1984b,1985a,1985b ) have conclusively shown that two nondimensional parameters must be introduced to take account of the characteristics of the viscosity law. Solomatov and Zharkov (1990) have recently confirmed this result with a semi-analytical theory. The first parameter is the "surface" Rayleigh number:

g p A T d 3 Rao - (20)

k r/0

here 7/o = 7/(T = 0; z = 0) (Solomatov and Zharkov, 1990). The second Rayleigh number is defined as:

a g p A T d 3 R a T = (21)

k ~/T m

here ?IT = ~(T = T; z = d/2) and T is the mean value of T in the layer. The relationship N u - Ra can now be expressed as:

= a Rao 9° R a ¢ T (22)

The values of a,/30 and/3T depend on boundary conditions and on the characteristics of the fluid: Newtonian or non-Newtonian. N u is more tightly dependent on RaT.


All the previous results strictly refer to plane layer models, in contrast to the geometry of convection in a planetary body that is curvilinear, with the upper boundary having a larger surface area than does the lower boundary. This results in an asymmetry both in temperature drop and in thickness of the upper and lower thermal boundary layers which is not found in planar geometry (Jarvis and Peltier, 1989; Bercovici et al., 1989; Jarvis, 1993). Using a 2D time-dependent numerical model of convection in a cylindrical annulus and the boundary layer theory, Jarvis (1923) has shown that the effects of curvature can be paranleterized in terms of the fraction f~ of the inner to the outer radii of the cylindrical bounding surface. In particular the relationship (15) may be rewritten as:

N u = a g e ( L ) R az (23)

where ge is the geometrical factor. In the range of fr between 0.2 and 0.6, the geometric factor ranges between 0.28 and 0.38 in Cylindrical geometry (Jarvis, 1993). It can be shown that the geometrical factor for a spherical axisymmetric geometry ranges between 2.2 and 1.1.

Using Equation (19) as a parameterization of the convective heat transport, thermal histories of icy objects have been obtained in different ways. One approach implies the solution of the heat transport equation, only in radial direction, introduc- ing an enhanced thermal conductivity, that is the product of the usual lattice thermal conductivity K and Nusselt number N u , to mimic convection (Kaula, 1979; Thurber et al., 1980; Coradini et al., 1982). The enhanced conductivity must be used inside the shell when the Rayleigh number is above the critical one. The energy equation, neglecting viscous dissipation and compressional work, becomes

PCv Ot - r 2 Or r 2 K N u + p H (24)

Another approach considers the balance of energy inside the convective shell, taking into account the basal heat flux entering the shell from the core and of the heat flux leaving the shell through the outer surface. Obviously this holds when a core is present; in the absence of a core, i.e. for an undifferentiated structure, the basal heat flux is zero. The cooling of the planetary objects produces the thickening of the lithosphere, defined as an outer shell whose thickness, L( t ) , is fixed by a temperature, the lithosphere temperature TL, below which the satellite material is rigid on geologic time scale. Using the second method, the growth of the lithosphere is obtained by balancing the heat fluxes in and out of the lithosphere with the growth rate of the lithosphere itself (Schubert et al., 1979; Schubert et al., 1981) i.e.

dL PSatellite cp [Tc(t ) - TL]-- ~- = qL(t) -- qt(t) (25)

here qt is the heat flux out of the convecting layer into the overlying lithosphere and qL is the heat flux coming out from the lithosphere. The temperature within

566 a . CORADINI ET AL.

the convecting region is assumed to be spatially uniform at the value Tc(t) except for the thermal boundary layers where the temperature varies linearly with radius. The boundary layer thickness for a vigorous convective system is

( Racr'~ ~ * : [ D ( t ) - L(t)] ~, Ra J (26)

where D(t) - L(t) is the thickness of the convective layer. The thermomechanical problem of convection, applied to icy satellites, has been

solved for Dione, using a finite element numerical simulation (Forni et al., 1991) and in this work for a Ganymede-like satellite, using a finite difference numerical method. The method used to numerically solve the set of equations and the results obtained will be reported below.

4. Satellites of Saturn and Uranus

The small Saturnian satellites, as the images returned from Voyager 1 and 2 encoun- ters with the Saturnian system have shown, have surfaces that are not uniformly cratered and are hence made up of units of different ages; the exception is Mimas whose surface is basically unmodified, its surface being uniformly cratered. The observed surface heterogeneity suggests that internal materials may be in motion as the result of thermally generated differences in density, so that convection is an important heat transfer mechanism in the interiors of these satellites. As a consequence the internal structure and the thermal profile are closely related and the readjustments of the interiors can cause compression and/or expansion of the overall satellite. Ellsworth and Schubert (1983), and Federico and Lanciano (1983) have calculated thermal and structural evolution of Saturn's small satellites taking into account both the gravitational energy gained by the satellite during its formation process, and the energy delivered by the decay of the radioactive isotopes, assumed to be in chondritic abundance.

As far as the initial temperature profile is concerned, it has been shown (Ellsworth and Schubert, 1983; Federico and Lanciano, 1983) that the initial heating is not negligible, but the temperatures are not high enough to reach the melting temperature of water ice. Thus primordial ice-rock differentation in these objects has not occurred, and the small Saturnian satellites are nondifferentiated, homogeneous bodies. If ammonia hydrate is present in these bodies there will be a limited amount of melting. In fact NH3H20-H20 ices have a eutectic melting point at 173 K and melting can occur, but, following Lewis (1972), ammonia hydrate is only a minor component, representing at most 20% of the total mass of ice, and partial melting of a minor constituent could not trigger the complete differentiation of a satellite. Moreover, as is shown in Forni et al. (1991), liquid ammonia hydrate can easily migrate through the satellite structure, reaching the surface in a very short


time (10 4 years). Migration of liquid ammonia hydrate and its penetration through the lithosphere can produce surface flooding and resurfacing (Stevenson, 1982).

