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The higher order gravitational moments of a differentially rotating planetare greatly affected or even dominated by the near surface differential flow.Unlike the contribution from rigid body rotation, this part of the gravityfield can be well estimated by linear theory when the differential rotation ison cylinders and the corresponding gravity field arises from the higher ordermoments directly introduced by that flow. In the context of an n = 1 polytrope,we derive approximate analytical formulas for the gravity moments. We find that$\Delta J_{2\mathrm{n}}$ is typically at most a few times $(-1)^{n+1} \; a \; q\; d^{5/2}$, where $a$ is the amplitude of the differential rotation (as afraction of the background rigid body rotation), $q = \Omega^2 R^3 / G M$ isthe usual dimensionless measure of rotation for the planet (mass $M$, radius$R$), and $d << 1$ is the characteristic depth of the flow as a fraction of theplanetary radius. Applied to Saturn, with $a$ set by the observed surface windamplitude, we find first that the observed signs of the $\DeltaJ_{2\mathrm{n}}$ are a trivial consequence of the definition of thecorresponding Legendre polynomials, but the {\it Cassini} observations can notbe explained by a simple exponentially decaying flow and instead require asubstantial retrograde flow at depth and a larger $d$ than the simple scalingsuggests. This is consistent with the results reported by \citet{Iess2019}.However, there is no fluid dynamical requirement that the flows observed in theatmosphere are a guide to the flows thousands of km deeper. We explore a widerange of flow depths and amplitudes which yield values for $\DeltaJ_{\mathrm{n}}$ that are acceptable within the error estimates and thushighlight the inherent non-uniqueness of inferences made from the higher ordergravity moments.

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A Linear Approximation for the Effect of Cylindrical Differential  Rotation on Gravitational Moments: Application to the Non-Unique  Interpretation of Saturn's Gravity