Regular article
Formation of the regular satellites of giant planets in an extended gaseous nebula I: subnebula model and accretion of satellites
Introduction
The regular satellites of Jupiter and Saturn generally have low inclinations and eccentricities. Perhaps most striking is the progression of satellite density in the Galilean system. Also, the ratios between the satellite systems and the parent bodies of mass and angular momentum are quite similar (Pollack et al., 1991), which suggests a common origin in an accretion disk present about the protoplanets at a late stage of their formation. These properties, taken together with the tantalizing ratio of the largest satellite of each system to its primary Ms/MP ∼ 10−4 (not too dissimilar from the ratio of giant planet to Sun), lead one to think of the Galilean satellite system as a kind of scaled-down solar system.
Given their similarities in distances, masses, and densities, an issue we wish to focus on is how to view Titan in light of the Galilean satellite system, especially Ganymede and Callisto. Yet, the differences between these three satellites are as intriguing as their similarities. The Galileo mission moment of inertia data are consistent with a fully differentiated Ganymede, but only a partially differentiated Callisto (Anderson et al., 1998). Moreover, Callisto shows no evidence of tectonic activity. Also, the association of craters with the presence of CO2 in Callisto but not Ganymede (Hibbitts et al., 2000) as well as the degradation of craters presumably due to the sublimation of CO2 in Callisto but not Ganymede (Moore et al., 1999), which is consistent with the presence of a CO2 atmosphere in Callisto (Carlson, 1999), seems to require that Callisto be assembled with and retain oxidized ices more volatile than H2O. In the case of Titan, it is probably the presence of methane in the atmosphere that has received the most attention Lunine 1989, Coradini et al 1989, Prinn and Fegley 1989.
Recently, Anderson et al. (2001) have investigated two and three layer models for Callisto’s internal structure assuming hydrostatic equilibrium. For the two layer models these authors find two limiting cases: a relatively pure ice shell about ∼300 km overlying a mixed ice and rock-metal interior, and a thick ≳1000-km ice and rock-metal outer shell overlying a rock-metal core. Since it is difficult to reconcile a metallic core with a partially differentiated state the former solution appears more likely. Given that accreting bodies allocate a fraction of their energy as surface heat Schubert et al 1981, Coradini et al 1982, fast satellite accretion would melt the water ice and lead to rock separation and runaway differentiation (Friedson and Stevenson, 1983). Previous attempts to explain an undifferentiated Callisto have relied on fine-tuning parameters Schubert et al 1981, Coradini et al 1982, Lunine and Stevenson 1982. Although it is possible that nonhydrostatic effects in Callisto’s core could be large enough to allow for complete differentiation of this satellite and still be sufficiently small in Ganymede’s core to have avoided detection, we regard this possibility as unlikely. Instead, we favor a model that makes Callisto slowly.
Other issues also seem difficult to explain. For instance, one might expect the outermost Galilean satellite to have significantly less angular momentum than the preceding satellite. It would seem unlikely that the satellite disk would have enough surface density to make a satellite the size of Callisto at 26RJ, but form no smaller objects outside its orbit. Furthermore, the separation between Ganymede and Callisto (∼10RJ) is so large that one is led to wonder why there are no satellites in between at ∼20RJ (see Mosqueira and Estrada (2003), hereafter Paper II, for a brief discussion of orbital stability). One can always argue serendipity, but the Galilean satellite system is sufficiently regular that we reserve this explanation as a last resort.
A related point can be made concerning Titan and Iapetus. If we form the satellites out of a continuous, smoothly varying accretion disk, it would seem difficult to explain why there are no large satellites between Titan at ∼20RS and Iapetus at ∼60RS (Hyperion does not have enough mass to affect this argument).
Also, one must account for the differences between the satellite systems of Jupiter and Saturn. In the case of Saturn’s satellite system, the concentration of mass in Titan needs to be addressed. But perhaps the most perverse difference between the two satellite systems is the fact that whereas the Galilean satellites get rockier closer to the planet, the inner satellites of Saturn appear to be made mostly of ice! Even so, we attempt a combined model for both Jupiter and Saturn (as well as Uranus).
If we take the satellite systems of Jupiter and Saturn and add the amount of gas necessary to create a solar composition mixture the resulting disks have a total angular momentum comparable to the spin angular momentum of the parent planet (Stevenson et al., 1986). The issue arises whether or not one would expect the circumplanetary disk to exhibit a solar mixture of elemental abundances of water and ice bearing materials. One can think of several processes that modified the abundances of rock and ice from their solar abundances. Yet, the fact that the similarly sized Ganymede, Callisto, and Titan all deviate from solar mixture by the same proportion (∼60% rock, ∼40% ice by mass) seems to indicate that one should be guided by solar mixtures and investigate mechanisms for deviation from them, such as size-dependent water vaporization on one end and water enrichment by composition selective mechanisms on the other. If so, one might calculate models with “minimum mass” by augmenting the mass of the satellites by some factor (typically ∼100; Pollack et al., 1994), corresponding to the mass ratio of gas to rock-metal/ice in the solar nebula. This factor might be decreased somewhat in view of the heavy-element enrichment of the giant planets or increased in view of the possible loss of some of the accreting materials as a result of the specifics of the process used to make the planet and satellites.