Calculations on the parameterized thermal evolution of the small satellites of Saturn have been perforined independently by Ellsworth and Schubert (1983), Fed- erico and Lanciano (1983). Even if there are differences in the initial assumptions, in the values of the physical quantities involved and in parameterizing convection, the results are in quite good agreement. In the case of Rhea, convection starts almost immediately after its formation and lasts for about 3 Ga, reaching its maximum vigour in the first 0.3 Ma. The maximum duration of the convection has been obtained for a model in which the surface is covered by a low thermal conductivity layer, as suggested by Stevenson (1982). After the end of convection, the body cools conductively. The mass of the satellite being constant, the time evolution of the internal temperatures causes changes of the internal densities and hence of the total radius of the object. In the case of Rhea the first stage corresponds to a radius increase due to thermal expansion, and subsequently, as the deep interior progressively cools, to a radius decrease. This contraction can be large, of the order of 10 km (~ 10%) if, during the last one billion years, an ice II core is formed. The geology of Rhea, on the contrary, is characterized by extensional tectonics; this is in conflict with the formation of a large ice II core and can imply that Rhea is warmer. A warmer interior can be obtained by adding another source of energy and Schubert et al. (1986) suggest tidal heating, or taking into account the presence of a low thermal conductivity regolith on the surface. Federico and Lanciano (1983) have shown that, when the regolith is considered, the early expansion is prolonged in time, reaching its maximum at 1 Ga. The subsequent evolution is a very slow cooling with a negligible shortening of the radius. For Mimas, the smallest among the regular Saturnian satellites, the model thermal history is basically conductive. In between these two extreme cases, convection lasts more or less depending on the radius of the satellite and on the silicate content. It should be noted that the fraction of silicates can not be uniquely determined when only the mean density of the satellite is known. The model thermal evolution of Iapetus is similar to that of Rhea with a convective period that lasts for about 2 Ga. Dione and Tethys have convective periods that last for about 1 Ga and 2 Ga respectively and only Dione undergoes an expansion stage during the first 0.5 Ma of its life.

From a geological point of view Dione is a complex satellite and Plescia (1983) claims that geological processes on this satellite were still active after the cessation of the heavy bombardment. These processes lasted approximately 1 Ga with a predominant extensional tectonics. For these reasons Fomi et al. (1991) carried out a finite-element numerical simulation of the thermal convection inside Dione in order to assess the characteristics of the convective motions and the induced stress distribution in the lithosphere. Taking into account the non-Newtonian viscosity of ice, the dependence of viscosity on temperature and pressure, and the internal heating, Forni et al. (1991) have shown that the convective vigour first increases, during the initial delivery of the accumulation energy, and then, after about 1


Ga, sustained only by the radioactive energy, it slowly decreases. The Rayleigh number returns to the critical value after 3 Ga. During this period of convective activity the stresses induced in the lithosphere reach their maxima, and the values are large enough to allow fracturing both in extensional and in compressional regimes. This stress period lasts for about 3 Ga because [he increase in viscosity, as the temperature decreases, is balanced by the decrease in the strain rate, due to the slow turning off of the convective activity. This result, obtained through the solutions of the thermomechanical equations, shows a longer duration of the endogenic activity and of the "high" stress period and is in good agreement with the geological observations (Forni et al., 1991).

No specific thermal evolution has been computed for the satellites of Uranus but analogous results can be obtained for objects similar in size to the Saturnian ones. For Miranda mantle convection has been invoked to tentatively explain the corona, that is extremely different from all the other geological structures observed on the surface of small icy satellites. It has been proposed that the mantle convection is driven by mass anomalies within the mantle, either as result of late accretion of large silicate bodies sinking toward the satellite center, or as compositional induced low density rising diapir. No calculation has been performed to predict the stresses induced in the lithosphere by such a kind of mantle convection, except for the first model (the "sinker" model). Janes and Melosh (1988) calculated the stresses induced in Miranda's lithosphere by an anomalously dense mass sinking through its mantle. The predicted stress patterns are consistent with the observed corona if the silicate sinkers are several tens of kilometers in diameter and if they are located near the top of the mantle beneath a relatively thin lithosphere: 5 km thick.

5. Tidally Heated Satellites

Tidal heating has been invoked as an additional source of energy able to explain unusual thermal activity at least on Io, Europa, Enceladus and probably on Triton. For Io, whose intense volcanic activity was predicted by Peale et al., (1979) and confirmed by Voyager observations, tidal dissipation is definitely strong enough to justify the large measured heat radiation rate (>1013 W, i.e. a mean surface heat flux of about 2 W/m2). The mean tidal dissipation rate, i n fact, strongly depends on the internal structure and rheology of the satellite and Segatz et al. (1988) have sfiown that for a viscoelastic model of Io it turns out to be as large as 3 × 1015 W for reasonable choices of mantle rheology parameter values. In constructing these models steady state equilibrium has been assumed where tidal heating equals the heat conducted through the lithosphere. The validity of the model requires that the calculated surface heat flow is equal to the measured one. Two types of model have been proposed: in the "deep mantle model" tidal heating occurs deep in the low Q, near-solidus mantle and local regions of partial melt are proposed to account for volcanic activity (Ross and Schubert, 1985; Ojakangas and Stevenson, 1986;


Segatz et al . , 1988; Fischer and Spohn 1990). In the "asthenospheric model", first proposed by Schubert et al. (1981), the mantle is decoupled from the lithosphere by a thin, partially molten asthenosphere where tidally forced viscous dissipation largely occurs. In this case magma migration through the lithosphere from the asthenosphere is the possible cause of volcanic activity (Ross and Schubert, 1985; Segatz et al . , 1988). The internal distribution of tidal heating is different in these two models: in the first one tidal heating is concentrated beneath the polar regions, while in the second it is concentrated beneath equatorial regions. These different distributions should appear in the surface structures, even if a convective heat transfer process would in some extent modify the obtained patterns.

