Abstract
In this paper, motions around the triangular libration point L4 in the planar elliptic restricted three-body problem are considered. Polar coordinates of the test particle centered at the first primary are adopted. Explicit expansion of the potential function in terms of the polar coordinates is carried out in two ways, by a recursive relation or by a direct expansion. Using the Lindstedt–Poincaré method, the explicit solution based on the polar coordinates is given for the first time. A comparison between the explicit solutions based on the Cartesian coordinates and on the polar coordinates truncated at the same order is made, and the result shows that the latter is at least one order of magnitude better in accuracy than the former. Based on the explicit solution, some analysis on the dynamics around the triangular libration points are made, including the secondary resonance structures on the \(\mu -e\) plane, the practical stable region described by the amplitudes of our explicit solution, and the resonance structures revealed by LCE maps and numerical frequency analysis.
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Acknowledgements
This work is supported by National Natural Science Foundation of China (No. 11773017, 11973010). The authors thank Prof. Hanlun Lei for providing the test examples (see Fig. 4) of the explicit solution based on Cartesian coordinates given by them in Lei and Xu (2014). The numerical frequency analysis conducted in Sects. 6 and 7.3 used the software NAFF_UV available at https://github.com/kskoufar/NAFF_UV.
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Tan, P., Hou, XY. & Liao, XH. Explicit solution and resonance dynamics around triangular libration points of the planar elliptic restricted three-body problem. Celest Mech Dyn Astr 133, 31 (2021). https://doi.org/10.1007/s10569-021-10028-6
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DOI: https://doi.org/10.1007/s10569-021-10028-6