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Multipole Expansion: Unifying Formalism for Earth and Planetary Gravitational Dynamics

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Abstract

The powerful method of multipole expansion has found wide utility in classical electromagnetics and quantum-mechanics. In contrast, the gravitational mechanics traditionally have only seen peripheral mentions of the corresponding (mass-density) multipoles in specific applications. In this paper, we develop the general theory of the multipole formalism for the classical two-body gravitational dynamics, a most prevalent subject in planetary (and global Earth) dynamics. We treat two general, well-encountered configurations: (i) the interior “mantle-inner core gravitational (MICG)” type of configuration consisting of concentric bodies; and (ii) the exterior type of configuration of “planet + satellite” separate bodies in the general form of tidal interactions. We derive concise and exact formulas in terms of multipole expansions; by retaining the leading term(s) of relevance, typically up to the quadrupolar terms that include the planetary triaxiality, one can evaluate to the precision desired while bearing in mind that higher subtleties are readily available in the higher multipole terms. The two-body problems that are so formulated range from potential energy, satellite orbit, to tides, librations of sorts and precession/nutation, to planetary rotational normal modes and wobbles. We demonstrate that, via its various symmetry properties, the multipole formalism provides a theoretical unification framework for these gravitational phenomena that are conventionally treated topic-by-topic in textbook literature.

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Acknowledgements

We thank the two anonymous reviewers for substantial improvements of the paper. This work is supported by the Ministry of Science and Technology of Taiwan via Grant #109-2116-M-001-028. The content is theoretical in nature and involves no observational data. The authors have no conflicts of interest to declare that are relevant to the content of this article.

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Authors and Affiliations

  1. Institute of Earth Sciences, Academia Sinica, Taipei City, Taiwan, 115, ROC

    Benjamin F. Chao & Sheng-An Shih

  2. Université Catholique de Louvain, 1348, Ottignies-Louvain-la-Neuve, Belgium

    Sheng-An Shih

  3. Royal Observatory of Belgium, 1180, Uccle, Belgium

    Sheng-An Shih

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  1. Benjamin F. Chao
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SAS is instrumental in the verification and interpretations of all the equations, and initiated the enlargement to planetary dynamic problems.

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Correspondence to Benjamin F. Chao.

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Chao, B.F., Shih, SA. Multipole Expansion: Unifying Formalism for Earth and Planetary Gravitational Dynamics. Surv Geophys 42, 803–838 (2021). https://doi.org/10.1007/s10712-021-09650-8

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