Abstract
We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).
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Perdomo, O. On the period of the periodic orbits of the restricted three body problem. Celest Mech Dyn Astr 129, 89–104 (2017). https://doi.org/10.1007/s10569-017-9766-8
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DOI: https://doi.org/10.1007/s10569-017-9766-8