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Investigating the non-inertial R2BP in case of variable velocity \(\vec{\mathbf{V}}\) of central body motion in a prescribed fixed direction

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Abstract

In this analytical study, we have presented a new type of solving procedure with the aim to obtain the coordinates of small mass m, which moves around primary MSun, referred to non-inertial frame of restricted two-body problem (R2BP) with a modified potential function (taking into account the variable velocity \(\vec{V}\) of central body MSun motion in a prescribed fixed direction) instead of a classical potential function for Kepler’s formulation of R2BP. Meanwhile, system of equations of motion has been successfully explored with respect to the existence of an analytical way of presenting the solution in polar coordinates {r(t), φ(t)}. We have obtained an analytical formula for function t = t(r) via an appropriate elliptic integral. Having obtained the inversed dependence r = r(t), we can obtain the time dependence φ = φ(t). Also, we have pointed out how to express components of solution (including initial conditions) from cartesian to polar coordinates as well.

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Acknowledgements

Sergey Ershkov is thankful to Prof. Nikolay Emelyanov for valuable advice during fruitful discussions in the process of preparing this manuscript. Authors appreciate efforts of both Reviewers, including the advice of Reviewer 2 to cite refs. [22-26] which are beyond celestial mechanics but within the framework of the analytical approach to the study of mathematical models in dynamics of rigid body rotation.

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Authors and Affiliations

  1. Plekhanov Russian University of Economics, Scopus Number 60030998, Moscow, Russia

    Sergey Ershkov

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

    Sergey Ershkov

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

    Dmytro Leshchenko

  4. Institute of Engineering Science UB RAS, Ekaterinburg, Russia

    E. Yu. Prosviryakov

Authors
  1. Sergey Ershkov
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  2. Dmytro Leshchenko
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  3. E. Yu. Prosviryakov
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Contributions

In this research, S.E. is responsible for the general ansatz and the solving procedure, simple algebra manipulations, calculations, results of the article in Sections 1–3 and also is responsible for the search of approximated solutions. D.L. is responsible for theoretical investigations as well as for the deep survey in the literature on the problem under consideration. E.P. is responsible for obtaining numerical solutions related to approximated ones (including their graphical plots). All authors reviewed the manuscript.

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Correspondence to Sergey Ershkov.

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Ershkov, S., Leshchenko, D. & Prosviryakov, E.Y. Investigating the non-inertial R2BP in case of variable velocity \(\vec{\mathbf{V}}\) of central body motion in a prescribed fixed direction. Arch Appl Mech 94, 767–777 (2024). https://doi.org/10.1007/s00419-023-02535-9

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