Theory of Figures to the 7th order and the interiors of Jupiter and Saturn

Interior modeling of Jupiter and Saturn has advanced to a state where thousands of models are generated that cover the uncertainty space of many parameters. This approach demands a fast method of computing their gravity field and shape. Moreover, the Cassini mission at Saturn and the ongoing Juno mission delivered gravitational harmonics up to J12. Here, we report the expansion of the Theory of Figures, which is a fast method for gravity field and shape computation, to the 7th-order (ToF7), which allows for computation of up to J14. We apply three different codes to compare the accuracy using polytropic models. We apply ToF7 to Jupiter and Saturn interior models in conjunction with CMS-19 H/He-EOS. For Jupiter, we find that J6 is best matched by a transition from He-depleted to He-enriched envelope at 2-2.5 Mbar. However, the atmospheric metallicity reaches 1xtimes solar only if the adiabat is perturbed toward lower densities, or if the surface temperature is enhanced by ~14 K from the Galileo value. Our Saturn models imply a largely homogeneous-in-Z envelope at 1.5-4x solar atop a small core. Perturbing the adiabat yields metallicity profiles with extended, heavy-element enriched deep interior (diffuse core) out to 0.4 RSat, as for Jupiter. Classical models with compact, dilute, or no core are possible as long as the deep interior is enriched in heavy-elements. Including a thermal wind fitted to the observed wind speeds, representative Jupiter and Saturn models are consistent with all observed Jn values.