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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

Chachan, Yayaati and Stevenson, David J. (2019) A linear approximation for the effect of cylindrical differential rotation on gravitational moments: Application to the non-unique interpretation of Saturn’s gravity. Icarus, 323 . pp. 87-98. ISSN 0019-1035. http://resolver.caltech.edu/CaltechAUTHORS:20190225-084353057

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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 ΔJ_(2n) 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=Ω^2R^3/GM 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 ΔJ_(2n) are a trivial consequence of the definition of the corresponding Legendre polynomials, but the 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 Iess et al. (2019). 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 ΔJ_n that are acceptable within the error estimates and thus highlight the inherent non-uniqueness of inferences made from the higher order gravity moments.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.icarus.2018.12.020DOIArticle
ORCID:
AuthorORCID
Chachan, Yayaati0000-0003-1728-8269
Stevenson, David J.0000-0001-9432-7159
Additional Information:© 2018 Elsevier Inc. Received 2 August 2018, Revised 30 November 2018, Accepted 6 December 2018, Available online 25 February 2019. Y.C. thanks Hao Cao for stimulating discussions, Steve Markham for help with Mathematica ©, and Heather Knutson for discussions on MCMC. We are grateful to the reviewers for their thoughtful comments and suggestions which improved this manuscript. This work uses matplotlib (Hunter, 2007), scipy (Jones et al., 2001), and sympy (Meurer et al., 2017).
Record Number:CaltechAUTHORS:20190225-084353057
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190225-084353057
Official Citation:Yayaati Chachan, David J. Stevenson, A linear approximation for the effect of cylindrical differential rotation on gravitational moments: Application to the non-unique interpretation of Saturn’s gravity, Icarus, Volume 323, 2019, Pages 87-98, ISSN 0019-1035, https://doi.org/10.1016/j.icarus.2018.12.020. (http://www.sciencedirect.com/science/article/pii/S0019103518305074)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:93199
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:25 Feb 2019 16:51
Last Modified:05 Mar 2019 23:36

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