The existence of an equilibrium state has been questioned on the ground of dynamical considerations of the Jupiter satellites system. Ojakangas and Stevenson (1986), considering the deep mantle model of a solid Io, have examined the stability of a model where the thermal and orbital evolution of a tidally heated satellite in an orbital resonance are coupled. The heat being transferred mainly by convection, the satellite's thermal evolution is governed by a heat balance where the time rate of change of internal energy equals the tidal energy dissipation rate minus the surface heat flux. When the Rayleigh number is larger than the critical one the relationship N w - R a is used (see section 3.3), and the ratio k 2 / Q is a function of temperature. This model is successful in allowing values of orbital eccentricity and surface heat flow for part of the time, consistent with current observations. Due to the inverse temperature dependence of the dissipation factor, runaway tidal heating occurs supporting the plausibility of periodic variations in heat flow and orbital eccentricities.

Fischer and Spohn (1990) studied evolutionary thermal and orbital histories of a viscoelastic partially molten Io, assuming a deep mantle model. The assumption of a partially molten interior was made on the basis of the stationary model results of Ross and Schubert (1985) and Ross et al. (1985), taking into account of the small magma removal rate (10 6 kg/s), obtained by Ojakangas and Stevenson (1986). In fact to justify the observed 10 mm year-1 of resurfacing rate (Johnson et al . , 1979), a magma removal rate of 4 × 107 kg s -1 is required (Fischer and Spohn, 1990). The theory of Yoder and Peale (1981) was used to calculate the orbital evolution in the resonance and the N u - R a relation to calculate the convective heat flux. The results of Fischer and Spohn (1990) show that at present there is an approximate equilibrium between tidal heat production and heat loss for reasonable values of mantle viscosity and shear modulus. The oscillations of thermal and orbital variables are a characteristic feature of the satellite evolution. The time of the onset of the oscillations depends on Io rheology.

Thermal history calculations of Io by Kawakami and Mizutani (1987) take explicitly into account the heat transfer associated with magmatic activity (advective heat transfer) and the radial distribution of the tidal heating. Radial tidal distribution is calculated using the procedure of Peale and Cassen (1978) for a two layers object, and for the advective heat transfer it is assumed that hot material


can be poured out onto the surface from magma sources shallower than 1000 km, where the pressure is the same as that at 150 km depth on Earth. Convective heat transfer is considered as the heat transport mechanism and it is modeled through the parameterized scheme (sec. 3.3). The calculations of Kawakami and Mizutani (1987) show that the total tidal heating energy is almost equal to the advectively transferred heat and that the permissible range of QSatellite constrains the radius of the liquid mantle in Io in the range from 1000 km to 1700 km. Among the models studied by Kawakami and Mizutani (1987) there is one in which the effects of the initial temperature profile are investigated. The initial temperature distribution is assumed to be 'hot', i.e. the upper mantle temperature is above the solidus and the temperatures in the central part are lower than 500 K. The result is that the hot outer region rapidly cools by the advective transport and the central part is heated by tidal heating. Subsequently, the temperature in the deep interior decreases because of the convective heat transfer, even if radioactivity is present. Altough the temperature evolution in the deep interior strongly depends on the initial profile, the isotherm in the upper mantle is not substantially different from those obtained with different initial assumptions.

Attempt to infer the Io tidal heating mode from observations is very difficult and different methods have been undertaken. Gaskell et al. (1988), for example, pointed out that the distribution of the equatorial topography qualitatively resembles the equatorial pattern of the tidal heating production of the asthenospheric model. Subsequnetly, Ross et al. (1985) examined the correlation between the topography of Io and model heat flow, and they combined detailed calculations of tidal heating with an isostatic model of topography. In this case not only equatorial topography but also polar topography is taken into account. By comparing observed and modeled topography, Ross et al. (1985) were able to construct a model of the internal structure of Io, able to give the necessary surface heat flux. In this model tidal dissipation occurs about two-thirds in the asthenosphere and the remaining one-third in the deep mantle, whose viscosity must lie between 1016 and 4 × 1017 Pa s, consistent with a solid mantle with some partial melting inside. The asthenosphere has a thickness between 50 km and few hundred kilometers with a viscosity that must be larger than 108 Pa s and less than 1012 Pa s, suggesting a crystal-rich magma. The lithosphere, whose density is about 7% less than that of the asthenosphere, has a thickness between 24 km and 50 km with a mean value of about 35 km. The core has a radius of 980 km with a Fe-FeS composition (Segatz et al., 1988). Here it has been briefly outlined how it is possible to obtain a reasonable internal structure, using thermal history models. A more comprehensive summary of the problems about Io can be found in Nash et al. (1986).

The mean density of Europa indicates a silicate composition with an H20 mass fraction of about 6% in order to dilute the density of an Io type body to Europa's value. The surface of the Jovian moon Europa is characterized by a lack of impact features with diameter larger than 30 km, a subdued topography and the presence of H20, implying recent resurfacing by water from a liquid layer (Cassen et al.,


1982; Squyres et al., 1983) or from dehydration of the interior (Finnerty et al., 1981). Curvilinear fracture-like features have orientations broadly consistent with tidally controlled tectonic activity (Helfenstein and Parmentier, 1980) and can also be generated by the global expansion due to dehydration (Finnerty et al., 1981).

Cassen et al. (1982) and Squyres et al. (1983) have shown that if a silicate core is overlain by an ocean of liquid water and a thin shell of ice I, tidal heating acting in both core and shell is sufficiently intense to stabilize an ice lithosphere of about 16 km (Squyres et al., 1983), where heat transport is by conduction in solid ice. In the 'internal ocean' model of Squyres et al. (1983) where the density is constant and only the shear moduli are different in the layers, the mean surface heat flow from radiogenic source is about 8 mW/m 2 and from a tidal source it is about 44 mW/m 2, so the energy sources are large enough to completely dehydrate silicates, also because accretional heating could have been important.

Petrological considerations, on the other hand, suggested to Ransford et al.

