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Abstract 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 J 12 . Here we report the expansion of the theory of figures, which is a fast method for gravity field and shape computation, to the seventh order (ToF7), which allows for computation of up to J 14 . 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 equation of state. For Jupiter, we find that J 6 is best matched by a transition from an He-depleted to He-enriched envelope at 2–2.5 Mbar. However, the atmospheric metallicity reaches 1 × 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 1.5–4 × solar atop a small core. Perturbing the adiabat yields metallicity profiles with extended, heavy-element-enriched deep interior (diffuse core) out to 0.4 R Sat , 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 J n values. 

We present the detection of three exoplanets orbiting the early M dwarf TOI-663 (TIC 54962195;V= 13.7 mag,J= 10.4 mag,R_★= 0.512 ± 0.015R_⊙,M_★= 0.514 ± 0.012M_⊙,d= 64 pc). TOI-663 b, c, and d, with respective radii of 2.27 ± 0.10R_⊕, 2.26 ± 0.10R_⊕, and 1.92 ± 0.13R_⊕and masses of 4.45 ± 0.65M_⊕, 3.65 ± 0.97M_⊕, and <5.2M_⊕at 99%, are located just above the radius valley that separates rocky and volatile-rich exoplanets. The planet candidates are identified in two TESS sectors and are validated with ground-based photometric follow-up, precise radial-velocity measurements, and high-resolution imaging. We used the software package juliet to jointly model the photometric and radial-velocity datasets, with Gaussian processes applied to correct for systematics. The three planets discovered in the TOI-663 system are low-mass mini-Neptunes with radii significantly larger than those of rocky analogs, implying that volatiles, such as water, must predominate. In addition to this internal structure analysis, we also performed a dynamical analysis that confirmed the stability of the system. The three exoplanets in the TOI-663 system, similarly to other sub-Neptunes orbiting M dwarfs, have been found to have lower densities than planets of similar sizes orbiting stars of different spectral types.