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Saturn’s fast spin determined from its gravitational field and oblateness



The alignment of Saturn’s magnetic pole with its rotation axis precludes the use of magnetic field measurements to determine its rotation period1. The period was previously determined from radio measurements by the Voyager spacecraft to be 10 h 39 min 22.4 s (ref. 2). When the Cassini spacecraft measured a period of 10 h 47 min 6 s, which was additionally found to change between sequential measurements3,4,5, it became clear that the radio period could not be used to determine the bulk planetary rotation period. Estimates based upon Saturn’s measured wind fields have increased the uncertainty even more, giving numbers smaller than the Voyager rotation period, and at present Saturn’s rotation period is thought to be between 10 h 32 min and 10 h 47 min, which is unsatisfactory for such a fundamental property. Here we report a period of 10 h 32 min 45 s ± 46 s, based upon an optimization approach using Saturn’s measured gravitational field and limits on the observed shape and possible internal density profiles. Moreover, even when solely using the constraints from its gravitational field, the rotation period can be inferred with a precision of several minutes. To validate our method, we applied the same procedure to Jupiter and correctly recovered its well-known rotation period.

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Figure 1: An example of the statistical distribution of solutions for Saturn’s rotation period.
Figure 2: Solutions for Saturn’s rotation period.
Figure 3: Solutions for Jupiter’s rotation period.
Figure 4: Solutions for the rotation periods of Saturn and Jupiter when assuming improved gravity data.

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We thank G. Schubert, M. Podolak, J. Anderson, L. Bary-Soroker, and the Juno science team for discussions and suggestions. We acknowledge support from the Israel Space Agency under grants 3-11485 (R.H.) and 3-11481 (Y.K.).

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Authors and Affiliations



R.H. led the research. R.H. and Y.K. initiated the research and wrote the paper. E.G. designed the optimization approach and executed all the calculations. R.H. computed interior models and defined the parameter space for the figure functions. All authors contributed to the analysis and interpretation of the results.

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Correspondence to Ravit Helled.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 The flattening parameters calculated by the model without constraining the figure functions.

Shown are f, k, h for Saturn with ΔR = 50 km.

Extended Data Figure 2 An example of our statistical optimization model for deriving the rotation period.

The results are shown for a case for which the range of allowed rotation period is between 10 h 24 min and 10 h 54 min. The solution is based on a combination of 2,000 individual sub-cases, each of them representing a case with specific random initial conditions within the defined parameter space. ac, A scatter plot (similar to Fig. 1a) of the distribution of solutions on the plane of the calculated rotation period Pcalc minus Pvoy versus each of ΔJ2, ΔJ4 and ΔJ6. Each blue dot represents one sub-case converged solution. In a the inner and outer black circles show the first and second standard deviations, respectively. dj, The distribution of solutions for the figure functions K2, K3, H3, F1, F2 and F3, respectively.

Extended Data Figure 3 Saturn’s calculated rotation period versus the uncertainty in the assumed rotation period and radius.

Pcalc shown (colour scale) as a function of ΔP (h) and ΔR (km). The dashed line presents the transition from the regime where the constraint on the rotation period (above the dashed line) to the regime where the constraint on the shape is more dominant.

Extended Data Figure 4 Radial density profiles for two different interior models for Saturn (top) and Jupiter (bottom).

The black curves correspond to models with very large cores and the blue curves are no-core models in which the density profile is represented by 6th-order polynomials. For the massive-core case we assume a constant core density of 1.5 × 104 kg m−3, reaching 20% of the planet’s radius. The density profiles are constrained to match the planetary mass, J2, J4, J6, mean radius, and the atmospheric density and its derivative at 1 bar (see details in refs 8, 13 and 18). We then use the difference in the values of the figure functions in the two limiting cases to limit their values.

Extended Data Figure 5 The calculated flattening parameters when the figure functions are limited by interior models.

ac, A scatter plot (similar to Fig. 1a) of the distribution of solutions on the plane of the calculated rotation period Pcalc minus Pvoy versus ΔJ2, ΔJ4 and ΔJ6, respectively. Each blue dot represents one sub-case converged solution. The calculated mean radius was set to be within 20 km of Saturn's observed mean radius. In a the inner and outer black circles show the first and second standard deviations, respectively. dj, The distribution of solutions for the figure functions K2, K3, H3, F1, F2 and F3, respectively.

Extended Data Table 1 The physical properties of Saturn and Jupiter used in the analysis
Extended Data Table 2 The calculated figure functions based on interior models of Saturn and Jupiter

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Helled, R., Galanti, E. & Kaspi, Y. Saturn’s fast spin determined from its gravitational field and oblateness. Nature 520, 202–204 (2015).

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