The Thermal Evolution and Internal Structure of Saturn's Mid-Sized Icy Satellites

Abstract

The Cassini-Huygens mission is returning new geophysical data for the midsize, icy satellites of Saturn (i.e., satellites with radii between 100 and 1,000 km). These data have enabled a new generation of geophysical model studies for Phoebe, Iapetus, Rhea, Mimas, Tethys, Dione, as well as Enceladus (which is addressed in a separate chapter in this book). In the present chapter we consider the new model studies that have reported significant results elucidating the evolutionary histories and internal structures of these satellites. Those results have included their age, the development of their internal structures and mineralogies, which for greatest fidelity must be done concomitantly with coupled dynamical evolutions. Surface areas, volumes, bulk densities, spin rates, orbit inclinations, eccentricities, and distance from Saturn have changed as the satellites have aged. Heat is required to power the satellites' evolution, but is not overly abundant for the midsized satellites. All sources of heat must be evaluated and taken into account. This includes their intensities and when they occur and are available to facilitate evolution, both internal and dynamical. The mechanisms of heat transport must also be included. However, to model these to high fidelity the material properties of the satellite interiors must be accurately known. This is not the case. Thus, much of the chapter is devoted to discussion of what is known about these properties and how the uncertainties affect the estimation of heat sources, transport processes, and the consequential changes in composition and evolution. Phoebe has an oblate shape that may be in equilibrium with its spin period of ~9.3 h. Its orbital properties suggest that it is not one of the regular satellites, but is a captured body. Its density is higher than that of the other satellites, consistent with formation in the solar nebula rather than from material around Saturn. Oblate shape and high density are unusual for objects in this size range, and may indicate that Phoebe was heated by ²⁶Al decay soon after its formation, which is consistent with some models of the origin of Kuiper-Belt objects. Iapetus has the shape of a hydrostatic body with a rotation period of 16 h. It subsequently despun to its current synchronous rotation state, ~79 day period. These observations are sufficient to constrain the required heating in Iapetus' early history, suggesting that it formed several My after CAI condensation. Since Saturn had to be present for Iapetus to form, this date also constrains the age of Saturn and how long it took to form. Both shape and gravitational data are available for Rhea. Gravity data were obtained from the single Cassini flyby during the prime mission and within the uncertainties cannot distinguish between hydrostatic and non-hydrostatic gravitational fields. Both Dione and Tethys display evidence of smooth terrains, with Dione's appearing considerably younger. Both are conceivably linked to tidal heating in the past, but the low rock abundance within Tethys and the lack of eccentricity excitation of Tethys' orbit today make explaining this satellite's geology challenging.

18.1 Appendix: Glossary of Symbols

• A = constant (Eq. 18.9)

• C = polar moment of inertia of the satellite (Eq. 18.4)

• C ₀ = initial concentration of radiogenic elements. (Eq. 18.2)

• C _p = temperature-dependent specific heat (Eqs. 18.3, 18.6, 18.8)

• C₂₂ = second degree gravitational harmonic (i.e., ellipticity of the equator) (Eq. 18.1)

• D = the semi-major axis of the orbit (Eq. 18.4)

• e = eccentricity (Eq. 18.5)

• dE/dt = average tidal heat produced during one orbit (Eq. 18.5)

• g = gravity (Eq. 18.12)

• G = universal gravitational constant (Eqs. 18.1, 18.4 through 18.7)

• h _a = fraction of mechanical energy retained as heat (Eq. 18.6)

• H _R = volumetric radiogenic heating rate (Eq. 18.2)

• Hm = internal heating rate (radiogenic, tidal dissipation) (Eq. 18.8)

• M _o,i = initial power produced by radiogenic decay per unit mass of element i (Eq. 18.2)

• J ₂ = second degree gravitational harmonic. = −(C ₂₀) (i.e., oblateness) (Eq. 18.1)

• k = thermal conductivity (Eqs. 18.8 and 18.13)

• k ₂ = the periodic, potential, tidal Love number (Eqs. 18.4 and 18.5)

• M = satellite mass (Eq. 18.1)

• M _p = Saturn's mass (Eq. 18.4)

• n = number of radiogenic elements included in the sum (Eq. 18.2)

• n = mean orbital motion (Eq. 18.5)

• q _conv = convective heat flux (Eq. 18.13)

• Q = the dissipation factor (Eqs. 18.4, 18.5, 18.10, 18.11)

• Q_act = the activation energy (typically 60kJ/mol for ice) (Eq. 18.9)

• r = distance from the center of the satellite (Eqs. 18.1, 18.6, and 18.8)

• R = satellite radius (Eqs. 18.1, 18.3, and 18.7)

• R _eq = equatorial radius of the satellite (Eqs. 18.4 and 18.5)

• R_gas = the perfect gas constant (Eqs. 18.9 and 18.11)

• R _a = Rayleigh number (Eq. 18.13)

• Ra _TBL = thermal boundary layer Rayleigh number (Eqs. 18.12 and 18.13)

• t = time (Eq. 18.8)

• t _0-CAIs = time since CAIs formation (Eq. 18.2)

• T = temperature (Eq. 18.8)

• T _i = temperature of the planetesimals (Eq. 18.6)

• ΔT = increase in the internal temperature (Eq. 18.3)

• ΔT _η = viscous temperature scale is then defined by (Eq. 18.10)

• ΔTt _TBL = temperature variation across thermal boundary layer (Eqs. 18.11 and 18.13)

• T _m = temperature of convective interior (Eq. 18.11)

• T _c = temperature at the base of the conductive lid (Eq. 18.11)

• T _m = temperature of the convective interior (Eq. 18.10)

• T(r) = temperature profile resulting from accretion (Eq. 18.6)

• V _eq = second degree equatorial gravitational potential (Eq. 18.1)

• x _s = mass fraction of silicates (Eq. 18.2)

• α = thermal expansion coefficient (Eq. 18.12)

• δ = thickness of thermal boundary layer (Eqs. 18.12 and 18.13)

• γ = moment of inertia (Eq. 18.3)

• η = viscosity (Eqs. 18.9 and 18.12)

• κ = thermal diffusivity (Eq. 18.12)

• λ = longitude (Eq. 18.1)

• λ _i decay constant of radiogenic element i (Eq. 18.2)

• <V> = mean satellitesimal encounter velocity

• Ω = the initial angular rate (Eq. 18.3)

• ρ = the density (Eqs. 18.2,18.6, 18.8, 18.12)

• ρ¯ = the satellite's mean density (Eq. 18.7)

• Ψ = porosity (Eq. 18.7)