Anderson and Schubert [2007. Saturn's Gravitational field, internal rotation, and interior structure. Science 317, 1384-1387 (paper I)] proposed that Saturn's rotation period can be ascertained by minimizing the dynamic heights of the 100 mbar isosurface with respect to the geoid; they derived a rotation period of 10 h 32 m 35 s. We investigate the same approach for Jupiter to see if the Jovian rotation period is predicted by minimizing the dynamical heights of its isobaric (1 bar pressure level) surface using zonal wind data. A rotation period of 9 h 54 m 29.7 s is found. Further, we investigate the minimization method by fitting Pioneer and Voyager occultation radii for both Jupiter and Saturn. Rotation periods of 9 h 55 m 30 s and 10 h 32 m 35 s are found to minimize the dynamical heights for Jupiter and Saturn, respectively. Though there is no dynamical principle requiring the minimization of the dynamical heights of an isobaric surface, the successful application of the method to Jupiter lends support to its relevance for Saturn. We derive Jupiter and Saturn rotation periods using equilibrium theory to explain the difference between equatorial and polar radii. Rotation periods of 9 h 55 m 20 s and 10 h 31 m 49 s are found for Jupiter and Saturn, respectively. We show that both Jupiter's and Saturn's shapes can be derived using solid-body rotation, suggesting that zonal winds have a minor effect on the planetary shape for both planets. The agreement in the values of Saturn's rotation period predicted by the different approaches supports the conclusion that the planet's period of rotation is about 10 h 32 m.
- Rotation period