(1981), that heat may be transported within Europa by sub-solidus convection throughout much of the satellite's history. Sub-solidus convection can occur because the viscosity of water-rich silicates may decrease significantly at the dehydration temperature. Convection inside Europa transports heat at a rate sufficient to man- tain a large part of the interior below the dehydration temperature (Ransford et

al., 1981). In the Ransford et al. (1981) model, a region of convecting anhydrous silicates is overlain by a lithosphere, the upper 200-300 km of which is composed of hydrous minerals. The ice crust is much thinner, only a few kilometers, and it is expected that Some free water gains access to the surface during the satellite history, but not in vast quantities. In this model the water is retained in hydrated minerals.

Viscous relaxation rates of craters have been determined by Thomas and Schu- bert (1986) to discriminate between models of Europa interior. Thomas and Schu- bert (1986) have shown that only for the 'internal ocean' model are the relaxation times for large craters (diameter about 100 km) of the order of 108 years, sufficiently srffall to account for the essentially uncratered appearance of the surface of Europa. For this reason, Ross and Schubert (1987) calculated the tidal heating for a more realistic three-layer model of Europa. The model consists of an ice 1 (p = 940 kg/m 3) elastic lithosphere, an underlying water layer (p = 1000 kg/m3), and an elastic silicate core. This new model takes account of the different densities of the layers, assumes a shear modulus equal to 4 × 109 Pa for the lithosphere and up to 60 × 109 Pa for the core and decouples the lithosphere from the core, through a water ocean 100 km deep. Thus the calculated tidal distortion is substantially less than what has been previously assumed and the stability of the internal ocean, at the current value of the orbital eccentricity, requires a very low QSateltite (Ross and Schubert, 1987).

Eneeladus has a wide diversity of terrains, ranging from substantially cratered to uncratered. In addition the craters themselves display various degrees of relaxation from one region to another. The crater free terrains are almost younger than 109


years and there have been emplacements of mobile material. Probably the existence of Saturn's E ring, which has a peak in its intensity near the orbit of Enceladus, needs an explanation, indicating that this satellite is currently active (Passey, 1983; Squyres et al., 1983b; Morrison et al., 1986; Ross and Schubert, 1989). A large internal heat source is required to explain the nature of Enceladus and tidal heating has been suggested. Poirier et al., (1983) have argued that tidal effects could not provide the required heating unless the past eccentricity was substantially greater than at present. Ross and Schubert (1989) showed that tidal heating for a Maxwell Enceladus model has a peak of 9.2 × 1011 W in contrast with the value of 8.4 × 108 W, obtained by Poirier et al., (1983). Using the Maxwell viscoelastic rheology, Ross and Schubert (1989) showed that a two and three layer model of Enceladus is able to justify thermal activity since Enceladus entered the commensurability with Dione. These models have surface heat flows of 5 mW/m 2 and lithospheres about 10 km thick. It should be noted that in a three-layer model a liquid ocean beneath the lithosphere is present and in this case the liquid is melted NH3-H20 eutectic. Dissipation is likely to occur mainly in the solid ice core beneath the fluid layer or in the liquid layer through turbulent flow. The uncertainty in the estimate of the turbulent stress does not rule out the possibility of freezing of the ocean.

Ross and Schubert (1990) also studied the post-capture orbital and thermal evolution of Triton. They assume the differentiated internal structure of Triton, proposed by Smith et al. (1989), consistent in a silicate core (3361 kg/m 3) covered by an ice I shell (980 kg/m3). They show that only a model in which the dissipation factor (k2/Q)Satellite is temperature dependent produces reasonable results. Depending on the initial temperature and magnitude of radiogenic heating, most of the changes in Triton orbital and thermal states take place during a sudden event which may occur several billion years after capture. This work also shows that global melting is unavoidable.

6. Large Satellites: Ganymede and Callisto

Inside a large icy satellite different ice phases can be present and their effects upon solid-state convection cannot be ignored. Phase transitions interact with convection via two principal mechanisms: distortion of the phase change boundary and buoyancy through thermal expansion. Thurber et al. (1980), in the case of Ganymede and Callisto, investigated the stability of phase changes through the dimensionless parameters S and RQ. These parameters have been introduced by Turcotte and Schubert (1971) and Schubert et al. (1975) in discussing the role of phase transitions in the Earth mantle. S is a measure of the density difference across the phase transition and RQ is a Rayleigh number associated with the phase transition. Thurber et al. (1980), using parameter values appropriate for Ganymede and Callisto, an icy-dominant rheology, and a linear stability analysis, estimated that the ice II-ice V and ice II-ice VI transitions are stable against penetrative


convection. Thus these phase transitions may have stabilizing effects on thermal convection. The style of convection would be two-cell convection and the heat transport will not be as efficient as that without a phase boundary. Bercovici et al. (1986) carried out linear stability analysis for an ice structure where the viscosity is a function of the temperature and in which two exothermic phase transitions (ice 1-ice II, ice VI-ice VIII) and two endothermic transitions (ice I- ice III, ice II-ice V) are present. The results of Bercovici et al. (1986) show that exothermic phase changes can either impede or enhance whole-layer convection while the endothermic phase changes always impede whole-layer convection. Sotin and Parmentier (1989), applied linear stability analysis to a differentiated satellite with a pure ice mantle containing the ice I-ice II (exothermic) and the ice II- ice V (endothermic) phase transitions in order to examine the effects of different thickness and viscosity on the flow. In the model of Sotin and Parmentier (1989) the ice 1-ice II transition occurs at 150 km depth, and the total thickness of the layer, composed by the two phases, varies from 210 km to 500 km depending on the temperature at the bottom of the ice II shell. Considering the relative thickness of the ice I shell compared to the ice II shell, the results of Sotin and Parmentier (1989) show that a whole-layer convection occurs at the minimum Rayleigh number for any value of the thermal gradient. Such strong values of S characterize the ice II- ice V phase transition, and layered convection is predicted by Sotin and Parmentier (1989) no matter what the ratio of the upper layer thickness to the whole fluid thickness is. These results need to be confirmed both because they depend on the assumed profiles of temperature and pressure inside the satellite, and because, as underlined by Sotin and Parmentier (1989), convection patterns at high Rayleigh numbers may be different from those predicted at the minimum critical Rayleigh number of the linear stability analysis. Moreover, finite amplitude calculations show that if, during the evolution of the satellite, a change in the convection pattern takes place and/or an intemaediate state is present, it may have important effect on the evolution of the mantle of a large icy satellite.

