The Problem of Small Divisors in Planetary Motion

Williams, Carol A.

doi:10.1007/978-94-009-5331-4_33

The Problem of Small Divisors in Planetary Motion

Carol A. Williams⁴

Conference paper

112 Accesses

Abstract

Small divisors caused by certain linear combinations of frequencies appear in all analytical planetary theories. With the exception of the deep resonance between Neptune and Pluto, they can be removed at the expense of introducing secular and mixed secular terms, limiting the domain in which the solution is valid. Because of them classical solutions are known not to converge uniformly; Poincaré referred to them as asymptotic. The KAM theory shows that if one is far enough from exact commensurability and has small enough planetary masses, expansions exist which will converge to quasi-periodic orbits. Solutions showing very small divisors are excluded from this region of convergence. The question of whether they are intrinsic to the problem or are just manifestations of the method of solution is not settled. Problems with a single commensurabily that can be isolated from the rest of the Hamiltonian may have solutions with no small divisors. The problem of two or more commensurabilities remains unsolved.

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Department of Mathematics, University of South Florida, Tampa, Florida, 33620, USA
Carol A. Williams

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Carol A. Williams
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Department of Aerospace Engineering, University of Texas, Austin, Texas, USA
R. L. Duncombe
Institut für Astronomie, Universität Wien, Wien, Österreich
R. Dvorak
Department of Applied Mathematics and Theoretical Physics, Liverpool University, England
P. J. Message

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Williams, C.A. (1984). The Problem of Small Divisors in Planetary Motion. In: Duncombe, R.L., Dvorak, R., Message, P.J. (eds) The Stability of Planetary Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5331-4_33

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DOI: https://doi.org/10.1007/978-94-009-5331-4_33
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8854-1
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