Linking low- to high-energy dynamics of invariant manifolds, transit orbits, and singular collision orbits in the planar circular restricted three-body problem

  • Kenta OshimaEmail author
Original Article


The first part of the present paper reveals interplays among invariant manifolds emanating from planar Lyapunov orbits around the three collinear Lagrange points \(L_1, L_2\), and \(L_3\) for high energies. Once the energetically forbidden region vanishes, the invariant manifolds together form closed separatrices bounding transit orbits in the phase space, deviating from the low-energy picture of invariant manifold tubes. Though the qualitatively different behavior of invariant manifolds emerges for high energies, associated transit orbits possess a common feature generalized from that of low-energy transit orbits. The second part extends our previous proposal of using singular collision orbits associated with the secondary to find trajectories reaching the vicinity of the secondary to low energies. Statistical analyses indicate that singular collision orbits are useful to find such transfer trajectories except for the very low-energy regime. These results are numerically obtained in the Earth–Moon and Sun–Jupiter planar circular restricted three-body problems.


Invariant manifolds Collinear Lagrange points Transit orbits Singular collision orbits Poincaré section Planar circular restricted three-body problem 



This study has been partially supported by Grant-in-Aid for JSPS Fellows No. 18J00678.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflicts of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.National Astronomical Observatory of JapanTokyoJapan

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