A calculation of Saturn's gravitational contraction history

Abstract

A calculation has been made of the gravitational contraction of a homogeneous, quasi-equilibrium Saturn model of solar composition. The calculations begin at a time when the planet's radius is ten times larger than its present size, and the subsequent gravitational contraction is followed for 4.5 × 10 9 years. For the first million years of evolution, the Saturn model contracts rapidly like a pre-main sequence star and has a much higher luminosity and effective temperature than at present. Later stages of contraction occur more slowly and are analogous to the cooling phase of a degenerate white dwarf star. Examination of the interior structure of the models indicates the presence of a metallic hydrogen region near the center of the planet. Differences in the size of this region for Jupiter and Saturn may, in part, be responsible for Saturn having a weaker magnetic field. While the interior temperatures are much too high for the fluids in the molecular and metallic regions to become solids by the current epoch, the temperature in the outer portion of the metallic zone falls below Stevenson's [ Phys. Rev. J. (1975)] phase separation curve for helium after 1.2 billion years of evolution. This would lead to a sinking of helium from the outer to the inner portion of the metallic region, as described by Salpeter [ Astrophys. J. 181, L83–L86 (1973)]. At the current epoch, the radius of the model is about 9% larger, while its excess luminosity is comparable to the observed value of Rieke [ Icarus 26, 37–44 (1975)], as refined by Wright [Harvard College Obs. Preprint No. 480 (1976)]. This behavior of the Saturn model may be compared to the good agreement with both Jupiter's observed radius and excess luminosity shown by an analogous model of Jupiter [Graboske et al., Astrophys. J. 199, 255–264 (1975)]. The discrepancy in radius of our Saturn model may be due to errors in the equations of state and/or our neglect of a rocky core. However, arguments are presented which indicate that helium separation may cause an expansion of the model and thus lead to an even bigger discrepancy in radius. Improvement in the radius may also foster a somewhat larger predicted luminosity. At least part and perhaps most of Saturn's excess luminosity is due to the loss of internal thermal energy that was built up during the early rapid contraction, with a minor contribution coming from Saturn's present rate of contraction. These two sources dominate Jupiter's excess luminosity. If helium separation makes an important contribution to Saturn's excess luminosity, then planetwide segregation is required. Finally, because Saturn's early high luminosity was about an order of magnitude smaller than Jupiter's, water-ice satellites may have been able to form closer to Saturn to Jupiter.