6.1. THE MODEL

In order to describe thermal convection in an axisymmetrical configuration in a large, icy satellite, a system of polar coordinates with the origin at the centre of the satellite has been assumed. The polar axis does not necessarily coincide with the axis of rotation. Even if a 3D model of compressible mantle convection with phase change has recently been put forward, 2D geometries are able to indicate complex new phenomena and modes of time-dependence in the convective flow (Tackley et

al., 1993). The medium is compressible and isoviscous and the physical variables do not

depend on the azimuthal angle (longitude) so that they are functions of r and 0 only. The motion will then take place in meridional planes and the velocity vector will have only two components, the radial one u~. and the zonal one uo. The


TABLE V

Physical parameters for convection

Symbol Value Parameter

R (m) 2.5 x 106 Total radius M (kg) 1.25 x 1023 Total mass d (m) 1.5 x 106 Shell thickness rc (m) 1.0 X 106 Core radius g (m/s 2) 1.334 Gravity acceleration To (K) 130 Surface temperature p0 (kg/m 3) 1400 Surface density p.~ (kg/m 3) 1600 Mid depth density

(K-1) 1.0 x 10 -4 Thermal expansivity t(s (MPa) 1.0 x 104 Isentropic modulus

of bulk compression I f (W/m K) 2.2 Thermal conductivity k (m2/s) 8.3 x 10 -7 Thermal diffusivity cp (J/kg K) 1800 Specific heat H (W/kg) 5.4 x 10 -12 Heating rate /? (W/m 2) 5.4 × 10 -3 Core heat flux D i = cegd/c p 0.12 Dissipation number Gr = c~K~/pocp 0.43 Griineisen parameter

numerical model solves the dynamical equations of mass, momentum and energy conservation in a spherical shell, extending from the surface to the core-mantle discontinuity, in presence of phase transitions.

Following the method outlined by Jarvis and Mc Kenzie (1980), Jarvis and Peltier (1989), and Machetel and Yuen (1989), the compressibility will be modeled using the inelastic liquid approximation and adopting as reference state the state with hydrostatic pressure, PH, and adiabatic temperature distribution, Ts. The inelastic approximation assumes in the equation of mass conservation that the convective velocities are small compared to the local sound speed and the medium undergoes compressions and expansions only on a convective time-scale and not on a sound wave time scale, so that the time derivative of the density can be neglected, leading to a much less severe Courant condition on the size of the numerical time-step. The density is assumed to be equal to the reference density, except for the buoyancy term of the momentum equation. Since s A T < < 1, the time rate of change of entropy is only given in terms of the temporal variation of


TABLE VI

Physical Parameters of the Phase Transitions

Phase Transition 3' Ap t~ Po MPa/K a kg/m 3 kg/m 3 MPa

Ice I-Ice II 0.8 340 1450 0.0

Ice H-Ice V - 9 . 0 100 1600 2000

Ice II-Ice VI - 1 . 2 140 1600 900.0

Ice VI-Ice VIII 3.5 300 1750 1000

temperature, neglecting that of pressure. The reference density profile/5 is given by:

= p ~ + ~ Ap~r~ (27) i

Pa~ = poexp [ (gp ,~IKs) ( r0 - r)] represents the Adams-Williamson density profile whose physical parameters are given in Table V.

The summation is extended over the phase transitions, the density jumps of which Api are given in Table VI. Fi gives the fraction of the denser phase present in the mantle material. The function F(P, T) is expressed, following Richter (1973), Christensen and Yuen (1984), Christensen and Yuen (1985), as a tanh-function and is a function of P0i, the phase transition pressures at zero temperature, fii, the mean geometrical density of the two phases, ei, the transition width, chosen equal to 15 km, and % the Clapeyron slope. The values of these physical parameters are reported in Table VI. The transition width has been assumed to be equal to 15 km and then much less than the thickness of the convecting layer; in this condition, as pointed out by Christensen (1989), the exact functional dependence of F(P, T) appears to be insignificant. Provided that fluid motions cause only small departures from the reference state, the equation of state may be written as a Taylor expansion around the reference state density distribution, #5. Under these assumptions, the linearized compressible equation of state in the presence of phase transitions is

p = f5 [ 1 - ~ [ c ~ ( T - T S ) ~ - I £ T I ( P - PH)]J+ [(on

7

(28)

where o, is the coefficient of thermal expansion and KT is the isothermal modulus of bulk compressibility.


The continuity, momentum and energy equations, using the vorticity - stream function formulation, become:

D2 ~ + 1 0t5 0~I ' _ fSco (29) p Or Or

[Di ] D2co + ~ + ~i 2APiri(le~pi - I'/) -b-r-r =0~

sin0 [po~RaR + Z 2Rb~-y~r~(1- r~)]N+ e~p~

4. Oi 0 10~ 5 [ ~ + Z 2zxP~r~( 1 ~ - r~) lN(~r~) (30)

22Rbl F t5[1+ D i ( T + T o ) ~ - ~ / i ~ i(1 - Fi)]

i

OT [ - ~ + u.grad r] =

Di V2T + RaR + fi#

1 ~ 2RbiF +~Di(T + To)ur[1 + F a R ) _ _ / ' 7 ~ ~(1 - Vi)] (31)

where

D2 0 2 1 0 1 0 2

= Or 2 r 2 tan 19 00 + r --2 00 .---5

Here co, ~ , I, and T denote vorticity, stream function, viscous dissipation and temperature, respectively. Following Machetel and Yuen (1989), the one-component vorticity vector is given by: co = r sin 0 rot (u). Moreover

