# On the period of the periodic orbits of the restricted three body problem

On the period of the periodic orbits of the restricted three body problem 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, $$2 T=k\pi +\int _\Omega g$$ 2 T = k π + ∫ Ω g where k is an integer, $$\Omega$$ Ω is the region enclosed by the periodic orbit and $$g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$$ g : R 2 → R is a function that only depends on the constant C known as the Jacobian constant; it does not depend on $$\Omega$$ Ω . 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 $$L_4$$ L 4 such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for $$L_5$$ L 5 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Celestial Mechanics and Dynamical Astronomy Springer Journals

# On the period of the periodic orbits of the restricted three body problem

, Volume 129 (2) – Apr 19, 2017
16 pages

/lp/springer_journal/on-the-period-of-the-periodic-orbits-of-the-restricted-three-body-1PUfaCryxn
Publisher
Springer Netherlands
Subject
Physics; Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics
ISSN
0923-2958
eISSN
1572-9478
D.O.I.
10.1007/s10569-017-9766-8
Publisher site
See Article on Publisher Site

### 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, $$2 T=k\pi +\int _\Omega g$$ 2 T = k π + ∫ Ω g where k is an integer, $$\Omega$$ Ω is the region enclosed by the periodic orbit and $$g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$$ g : R 2 → R is a function that only depends on the constant C known as the Jacobian constant; it does not depend on $$\Omega$$ Ω . 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 $$L_4$$ L 4 such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for $$L_5$$ L 5 .

### Journal

Celestial Mechanics and Dynamical AstronomySpringer Journals

Published: Apr 19, 2017

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