The use of vertical instability of \(L_1\) and \(L_2\) planar Lyapunov orbits for transfers from near rectilinear halo orbits to planar distant retrograde orbits in the Earth–Moon system

  • Kenta OshimaEmail author
Original Article


This paper highlights natural transport pathways between in-plane and out-of-plane states associated with the vertical instability of planar Lyapunov orbits around the Lagrange points \(L_1\) and \(L_2\) in the Earth–Moon circular restricted three-body problem. Computations of invariant manifolds associated with the vertical instability of planar periodic orbits, “vertically” stable and unstable manifolds, enable quantitative analyses of inclination changes. This study finds that multiple lunar flybys gradually change the orbital elements of vertically stable and unstable manifolds, and that the distributions of the affected orbital elements depend on the Jacobi constant and on the associated Lagrange point \(L_1\) or \(L_2\) of the planar Lyapunov orbits. As an application, this study uses the vertically stable manifolds of the planar Lyapunov orbits as initial guesses for optimizing transfers from near rectilinear halo orbits to planar distant retrograde orbits. Significant \({\varDelta }v\) savings as compared with the known solutions demonstrate the usefulness of the vertical instability in spacecraft trajectory designs.


Vertical instability Planar Lyapunov orbits Near rectilinear halo orbits Planar distant retrograde orbits Circular restricted three-body problem Optimization 



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

Compliance with ethical standards

Conflicts 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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