  The higher order gravitational moments of a differentially rotating planet
are greatly affected or even dominated by the near surface differential flow.
Unlike the contribution from rigid body rotation, this part of the gravity
field can be well estimated by linear theory when the differential rotation is
on cylinders and the corresponding gravity field arises from the higher order
moments 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 a
fraction of the background rigid body rotation), $q = \Omega^2 R^3 / G M$ is
the 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 the
planetary radius. Applied to Saturn, with $a$ set by the observed surface wind
amplitude, we find first that the observed signs of the $\Delta
J_{2\mathrm{n}}$ are a trivial consequence of the definition of the
corresponding Legendre polynomials, but the {\it Cassini} observations can not
be explained by a simple exponentially decaying flow and instead require a
substantial retrograde flow at depth and a larger $d$ than the simple scaling
suggests. This is consistent with the results reported by \citet{Iess2019}.
However, there is no fluid dynamical requirement that the flows observed in the
atmosphere are a guide to the flows thousands of km deeper. We explore a wide
range of flow depths and amplitudes which yield values for $\Delta
J_{\mathrm{n}}$ that are acceptable within the error estimates and thus
highlight the inherent non-uniqueness of inferences made from the higher order
gravity moments.