In order to arrive at a specific model for the formation of regular satellites in a gaseous medium we need to characterize the subnebular viscosity. It has been suggested that because of the stabilizing influence of a positive radial gradient in specific angular momentum, turbulence in a Keplerian disk is not self-sustaining unless a source of “stirring” is found Ryu and Goodman 1992, Balbus et al 1996, Stone and Balbus 1996. As a result, one needs to identify a specific mechanism that can maintain turbulence in the dense, high orbital frequency subnebula. One such suggestion is that convection drives turbulence Cameron 1978a, Lin and Papaloizou 1980, Ruden and Lin 1986; however, eventually particle growth may stop convection by diminishing the Rosseland mean opacity and weakening its temperature dependence (Weidenschilling and Cuzzi 1993). Given the fast dynamical time scale and the high particle density of the subnebula disk, coagulation and settling for sticky particles may take place on a time scale faster than disk evolution. Furthermore, if convection drives turbulence then angular momentum transport may be weak (α ∼ 10−5; Stone and Balbus, 1996) and directed inwards Ryu and Goodman 1992, Kley et al 1993, Stone and Balbus 1996, Cabot 1996, which would essentially terminate gas accretion onto the primary. Another possibility is that turbulence is driven by a magnetohydrodynamic (MHD) instability (Balbus and Hawley, 1991). But this is also unlikely to apply (Gammie, 1996) in the dense and relatively cool subnebula disk. Alternatively, there are a variety of ways that accretion itself, or the gravitational energy released by it, can provide the source of free energy that can drive turbulence. It has been pointed out (but not quantitatively explored) that a turbulent shear layer, where the angular momentum of the infalling gas is adjusted to the angular momentum of the Keplerian disk flow, exists below an accretion shock and may provide a localized viscosity Cassen and Moosman 1981, Cassen and Summers 1983. More recently it has been shown that a bump in the temperature profile of the disk, as may result from accretion, that leads to a strong radial entropy gradient generates Rossby waves and localized turbulence Lovelace et al 1999, Li et al 2000. Similarly, but more generally, Klahr and Bodenheimer (2001) study a global baroclinic instability as a source of turbulence and outward angular momentum transport in Keplerian accretion disks characterized by a negative radial entropy gradient.
To create a coherent scenario of satellite formation, the source of the solids that go into the satellite systems must be considered. The concentration of rock/ice to gas in the subnebula may depend on the ability of the protoplanet to disturb the orbits of planetesimals situated within a few AU of its orbit into ones that crossed its orbit. One would expect that in a time scale much shorter than the lifetime of the solar system virtually all the planetesimals located in the outer solar system would have their orbits perturbed into giant planet crossing orbits (Gladman and Duncan, 1990). What happens to such a planetesimal depends on the size of the planet at the time of crossing. If the giant planet’s envelope filled a fair fraction of its Hill radius, as it probably did during most or all of its gas accretion phase (unless significant amounts of gas accreted through the gap after gap-opening), then the distended atmosphere would have greatly increased the planet’s cross-section Bodenheimer and Pollack 1986, Pollack et al 1996. Early arriving (before the completion of planetary accretion) icy planetesimals of size <10 km (Zahnle, private communication) may break up in the contracting envelope of the giant planet, and their condensable content may then be entrapped in the gas and left behind in the form of a circumplanetary disk. On the other hand, most late arriving planetesimals may have been scattered to further regions of the solar system with some sent to the Oort cloud and some lost altogether. Thus, our model relies on early arriving planetesimals that break up or dissolve in the extended giant-planet envelopes to provide the bulk of the material that will eventually make the satellite systems, delivered to the satellite disk in a time scale given by the envelope collapse time.
This formation model is consistent both with a model that captures irregular satellites at a time when the proto-planetary envelope was collapsing rapidly and extended several hundred planetary radii (Pollack et al., 1979) and with a model that captures irregular satellites using a long-lived circumplanetary gas disk (Cuk and Burns, 2002). Here late arriving interplanetary debris plays a role in that it can threaten the survival of regular satellites close to their primary. Hence, the large disparity in masses between Titan and all other moons of Saturn may in part be the result of the break-up of satellites by high-velocity impacts (e.g., Lissauer, 1995; but note that gas would still be needed to clear up the collisional debris and prevent re-accretion). In contrast, a starved disk model (Stevenson, 2001) relies on the late arriving planetesimals or flow through the gap to form a disk around the planet out of which all the regular satellites will eventually accrete. One should keep in mind, however, that most planetesimals were probably scattered or the giant planets would have ended up with too much high-Z mass (Podolak et al., 1993) and that most of the mass in the nebula disk at late times is in the form of planetesimals Mizuno et al 1978, Weidenschilling 1997. Furthermore, the high specific angular momentum of gas arriving at late times may place it in orbit well outside the region where most of the satellite mass is found.
