Abstract
In this work, we investigate the Earth–Moon system, as modeled by the planar circular restricted three-body problem, and relate its dynamical properties to the underlying structure associated with specific invariant manifolds. We consider a range of Jacobi constant values for which the neck around the Lagrangian point \(L_1\) is always open, but the orbits are bounded due to Hill stability. First, we show that the system displays three different dynamical scenarios in the neighborhood of the Moon: two mixed ones, with regular and chaotic orbits, and an almost entirely chaotic one in between. We then analyze the transitions between these scenarios using the monodromy matrix theory and determine that they are given by two specific types of bifurcations. After that, we illustrate how the phase space configurations, particularly the shapes of stability regions and stickiness, are intrinsically related to the hyperbolic invariant manifolds of the Lyapunov orbits around \(L_1\) and also to the ones of some particular unstable periodic orbits. Lastly, we define transit time in a manner that is useful to depict dynamical trapping and show that the traced geometrical structures are also connected to the transport properties of the system.
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Notes
For the equations of motion in Hamiltonian form see, e.g., Belbruno (2004).
The direction of the bifurcation determines the stability of a new periodic orbit which appears outside \(\varSigma \) and hence it is not relevant to our analysis.
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Acknowledgements
VMO would like to thank Prof. Dr. J. D. Mireles James for his notes on Celestial Mechanics (Available at http://cosweb1.fau.edu/~jmirelesjames/notes.html). This study was financed in part by the Coordenação de Aperfeiçamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and the São Paulo Research Foundation (FAPESP, Brazil), under Grant No. 2018/03211-6.
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This article is part of the topical collection on Toward the Moon and Beyond Guest Editors: Terry Alfriend, Pini Gurfil and Ryan P. Russell.
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de Oliveira, V.M., Sousa-Silva, P.A. & Caldas, I.L. Order-chaos-order and invariant manifolds in the bounded planar Earth–Moon system. Celest Mech Dyn Astr 132, 51 (2020). https://doi.org/10.1007/s10569-020-09989-x
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DOI: https://doi.org/10.1007/s10569-020-09989-x