Determination of the Minimum Masses of Heavy Elements in the Envelopes of Jupiter and Saturn

We calculate the minimum mass of heavy elements required in the envelopes of Jupiter and Saturn to match the observed oversolar abundances of volatiles. Because the clathration efficiency remains unknown in the solar nebula, we have considered a set of sequences of ice formation in which the fraction of water available for clathration is varied between 0 and 100%. In all the cases considered, we assume that the water abundance remains homogeneous whatever the heliocentric distance in the nebula and directly derives from a gas phase of solar composition. Planetesimals then form in the feeding zones of Jupiter and Saturn from the agglomeration of clathrates and pure condensates in proportions fixed by the clathration efficiency. A fraction of Kr and Xe may have been sequestrated by the H+ 3 ion in the form of stable XeH+ 3 and KrH+ 3 complexes in the solar nebula gas phase, thus implying the formation of at least partly Xe- and Kr-impoverished planetesimals in the feeding zones of Jupiter and Saturn. These planetesimals were subsequently accreted and vaporized into the hydrogen envelopes of Jupiter and Saturn, thus engendering volatiles enrichments in their atmospheres, with respect to hydrogen. Taking into account both refractory and volatile components, and assuming plausible molecular mixing ratios in the gas phase of the outer solar nebula, we show that it is possible to match the observed enrichments in Jupiter and Saturn, whatever the clathration efficiency. Our calculations predict that the O/H enrichment decreases from ~ 5.5 to 5.1 times (O/H) in the envelope of Jupiter and from 15.2 to 14.1 times (O/H) in the envelope of Saturn with the growing clathration efficiency in the solar nebula. As a result, the minimum mass of ices needed to be injected in the envelope of Jupiter decreases from ~ 20.0 to 18.6 M {circled plus}, including a mass of water diminishing from 10.4 to 9.3 M {circled plus}. In the same conditions, the minimum mass of ices needed in the envelope of Saturn decreases from ~ 16.7 to 15.6 M {circled plus}, including a mass of water diminishing from 8.6 to 7.7 M {circled plus}. The accretion of planetesimals with ices to rocks ratios ~ 1 in the envelope of Jupiter, namely a value derived from the bulk densities of Ganymede and Callisto, remains compatible with the mass of heavy elements predicted by interior models. On the other hand, the accretion of planetesimals with similar ice-to-rock in the envelope of Saturn implies a mass of heav