RaR = c~ gpo(poHd + F)d 4 Kkr]

noindent is the Roberts-Rayleigh number,

gApid 3 Rbi -

k~

is the boundary Rayleigh number, and

poH d # -

F + poHd


TABLE VII

Scaling factors for different physical quantities

Symbol Quantity Non-dimensionalization factor

r, 0 Spherical Coordinates d Time d2 / k

u = (uT, uo) Velocity k id T Temperature (poHd 2 + F ) / k P Dynamic Pressure (kq)/d 2 ¢3 Reference Density P0 A pi Density Jump Po 7i ClapeyronSlope (poHd 2 + F)/(kpog)

represents the ratio of the internal heat flux to the total heat flux. In the modelling it has been assumed that the strength of the internal heat source is constant. The scaling factors to non-dimensionalize the equations are given in Table VII, and the meaning of the symbols is given in Tables V, VI and VII.

The Equations (29) and (30) are solved with a second-order, alternate direction, implicit finite difference scheme. We have used a conjugate gradient method, described in Auer (1991), to accelerate the convergence of the coupled stream function and vorticity equations. This convergence is tested before starting a new time step. The time step in Equation (31) satisfies the Courant-Lewis-Friedrichs criterion, based on the distance Ar between two adjacent radial points and the maximum radial velocity.

Two numerical tests have been performed in order to validate the numerical program. First the Roberts-Rayleigh convection has been studied, reproducing some of the calculations made by Peltier et al. (1989), and Solheim and Peltier (1990). In this case no phase transitions are present and the simulations were characterized by different values of the internal heating parameters. Namely: with RaR= 10 7, the numerical program has been run with the following values of the intensity of the internal heat source: # =0.25, # =0.5 and # = 1.0. A good agreement has been obtained on a grid with 120 radial points and 360 tangential points.

To test the program in the presence of phase transitions, the results obtained by Machetel and Weber (1991) in studying the effect of the 670 km spinel-perovskite endothermic phase transition in the Earth, have been reproduced. In this simulation two values of the Clapeyron slope for this phase transition have been used: 7 =

- 2.0 MPa K- 1 and 3' = -4.0 MPa K - 1 and the Benard-Rayleigh number was equal


to 2.1 106. Our results, obtained on a grid of 120 × 360 points, are the same as those obtained by Machetel and Weber (1991) and convection is completely layered with the large value of the Clapeyron slope and is intermittent with the Small value. The influence of phase transitions can be measured by the phase parameter Ph, defined as follows:

P h - 7 Ap crgd p2

From the numerical experiments of Christensen and Yuen (1984), 1985), Chris- tensen (1989) it results that an exothermic reaction (Ph > 0) slightly enhances convection; on the contrary, an endothermic reaction with a negative Ph value enforces layered convection. In the simulations performed by Machetel and Weber (1991) the values of Ph are respectively -0.23 and -0.11 and the behaviour of the convective flow is in agreement with the numerical results of Christensen (1989).

The internal structure of a partially differentiated large icy satellite would con- sist in a partially differentiated icy-silicate mantle overlying a silicate solid core (Mueller and McKinnon, 1988); also the crusts of Ganymede and Callisto are ice- rich as pit crater morphology indicates (Schenk, 1993). The size of the core depends on the degree of the differentiation, that is largely unknown. In the modelling, a core radius of 1000 km has been assumed in order to study the characteristics of the convective flow in presence of different phase transitions. The physical characteristics of the phase transitions taken into account in this study are listed in Table VI. The top temperature is 130 K and the reference temperature profile is such that at the core-mantle boundary (CMB) the temperature does not reach the melting value of ice. Two different Rayleigh numbers, RaR= 1.26 × 10 7 and 6.32 × l0 7,

depending on the choice of the viscosity value, are considered (Table VIII). For each Rayleigh number, calculations are made for both rigid and shear stress free upper boundary. The assumption of a solid core implies a rigid boundary condition at the CMB. On the axis, the mechanical equations are solved under the following boundary conditions, related to the symmetry of the problem:

uo=O

1 Our Ouo - - - - 0

r O0 Or

The flux at the CMB is chosen to take into account both internal heating and heating from the core. The abundance of the internal heat sources is assumed to be chondritic. The outer boundary of the mantle of an icy satellite is an isothermal surface and since the core is solid, at the inner boundary a constant heat flux, which has been computed to be 0.49 (nondimensional), has been imposed. Moreover, at the axis

OT - 0

O0


TABLE VIII

Structural and dynamical cases.

Core Radius Viscosity # RaR km Pa s

1000 5.0 x 10 ~° 0.51 1.26 x 107

1.0 × 102° 0.51 6.32 × 107

T i m e = 0 . 0 7 4 0 T i m e = 0 . 0 2 9 7

q, ~ T

Fig. 6. Stream function and temperature contours for the rigid surface boundary condition and for a core radius of 1000 km. On the left RaR is equal to 1.26 × 107 and on the right it is equal to 6.32 107. On the stream function hemisphere the bold line represents the zero value. The phase transition ice II-ice V is in this case stable against convection and a double layered convection develops. This temperature field is stable because its subsequent temporal variation are smaller than 1%.

because of symmetry. The initial temperature distribution is conductive and it has been randomly perturbated. An initial spherical domain of no flow has been assumed and a zero stream function was imposed on all the boundaries for sake of simplicity.

The temporal evolution of the different models has been followed for a time span of the order of 2000 Ma, when for both values of the Rayleigh number the temperature field reaches a "stability": afterwards the temperature field does not vary by more than 1% from one time step to the next.

