Eur. Phys. J. Plus (2020) 135: 603
https://doi.org/10.1140/epjp/s13360-020-00579-2

Regular Article

Solving procedure for the motion of infinitesimal mass in BiER4BP

¹ Plekhanov Russian University of Economics, Scopus number 60030998, Moscow, Russia
² Sternberg Astronomical Institute, M.V. Lomonosov’s Moscow State University, 13 Universitetskij prospect, Moscow, Russia 119992,
³ Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
⁴ Odessa I. I. Mechnikov National University, 2 Dvoryanskaya St., Odessa, Ukraine

^* e-mail: sergej-ershkov@yandex.ru

Received: 8 May 2020
Accepted: 29 June 2020
Published online: 27 July 2020

Abstract

In this paper, we present a new ansatz for solving equations of motion for the trapped orbits of the infinitesimal mass m, which is moving near the primary M ₃ in case of bi-elliptic restricted problem of four bodies (BiER4BP), where three primaries M ₁, M ₂, M ₃ are rotating around their common center of mass on elliptic orbits with hierarchical configuration M ₃ ≪ M ₂ ≪ M ₁. A new type of the solving procedure is implemented here to obtain the coordinates of the infinitesimal mass m with its orbit located near the primary M ₃. Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical or semi-analytical (approximated) way for presentation of the solution. We obtain as follows: (1) the solution for coordinate x is described by the key nonlinear ordinary differential equation of fourth order at simplifying assumptions, (2) solution for coordinate y is given by the proper analytical expression, depending on coordinate x and true anomaly f, (3) the expression for coordinate z is given by the equation of Riccati-type—it means that coordinate z should be quasi-periodically oscillating close to the fixed plane .