Allen S. Grossman

A calculation of Saturn's gravitational contraction history

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 planets radius is ten times larger than its present size, and the subsequent gravitational contraction is followed for 4.5 × 109 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 Stevensons [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 Jupiters 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 Saturns 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 Saturns present rate of contraction. These two sources dominate Jupiters excess luminosity. If helium separation makes an important contribution to Saturns excess luminosity, then planetwide segregation is required. Finally, because Saturns early high luminosity was about an order of magnitude smaller than Jupiters, water-ice satellites may have been able to form closer to Saturn to Jupiter.

Calculations of the evolution of the giant planets

Abstract Evolutionary calculations are presented for spherically symmetric protoplanetary configurations with a homogeneous solar composition and with masses of 10−3, 1.5 × 10−3, 2.85 × 10−4, and 4.2 × 10−4 M⊙. Recent improvements in equation-of-state and opacity calculations are incorporated. Sequences start as subcondensations in the solar nebula with densities of ∼10−10 to 10−11 g cm−3, evolve through a hydrostatic phase lasting 105 to 107 years, undergo dynamic collapse due to dissociation of molecular hydrogen, and regain hydrostatic equilibrium with densities ∼1 g cm−3. The nature of the objects at the onset of the final phase of cooling and contraction is discussed and compared with previous calculations.

The effect of dense cores on the structure and evolution of Jupiter and Saturn

Abstract We have calculated evolutionary and static models of Jupiter and Saturn with homogeneous solar composition mantles and dense cores of material consisting of solar abundances of SiO 2 , MgO, Fe, and Ni. Evolutionary sequences for Jupiter were calculated with cores of mass 2, 4, 6, and 8% of the Jovian mass. Evolutionary sequences for Saturn were calculated with cores of mass 16, 18, 20, and 22% of total mass. Two envelope mixtures, representative of the solar abundances were used: X (mass fraction of hydrogen) = 0.74, Y (mass fraction of helium) = 0.24 and X = 0.77 and Y = 0.21. For Jupiter, the observations of the temperature at 1 bar pressure ( T 1bar ), radius and internal luminosity were best fit by evolutionary models with a core mass of ∼6.5% and chemical composition of X = 0.77, Y = 0.21. The calculated cooling time for Jupiter is approximately 4.9 × 10 9 years, which is consistent, within our error bars, with the known age of the solar system. For Saturn, the observations of the radius, internal luminosity and T 1BAR can be best fit by evolutionary models with a core mass of ∼21% and chemical composition of X = 0.77, Y = 0.21. The cooling time calculated for Saturn is approximately 2.6 × 10 9 years, almost a factor 2 less than the present age of the solar system. Static models of Jupiter and Saturn were calculated for the above chemical compositions in order to investigate the sensitivity of the calculated gravitational moments, J 2 and J 4 , to the mass of the dense core, T 1BAR and hydrogen/helium ratio. We find for Jupiter that a model having a core mass of approximately 7% gives values of J 2 , J 4 , and T 1BAR that are within observational limits, for the mixture X = 0.77, Y = 0.21. The static Jupiter models are completely consistent with the evolutionary results. For Saturn, the quantities J 2 , J 4 , and J 6 determined from the static models with the most probable T 1BAR of 140°K, using modeling procedures which result in consistent models for Jupiter, are considerably below the observed values.

An evolutionary calculation of Jupiter

Abstract A preliminary evolutionary calculation has been made for a stellar object of mass 0.001 M ⊙ composed of pure hydrogen. The star undergoes the gravitational contraction from an initial radius of 35 times the present radius of Jupiter ( R Jup ). We assume the interior to be in convective equilibrium throughout the evolution. The evolution has been followed for 10 9 y at which time R = 2.8 R Jup and the central temperature is 18000 K. The log of the luminosity (in units of solar luminosity) and effective temperature due to the internal energy sources are log L / L ⊙ = −9.0 and log T e = 1.77.