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Numerical integration of the restricted three-body problem with Lie series

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Abstract

The aim of this work is to present some recurrence formulas for the equations of motion of an infinitesimal body in the planar restricted three-body problem which allow us to integrate numerically this problem via a Lie series approach. For doing this, the equations of motion of the problem are transformed to an origin at one of the libration points and the Lie operator and recurrence formulas for the terms of the Lie series are constructed. In addition, we provide an algorithm that allows us to find any number of Lie series terms and which gives successful calculations for the orbit of the infinitesimal body around one of the libration points. Furthermore, all our mathematical relations are performed under the effect of the zonal harmonic parameters of the bigger primary up to J4. Finally, a numerical application of these results is given to the case of the Earth–Moon system.

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Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. (857-71-D1435). The authors, therefore, acknowledge with thanks DSR technical and financial support. This work has been partially supported by MICINN/FEDER grant number MTM2011-22587.

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science and Arts (Khulais), King Abdulaziz University, Jeddah, Saudi Arabia

    Elbaz I. Abouelmagd

  2. Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Elbaz I. Abouelmagd

  3. Departamento de Matemática Aplicada y Estadística, Hospital de Marina, Universidad Politécnica de Cartagena, 30203, Cartagena, Región de Murcia, Spain

    Juan L. G. Guirao

  4. Mathematics Department, Faculty of Science, Ain Shams University, Cairo, Egypt

    A. Mostafa

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  1. Elbaz I. Abouelmagd
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  2. Juan L. G. Guirao
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  3. A. Mostafa
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Correspondence to Elbaz I. Abouelmagd.

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Abouelmagd, E.I., Guirao, J.L.G. & Mostafa, A. Numerical integration of the restricted three-body problem with Lie series. Astrophys Space Sci 354, 369–378 (2014). https://doi.org/10.1007/s10509-014-2107-4

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  • DOI: https://doi.org/10.1007/s10509-014-2107-4

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