6.2. RIGID BOUNDARY

Temperature and stream function contours are shown in Figure 6 for both Rayleigh numbers, when stability has been reached: in both cases the convection is double- layered. In the upper layer the Ice I-ice II phase transition is present and the

580

t

1 . 0 0

0.90

0.80

0.70

0.60

R a = 1 . 2 6

A. CORADINI ET AL.

1 0 7

' ' 1 . 0 0

0.90

0.80

m 0.70

0.60

R a = 6 . 3 2 i 0 ' ' ~ ' ' I ' ' ' I ' ' '

0.50 . ~ 0.50

\ 0.40 . . . . :'~- 0.40 , , ~'-._. ,---.~ , ,

0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 Mean T e m p e r a t u r e Mean T e m p e r a t u r e

Fig. 7. Mean temperature profiles for the "stable" configuration and for the two Rayleigh numbers. The solid lines refer to the rigid boundary condition and in this case a well developed boundary layer appears at the ice U-ice V endothermic phase transition. The rigid boundary profile is wanner than the shear stress free profile.

temperatures are in this case so high that the ice II-ice V endothermic phase transition is stable, being impermeable to the flow. This result is in agreement with the result obtained by Bercovici et al. (1986) and by Sotin and Parmentier (1989), who used linear stability analysis. There is also a good agreement with the results of Christensen (1989) because the endothermic phase transition is in this condition characterized by a phase parameter extremely negative (Ph = - 1.76).

For both the Rayleigh number values, the convective pattern is double layered, but the behaviour of the flow is different. In fact, even if the upper layer has the same pattern in both cases, the lower layer for the larger RaR presents a more instable behaviour, characterized by the presence of many thermal instabilities, that appear both at the bottom and at the top boundary layers. The top instabilities, appearing in the boundary layer that is formed at the endothermic phase transition, are due to the large amount of the internal heating that generates an asymmetry in the heating ((Peltier and Solheim, 1992)). In fact large, low-temperature currents can appear that were not present in the small Rayleigh number case.

The mean temperature profiles for the rigid boundary are shown in Figure 7 (solid lines). The mean temperature is relatively high because the removal of the

ORIGIN AND THERMAL EVOLUTION OF ICY SATELLITES 5 81

Time = 0.0422

¢ 0 T

Time = 0.0224 ,I, T

q

o

Fig. 8. Stream function and temperature contours for the shear stress free boundary condition and for a core radius of 1000 km. On the left RaR is equal to 1.26 107 and on the right it is equal to 6.32 107. On the stream function hemisphere the bold line represents the zero value. The two phase transitions ice I- ice II and ice VI-ice VIII are present while the ice N-ice VI is intermittent against convection. This temperature field is stable because its variations are smaller than 1%. Regions of possible melting are marked on the temperature hemisphere.

heat is less efficient when rigid boundaries are present. In fact the rigid boundary reduces the mean velocities in the thermal boundary layer and reduces the heat flow proportionally to the square root of this mean velocity. Consequently the boundary layer thickens and the temperature increases. Moreover, as it has been observed, the convection is layered. This produces a large boundary layer at the impermeable ice II-ice V endothermic phase transition.

In the case of large Rayleigh number the convective process is more efficient as it can be seen from the mean temperatures that are cooler and from the thickness of the boundary layers that is thinner than in the small Rayleigh number case. Nevertheless the boundary layer at the endothermic phase transition is as prominent as in the small Rayleigh number case.

6.3. SHEAR STRESS FREE BOUNDARY

Temperature and stream function contours for the shear stress free boundary are shown in Figure 8 for both Rayleigh numbers. In these cases in addition to ice I - ice II, the endothermic ice II-ice VI and the exothermic ice VI-ice VIII phase transitions are present because in this condition the temperature field is colder.

The behaviour of these last two phase transitions has been studied using linear stability analysis by Thurber et al. (1980) and Bercovici et al. (1986) respectively.

582 A . C O R A D I N I E T A L .

0~ Z~

2 .0

1 .5

1 .0

0 .5

Ra = 1 . 2 6 107

I I~

I ^ f l ~

1 I~ n 111, I '~

I 1 i i I

I 1 I I

" ; ivl ',! I i' ' ; ' ' ; " ' "

I

' 1 ' ' ' ' ' ' ' '

J

l I

/ t l II f

I" I -

, I

7!wAJ , , f i , , I y

. 0 i i i i , i i i i I I i i t i i i i i [ i i i l t i , , , 1 , . , , , . , i i

0 . 0 1 0 0 . 0 2 0 0 . 0 3 0 0 . 0 4 0 0 . 0 5 0 T i m e

Fig. 9. Normalized vertical flux for two depths where the transitions ice I-ice ]I (dotted line) and ice II-ice VI (solid line) are localized. Intermittently the flux from the ice II-ice VI phase transition is higher than the one from the ice I-ice II transition and the double layered convection disappears.

During the evolution the endothermic phase transition is intermittently permeable to the convective flow and this result is not in agreement with what has been argued by Bercovici et al. (1986). On the other hand, this intermittent feature can be justified by noting that the value of the phase parameter is only mildly negative (Ph -- -0.33). The intermittance of layered convection is depicted in Figure 9, where the normalized vertical mass flux across the two phase transitions ice I - ice II (dotted line) and ice II-ice VI (solid line) is shown. It can be observed that intermittently the mass flux across the endothermic phase transition (solid line) is higher than the mass flux across the exothermic phase transition ice I-ice II (dotted line). The convective flow is characterized by large descending penetrative cold plumes that, due to mass conservation, make the convection one-layered (Machetel and Yuen, 1989). The presence of these cold plumes is due to the great amount of internal heating which introduces an asymmetry into both the stream function and temperature fields. In presence of internal sources less heat is conducted into the bottom of the layer than is conducted away from the top. Consequently thermal instabilities, originating at the top boundary layer i.e. cold plumes, are more prominent than the instabilities rising from the bottom boundary layer i.e. hot plumes (Jarvis and Peltier, 1989).

