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On the Motion of Small Satellite near the Planet in ER3BP

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Abstract

In the current research, a novel approach or solving procedure for equations of the trapped motion for small mass m near the primary mplanet in case of the elliptic restricted problem of three bodies (ER3BP) is presented. We consider two primaries MSun and mplanet which are orbiting around their barycenter on elliptic orbits. Our aim of investigating such the class of motions is to obtain the coordinates of the aforementioned satellite (in the synodic co-rotating Cartesian coordinate system \( \overrightarrow{r} \)={x, y, z}) which will always maintain its orbit located near the second of these primaries, mplanet. We obtain a family of semi-analytical (approximated) solutions as follows: 1) equation for x is given via coordinate y and true anomaly f, with help of the additional variable parameter α, which determines a Riccati-type character of solution for coordinate z (which means possibility of sudden jumping of the magnitude of solution), 2) expression for y is given via coordinate x, true anomaly f, and the chosen parameter α, 3) coordinate z is to be quasi-oscillating with small amplitude depending on true anomaly f which is tracing the motion of small mass (satellite) around primary mplanet. Besides, the additional ways of semi-analytical solving equations of motion are illuminated.

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Acknowledgements

Authors are thankful to unknown esteemed Reviewer with respect to his valuable efforts and advice which have improved structure of the article significantly.

Remark Regarding Contributions of Authors as below

In this research, Dr. Sergey Ershkov is responsible for the general ansatz and the solving procedure, simple algebra manipulations, calculations, results of the article and also is responsible for the search of approximated solutions.

Dr. Dmytro Leshchenko is responsible for theoretical investigations as well as for the deep survey in literature on the problem under consideration.

Dr. Alla Rachinskaya is responsible for testing the initial conditions for the approximated solutions as well as for obtaining numerical solutions related to approximated ones (including their graphical plots, presented in [8]).

All authors agreed with results and conclusions of each other.

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Authors and Affiliations

  1. Plekhanov Russian University of Economics, Scopus number 60030998, 36 Stremyanny Lane, Moscow, 117997, Russia

    Sergey Ershkov

  2. Sternberg Astronomical Institute, M.V. Lomonosov’s Moscow State University, 13 Universitetskij prospect, 119992, Moscow, Russia

    Sergey Ershkov

  3. Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine

    Dmytro Leshchenko

  4. Odessa I. I. Mechnikov National University, 2 Dvoryanskaya St, Odessa, Ukraine

    Alla Rachinskaya

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  1. Sergey Ershkov
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  2. Dmytro Leshchenko
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  3. Alla Rachinskaya
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Correspondence to Sergey Ershkov.

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Ershkov, S., Leshchenko, D. & Rachinskaya, A. On the Motion of Small Satellite near the Planet in ER3BP. J Astronaut Sci 68, 26–37 (2021). https://doi.org/10.1007/s40295-021-00255-2

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