Lissajous and halo orbits in the restricted three-body problem by normalization method
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ORIGINAL PAPER
Lissajous and halo orbits in the restricted three-body problem by normalization method Tong Luo . Giuseppe Pucacco . Ming Xu
Received: 13 November 2019 / Accepted: 1 August 2020 Springer Nature B.V. 2020
Abstract We perform an analytical study of the Lissajous and halo orbits around collinear points L1 and L2 in a spatial circular restricted three-body problem of an arbitrary value of the mass ratio. Using a canonical transformation procedure, we generate complete and resonant normal forms through reduction to center manifolds. The coefficients in the normal forms are explicitly expressed as functions of mass ratio for the first time so that one can evaluate the energy level at which bifurcation of halo orbit takes place. Another contribution of this paper is giving the analytical solutions of Lissajous and halo orbits in the initial synodic reference system through the inverse transformation of normalization. The analytical results are the series form of normalized action-angle variables, and their coefficients are also explicitly
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11071-020-05875-1) contains supplementary material, which is available to authorized users. T. Luo M. Xu (&) School of Astronautics, Beihang University, Beijing 100191, China e-mail: [email protected] T. Luo e-mail: [email protected] G. Pucacco Department of Physics, University of Roma Tor Vergata, 00133 Rome, Italy e-mail: [email protected]
expressed as functions of mass ratio. Finally, comparison results demonstrate that the solutions for a Lissajous orbit derived through normalization method and Lindstedt–Poincare´ method are completely the same, while the solutions for a halo orbit derived through these two methods are different but have the roughly equal accuracy. Keywords Halo orbits Lissajous orbits Spatial circular restricted three-body problem Normalization method Lindstedt–Poincare´ method
1 Introduction Spatial circular restricted three-body problem (SCR3BP) is a fundamental model to describe the motion of a small body affected by the gravitational attraction of two primaries. Different values of the mass ratio l (0 \ l B 1/2) of the primaries correspond to different celestial systems, such as the Earth– Moon case and the Sun–Earth case. On the line connecting the two primaries exist three collinear equilibrium points [1], which are linearly unstable for arbitrary value of l. However, Lyapunov’s center theory [2] guarantees each collinear point can generate a pair of periodic orbit families, to which we refer as the planar and the vertical Lyapunov families. Quasiperiodic orbits, which form the so-called
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Lissajous family, are derived through the combination of the planar and the vertical Lyapunov orbits. In addition, well-known halo orbits arise at the first 1:1 bifurcation from the planar Lyapunov family [3, 4]. Halo orbits have been widely applied to many space missions than
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