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A multi-step method to calculate long-term quasi-periodic orbits around the Sun-Earth \(L_{1}\)/\(L_{2}\)

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Abstract

Quasi-periodic orbits around the libration point orbits have attracted significant attention since they can provide more opportunities for space missions. This paper proposes a variable step-size multistep method that can effectively calculate long-term quasi-periodic orbits around the Sun-Earth \(L_{1}\)/\(L_{2}\) in the Sun-Earth-Moon bicircular model. In this method, periodic orbits in the Sun-Earth \(L_{1}\)/\(L_{2}\) serve as the initial reference trajectories for quasi-periodic orbits. Then a two-level multiple shooting is introduced to transition periodic orbits to the short-term quasi-periodic orbits. Finally, a variable step-size multistep method is proposed to obtain long-term quasi-periodic orbits based on the approximately linear relationship between the two-level multiple shooting method and step size. Numerical results show that this method can effectively obtain a large set of long-term quasi-periodic orbits whether the quasi-periodic orbits have free or fixed initial positions. Furthermore, this method is also extended to design long-term quasi-periodic orbits with differential phase angles of the Moon.

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Acknowledgements

We acknowledge support from the National Natural Science Foundation of China (Grants No. 12102344) and the National Key Laboratory of Aerospace Flight Dynamics.

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Authors and Affiliations

  1. School of Astronautics, Northwestern Polytechnical University, Youyixilu 127, Xi’an, 710072, Shaanxi, China

    Minghu Tan, Bingbing Ma & Haoyu Li

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  1. Minghu Tan
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  2. Bingbing Ma
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Contributions

The main manuscript text was written by Minghu Tan. The analysis about the multiple shooting method was performed by Haoyu Li. Figures 3 and 6 were prepared by Bingbing Ma.

Fig. 3
figure 3

Illustration of the two-level multiple shooting method with a free initial position

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Fig. 4
figure 4

A quasi-periodic orbit with duration \(T_{f} = 10 \text{ years}\)

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Fig. 5
figure 5

Results of generating quasi-Lyapunov orbits with (a) \(T_{f} = 2 \text{ years}\); (b) \(T_{f} = 5 \text{ years}\); (c) \(T_{f} = 7 \text{ years}\); (d) \(T_{f} = 10 \text{ years}\)

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Fig. 6
figure 6

Illustration of the two-level multiple shooting method with a fixed initial position

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Fig. 7
figure 7

Results of generating quasi-Halo orbits using the multiple shooting method with a fixed initial position with (a) \(T_{f} = 1 \text{year}\); (b) \(T_{f} = 2 \text{ years}\); (c) \(T_{f} = 3 \text{ years}\)

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Fig. 8
figure 8

Results of generating quasi-Lyapunov orbits using the multiple shooting method with a fixed initial position with (a) \(T_{f} = 1 \text{year}\); (b) \(T_{f} = 2 \text{ years}\); (c) \(T_{f} = 3 \text{ years}\)

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Algorithm 1
figure 9

Procedure of the multistep method of calculating quasi-periodic orbits

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Fig. 9
figure 10

Quasi-Halo orbits with (a) \(m = 50\mu _{m}/N\); (b) \(m = 500\mu _{m}/N\)

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Algorithm 2
figure 11

Procedure of the variable step-size multistep method of calculating quasi-periodic orbits

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Fig. 10
figure 12

Quasi-Lyapunov orbits with free initial positions and (a) \(T_{f} = 20 \text{ years}\); (b) \(T_{f} = 50 \text{ years}\)

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Fig. 11
figure 13

Quasi-Halo orbits with free initial positions and (a) \(T_{f} = 20 \text{ years}\); (b) \(T_{f} = 50 \text{ years}\)

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Fig. 12
figure 14

Quasi-Lyapunov orbits with a fixed initial position and (a) \(T_{f} = 20 \text{ years}\); (b) \(T_{f} = 50 \text{ years}\)

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Fig. 13
figure 15

Quasi-Halo orbits with a fixed initial position and (a) \(T_{f} = 20 \text{ years}\); (b) \(T_{f} = 50 \text{ years}\)

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Fig. 14
figure 16

(a) Quasi-Lyapunov orbit with \(\theta _{m0} = \pi /2\); (b) Quasi-Lyapunov orbit with \(\theta _{m0} = \pi \); (c) Quasi-Halo orbit with \(\theta _{m0} = \pi /2\); (d) Quasi-Halo orbit with \(\theta _{m0} = \pi \)

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Correspondence to Haoyu Li.

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Tan, M., Ma, B. & Li, H. A multi-step method to calculate long-term quasi-periodic orbits around the Sun-Earth \(L_{1}\)/\(L_{2}\). Astrophys Space Sci 367, 101 (2022). https://doi.org/10.1007/s10509-022-04135-5

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