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

## TL;DR

This paper develops a linear theory to estimate how cylindrical differential rotation affects planetary gravity moments, revealing the non-uniqueness of interpreting Saturn's gravity data and implications for deep flow structures.

## Contribution

It introduces analytical formulas for gravity moments under differential rotation and demonstrates the non-uniqueness in inferring internal flows from gravity measurements.

## Key findings

- Linear approximation effectively estimates gravity moments from differential rotation.
- Observed gravity moments can be explained by various flow depths and amplitudes, indicating non-uniqueness.
- Simple exponential decay flow models are insufficient to explain Saturn's gravity data.

## Abstract

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.