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Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

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Abstract

The current methods for designing periodic orbits in the elliptic restricted three-body problem (ERTBP) have the disadvantages of targeting limited orbits and ergodic searches and considering only symmetric orbits. A universal method for designing periodic orbits is proposed in this paper. First, the homotopy classes of orbits are structured based on their topological structures. Second, a dynamic model based on homotopy classes, ranging from the circular restricted three-body problem (CRTBP) to the ERTBP, can be built using the homotopy method. Third, a multi- and a single-period orbit were selected based on the resonance ratios. Finally, the corresponding orbit in the ERTBP was computed by modifying the initial condition of the orbit in the CRTBP. This method, without an ergodic search, can extend to any orbit, including an asymmetric orbit in the CRTBP, to the ERTBP model, and the two orbits are of the same homotopy class. Examples of the Earth–Moon ERTBP are presented to verify the efficiency of this method.

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Abbreviations

a 12 :

semi-major axis of primaries

CRTBP:

circular restricted three-body problem

e :

eccentricity in the orbital model

ERTBP:

elliptic restricted three-body problem

f :

true anomaly of primaries

ME-Halo:

multi-revolution elliptic halo

r 12(f):

instantaneous distance between two primaries

RTBP:

restricted three-body problem

T C, T E :

orbital period in the dynamical model of CRTBP and ERTBP, respectively

W (γ, p):

winding number of an orbit about the point p

Θ:

cumulative angle of the rotating system turn relative to the inertial frame

μ :

mass ratio parameter

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Acknowledgements

The authors gratefully acknowledge financial support from the Natural Science Basic Research Program of Shaanxi (Grant No. 2021JQ-846) and the Science Research Fund of Xi’an Aeronautics University (Grant No. 2020KY2236).

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Authors and Affiliations

  1. School of Electronic Engineering, Xi’an Aeronautical Institute, Xi’an, 710077, China

    Yue Zheng

  2. School of Aircraft Engineering, Xi’an Aeronautical Institute, Xi’an, 710077, China

    Min Zhao

  3. School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Yue Zheng

Authors
  1. Yue Zheng
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  2. Min Zhao
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Correspondence to Yue Zheng.

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The authors have no competing interests to declare that are relevant to the content of this article.

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Yue Zheng received her Ph.D. degree from the School of Astronautics at Northwestern Polytechnical University, China. She is now a teacher in the School of Electronic Engineering at Xi’an Aeronautical Institute, China. Her research interests include aircraft orbit design, astrodynamics, and machine learning. E-mail: 2012100018@mail.nwpu.edu.cn

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Zheng, Y., Zhao, M. Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem. Astrodyn 8, 175–188 (2024). https://doi.org/10.1007/s42064-023-0188-y

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