Note on the trapped motion in ER3BP at the vicinity of barycenter
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O R I G I NA L
Sergey Ershkov
· Dmytro Leshchenko · Alla Rachinskaya
Note on the trapped motion in ER3BP at the vicinity of barycenter
Received: 22 March 2020 / Accepted: 25 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we present a new approach for solving equations of motion for the trapped motion of the infinitesimal mass m in case of the elliptic restricted problem of three bodies (ER3BP) (primaries MSun and m planet are rotating around their common centre of masses on elliptic orbit): a new type of the solving procedure is implemented here for solving equations of motion of the infinitesimal mass m in the vicinity of the barycenter of masses MSun and m planet . Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical way for presentation of the approximated solution. As the main result, equations of motion are reduced to the system of three nonlinear ordinary differential equations: (1) equation for coordinate x is proved to be a kind of appropriate equation for the forced oscillations during a long-time period of quasi-oscillations (with a proper restriction to the mass m planet ), (2) equation for coordinate y reveals that motion is not stable with respect to this coordinate and condition y ∼ 0 would be valid if only we choose zero initial conditions, and (3) equation for coordinate z is proved to be Riccati ODE of the first kind. Thus, infinitesimal mass m should escape from vicinity of common centre of masses MSun and m planet as soon as the true anomaly f increases insofar. The main aim of the current research is to point out a clear formulation of solving algorithm or semi-analytical procedure with partial cases of solutions to the system of equations under consideration. Here, semi-analytical solution should be treated as numerical algorithm for a system of ordinary differential equations (ER3BP) with well-known code for solving to be presented in the final form. Keywords Elliptic restricted three-body problem · Trapped motion · Riccati ODE S. Ershkov (B) Plekhanov Russian University of Economics, Scopus Number 60030998, Moscow, Russia E-mail: [email protected] S. Ershkov Sternberg Astronomical Institute, M.V. Lomonosov’s Moscow State University, 13 Universitetskij prospect, Moscow, Russia 119992 D. Leshchenko Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine E-mail: [email protected] A. Rachinskaya Odessa I. I. Mechnikov National University, 2 Dvoryanskaya St., Odessa, Ukraine E-mail: [email protected]
S. Ershkov et al.
1 Introduction, equations of motion In the restricted three-body problem (R3BP), the equations of motion describe the dynamics of an infinitesimal mass m under the action of gravitational forces effected by two celestial bodies of giant masses MSun and m planet (m planet < MSun ), which are rotating around their common centre of mass on Keplerian trajectories. In the current research, we will assume that the small mass m is
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