ORIGIN AND THERMAL EVOLUTION OF ICY SATELLITES 5 8 3

The mean temperature profiles for the shear stress free boundary are shown in Figure 7 (dotted lines). Due to the free surface condition the removal of the heat is much effective and the convection process is more efficient. The consequence is that the mean temperature is much lower than the temperatures obtained in the rigid surface case. Moreover, the convective flow being one-layered, the boundary layer at the endothermic phase transition is reduced. Nevertheless it still exists, and this transition is not fully permeable to the convective flow. The presence of the phase boundary barrier produces an overheated lower mantle in agreement with the results of Zhao et al. (1992).

In the simulations discussed above the non-linear effects of melting on the convective processes are not taken into account, but, when the temperature field is known, it is possible to show the locations where, the temperatures are higher than the melting temperature of ice during the simulations. The amount of melting can be used in this case as a diagnostic parameter in order to draw some planetological conclusions. In the rigi d boundary case, the temperatures beneath the ice II-ice V phase transition are such that all the lower mantle has a higher temperature than the melting temperature. Thus it is plausible that heavy component (silicate particles) starts to separate when such temperatures are attained. The study of the physics of this process is out of the context of this work, but when the body differentiates into a mantle and core an amount of energy becomes available for additional heating and could have surficial expressions in form of volcanic features. Large volcanic structures are not observed on the surface of Ganymede and Callisto, even if some volcanic features on Ganymede have been identified. It must be remembered that at the end of the accumulation the Ganymede thermal lithosphere could be only half as thick as the Callisto thermal lithosphere (Coradini et al., 1982), thus it can be argued that a stress free boundary condition is appropriate for Ganymede since the brittle crust is very thin. Following the same line of thinking, a rigid boundary condition would be appropriate for Callisto, assuming that the differentiation occured very early in its life prior to the formation of the present observable surface. This inference needs to be more carefully studied, taking into account also that different boundary conditions can work during different periods of the life of an active satellite.

7. Concluding Remarks

The problem of coupled origin and evolution of planetary satellites is far from being solved, owing to several severe limitations on our knowledge: first of all the data on the structure of the different bodies orbiting around the giant planets is poor, being mainly based on the flybys of the Voyager missions. Some satellites have been imaged incompletely: therefore the appearance of their surface it not well known. For all of them the resolution of images is no better than one kilometer and in many cases is much less. As far as the internal structure is concerned, the situation is even


worse, only the internal density being known, and that with a large uncertainty. Nothwithstanding these difficulties, our knowledge has improved lately because a great deal of work has been done in trying to identify the commonalities between the different systems and the peculiar aspects that are strictly related to the local history; at the same time the single bodies have been studied by means of ground based measurements that have complemented the Voyager data with some new information of the chemical composition of the different satellites. Therefore a more realistic modeling of the thermal and structural evolution of the satellites has been performed.

The emerging picture seems to be as follows: during the evolutive history of the giant planets the formation of a disk of gas and dust orbiting around them was a highly probable event. The nature of the material present in the disks is difficult to establish due to the concurrence of different physical mechanisms, such as: capture of material from the protosolar nebula, that was, in turn evolving; collisional processes that can lead either to coagulation with the formation of larger aggregates of particles, or to the destruction of pre-existing bodies; condensation of solid material from the gaseous component. This sequence of events is no different from what has been hyphothesized for the evolution of the protosolar nebula and for the formation of terrestrial planets, however, in the case of satellite formation it is clear that the role played by the volatile component and by the different ices was much more important. This is the reason why the mechanisms of formation of planetesimals in the protosolar disk must be applied to the satellite disks with some caution. In any case the different proportions of volatiles vs. refractory materials probably also determines the different evolutionary histories of the single satellites, as we have seen in the section on thermal evolution. An effort should be made to relate the chemical evolution of disks with the primordial composition of the different satellites. Some work has been done in that direction for Titan, where the presence of the atmosphere can be explained assuming its formation in a disk originally rich in CO. A larger percentage of volatiles vs. refractory material is also revealed by indirect proof: in all the satellite systems studied so far, some satellites bear indications of a long-lasting intemal evolution. This is difficult to explain if the role of different high volatile ices is not taken into account. Therefore, where the temperatures are lower, new volatile species start to freeze, drastically affecting the rheological properties of different materials, reducing their melting point and their viscosities. This allows a thermal and structural evolution of icy bodies similar to that of terrestrial bodies, with a much smaller energy input.

These speculations are supported only by indirect data. We can give only upper limits for many of the volatile species that, if they were present on the satellites, would have affected their thermal history. A typical example is ammonia ice, the presence of which has been postulated several times in order to explain the thermal history of Enceladus but which has never been measured. A better knowledge of the present chemical composition of planetary satellites and of their internal structure will allow us to better constrain the evolutionary models. In particular,

ORIGIN AND THERMAL EVOLUTION OF ICY SATELLITES 5 85

the thermal evolution and the development of convection is strictly related to the the rheological properties of the materials involved and to their thermodynamical state. The effect of phase transitions is also important.

Our study of the convection in the mantle of a large icy satellite has also revealed the influence of the surface boundary conditions. A rigid boundary condition means that the surface layers are immobile and the flow pattern does not favour heat transfer at the surface. Thus the internal temperatures stabilize the ice II-ice V phase transition. For a shear stress free boundary condition the surface is in motion, there is no shear, the heat flow toward the surface is favoured and the internal temperatures stabilize the ice II-ice VI phase transition. In general which of these two boundary conditions is more accurate in representing physical reality is not a solved problem, moreover the physical characteristics of the surface of the icy satellites are not completely known.

To conclude, measurements that allow us to improve our knowledge of planetary satellites are badly needed, both in order to understand their thermal history and to retrieve their formation processes inside the protoplanetary disks. Such information is expected from future planetary missions like Galileo and Cassini.

Acknowledgments

The authors thank Profs. V.S. Safronov, E.L. Ruskol and Dr. S. Erard for their careful reviews and constructive comments. This work was partly supported by Agenzia Spaziale Italiana (ASI) grant "Progetto Cassini". Two of the authors (C.E and O.E) have been financially supported by MURST (FONDI 40%) and PNP respectively.

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Origin and thermal evolution of icy satellites

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