The effect of the equatorially symmetric zonal winds of Saturn on its gravitational field

The penetration depth of Saturn’s cloud-level winds into its interior is unknown.A possible way of estimating the depth is through measurement of the effect of the winds on the planet’s gravitational field.We use a self-consistent perturbation approach to study how the equatorially symmetric zonal winds of Saturn contribute to its gravitational field.An important advantage of this approach is that the variation of its gravitational field solely caused by the winds can be isolated and identified because the leading-order problem accounts exactly for rotational distortion,thereby determining the irregular shape and internal structure of the hydrostatic Saturn.We assume that（i）the zonal winds are maintained by thermal convection in the form of non-axisymmetric columnar rolls and（ii）the internal structure of the winds,because of the Taylor-Proundman theorem,can be uniquely determined by the observed cloud-level winds.We calculate both the variation △J_n,n=2,4,6...of the axisymmetric gravitational coefficients J_n caused by the zonal winds and the non-axisymmetric gravitational coefficients △J_（nm） produced by the columnar rolls,where m is the azimuthal wavenumber of the rolls.We consider three different cases characterized by the penetration depth 0.36 R_S,0.2 R_S and 0.1 R_S,where R_S is the equatorial radius of Saturn at the 1-bar pressure level.We find that the high-degree gravitational coefficient （ J_（12）＋△J_（12）） is dominated,in all the three cases,by the effect of the zonal flow with ｜△J_（12）/J_（12）｜〉100%and that the size of the non-axisymmetric coefficientsdirectly reflects the depth and scale of the flow taking place in the Saturnian interior.

A model of Saturn inferred from its measured gravitational field

We present an interior model of Saturn with an ice-rock core,a metallic region,an outer molecular envelope and a thin transition layer between the metallic and molecular regions.The shape of Saturn’s 1 bar surface is irregular and determined fully self-consistently by the required equilibrium condition.While the ice-rock core is assumed to have a uniform density,three different equations of state are adopted for the metallic,molecular and transition regions.The Saturnian model is constrained by its known mass,its known equatorial and polar radii,and its known zonal gravitational coefficients,J_（2n）,n=1,2,3.The model produces an ice-rock core with equatorial radius 0.203 R_S,where R_S is the equatorial radius of Saturn at the 1-bar pressure surface;the core densityρ_c=10388.1 kgm^（3）corresponding to 13.06 Earth masses;and an analytical expression describing the Saturnian irregular shape of the 1-bar pressure level.The model also predicts the values of the higher-order gravitational coefficients,J_8,J_10 and J_12,for the hydrostatic Saturn and suggests that Saturn’s convective dynamo operates in the metallic region approximately defined by 0.2 R_S

Saturn＇s gravitational field induced by its equatorially antisymmetric zonal winds

The cloud-level zonal winds of Saturn are marked by a substantial equatorially antisymmetric component with a speed of about 50 m s^-1 which, if they are sufficiently deep, can produce measurable odd zonal gravitational coefficients △J2 k＋1, k = 1, 2, 3, 4. This study, based on solutions of the thermal-gravitational wind equation, provides a theoretical basis for interpreting the odd gravitational coefficients of Saturn in terms of its equatorially antisymmetric zonal flow. We adopt a Saturnian model comprising an ice-rock core, a metallic dynamo region and an outer molecular envelope. We use an equatorially antisymmetric zonal flow that is parameterized, confined in the molecular envelope and satisfies the solvability condition required for the thermal-gravitational wind equation. The structure and amplitude of the zonal flow at the cloud level are chosen to be consistent with observations of Saturn.We calculate the odd zonal gravitational coefficients △J2k＋1, k = 1, 2, 3, 4 by regarding the depth of the equatorially antisymmetric winds as a parameter. It is found that △J3 is-4.197 × 10^-8 if the zonal winds extend about 13 000 km downward from the cloud tops while it is-0.765 × 10^-8 if the depth is about 4000 km. The depth/profile of the equatorially antisymmetric zonal winds can eventually be estimated when the high-precision measurements of the Cassini Grand Finale become available.