In Section 2 we organize the satellite systems of the giant planets according to the Hill radius of the primary. In Section 3 we characterize the subnebulae of giant planets, especially that of Jupiter. In Section 4 we discuss the accretion of the Galilean satellites, reserving discussion of Callisto for Section 5. In Section 6 we turn to Saturn’s satellite system. In Section 7 we discuss the satellite system of Uranus. In Section 8 we make some comments on an alternative satellite accretion model that leads to a long accretion time scale for every satellite. In Section 9 we present our conclusions and discussion. In Paper II we turn to the migration and survival of full-sized satellites.
Section snippets
Regular satellites of giant planets
We begin with a brief comparative discussion of the satellite systems of the giant planets. We compare satellite positions mainly in terms of the Hill radius RH = a(MP/3M⊙)1/3 of the planet (and the concomitant centrifugal radius rc ≈ RH/48; see Sect. 3). In Fig. 1, we plot the locations of the regular satellites (solid circles) and the innermost irregular satellites (open circles) in units of the Hill radius of the giant planet. The bold dashed line describes the position of the centrifugal
The giant planet subnebula
The “minimum” mass subnebula we use here is one of solar nebula composition that provides just enough mass to form the observed satellite systems with the observed rock/ice mass ratio. Given Jupiter’s relative enrichment in heavy elements with respect to the solar nebula, the minimum mass subnebula is not a firm lower bound. On the other hand, inefficiencies in the satellite formation process and depletion of solids due to planetesimal formation mean that it is not a firm upper bound either.
Galilean satellite accretion and evolution
In analogy to gas-free planetary accretion, we begin the problem of satellite accretion by calculating characteristic sizes of satellitesimals and satellite embryos for our disk parameters assuming a satellitesimal density of ρs = 1.5 g cm−3. Though our problem differs markedly from one in which the satellites are accreted in the absence of gas, we will show later that the characteristic sizes one obtains in the presence of gas are roughly consistent with the ones we give below, which are meant
Slow formation of callisto
Our model has Callisto forming from an extended, low optical depth gas disk. We expect this gas disk to be largely quiescent with very low gas viscosity. This means that the dust and rubble layer will quickly settle down to the midplane within a scale-height much smaller than the gas scale-height. The size of the dust and rubble layer is determined by shear turbulence close to the midplane (Cuzzi et al., 1993).
First we calculate characteristic masses and lengths in analogy to the gas free
Saturn’s regular satellite system
In order to apply our model to Saturn we first need to constrain the nebula parameters for Saturn as we did for Jupiter. First, we note that the ratio of the reconstituted Galilean satellite masses to the saturnian satellite masses is ≈3.7. On the other hand, the ratio of the atmospheric envelopes of Jupiter to Saturn is ≈3.7 for giant planet core masses of ≈12 Earth masses, consistent with nominal values.
In Fig. 2, we plot two models. The first model simply assumes the same mass ratio (∼3.7)
Uranian satellite system
As we did in the case of Saturn, we begin our discussion of the satellite system of Uranus by comparing the ratio of masses of the atmospheric envelopes to the ratio of masses of the satellite systems. Assuming Saturn to have a ∼15 Earth mass core and Uranus a smaller ∼10 Earth mass core, the mass ratio of the envelopes for the two planets is ∼18. While substantially uncertain, this value compares favorably with the mass ratio of the satellite systems of the two planets ∼15.
All the regular
Starved disk model
A scenario in which the giant planet satellites accrete from a disk produced by the direct infall of gas and solids from heliocentric orbit, leading to long satellite assembly times ∼106 years despite the short disk accumulation times ∼102 years (Stevenson, personal communication), has numerous issues to overcome. Here we mention some of the outstanding ones and leave development of such a model (if viable) for later work.
First is the issue of satellite survival. Recent numerical simulations
Discussion and conclusions
We have used a consistent model for the accretion of regular satellites of Jupiter, Saturn, and Uranus. We do exclude coorbitals, small regular satellites found close to the giant planet, and the satellites of Neptune from consideration. The coorbitals will require a separate treatment, which we do not attempt here. Small satellites close to the planet are likely to have undergone significant collisional or tidal evolution after their formation, so they may not provide useful constraints on
Acknowledgements
We thank Jeffrey Cuzzi, Kevin Zahnle, Doug Lin, Peter Bodenheimer, Sandy Davis, Dale Cruikshank, and Jeff Moore for discussions. One of us (I.M.) had numerous, very helpful discussions with Dave Stevenson. We also thank Jeffrey Cuzzi, Kevin Zahnle, and Jack Lissauer for reading the manuscript and suggesting improvements, and Steve Squyres for discussions, comments on the manuscript, and generous support of this work. This research was supported by a grant from the Planetary Geology and
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