Solving procedure for 3D motions near libration points in CR3BP
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ORIGINAL ARTICLE
Solving procedure for 3D motions near libration points in CR3BP Sergey V. Ershkov1,2
· Dmytro Leshchenko3
Received: 28 January 2019 / Accepted: 7 November 2019 © Springer Nature B.V. 2019
Abstract In a novel approach for solving the equations of the Circular Restricted Three-Body Problem (CR3BP) first formulated in Ershkov (Acta Mech. 228(7):2719–2723, 2017a), we apply in this communication a procedure for solving the Euler-Poisson equations for the momentum equations of the CR3BP near the libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution in the vicinity of libration points. A new and elegant ansatz is suggested in this publication, whereby, in solving, the momentum equation is reduced to a system of three linear ordinary differential equations of first order in regard to the three components of the velocity of the infinitesimal mass m (dependent on time t). Under this premise, a proper elegant partial solution has been obtained due to the invariant dependence between temporary components of the solution. We conclude that the system of CR3BP equations does not have the analytical presentation of the solution (in quadratures) even in the vicinity of the libration points except of the generalized Jacobi integral.
B S.V. Ershkov
[email protected] D. Leshchenko [email protected]
1
Plekhanov Russian University of Economics, Scopus number 60030998, Stremyanny lane 36, Moscow, 117997, Russia
2
Sternberg Astronomical Institute, M.V. Lomonosov’s Moscow State University, 13 Universitetskij prospect, Moscow 119992, Russia
3
Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
Keywords Circular Restricted Three-Body Problem (CR3BP) · Poisson equations · Riccati equation · Jacobi integral
1 Introduction, equations of motion The equations of motion of the restricted three-body problem (R3BP) describe the dynamics of infinitesimal mass m under the action of gravitational forces affected by two celestial bodies of giant masses (in this problem MSun and mplanet , mplanet < MSun ), which are rotating around their common center of masses on Kepler trajectories. The aforesaid infinitesimal mass m is supposed to be moving (as a first approximation) inside the restricted region of space around the planet of mass mplanet or inside the so-called Hill sphere (Ershkov and Shamin 2018a) (where ap is the semimajor axis of the planet): rH = ap ·
mplanet MSun
1 3
(*)
It is worth to note that a large amount of previous and recent results concerning analytical development with respect to these equations exist; the fundamental work in Arnold (1978), Lagrange (1873), Szebehely (1967), Duboshin (1968), Bruns (1887), Chernikov (1970) should be mentioned particularly. We should especially emphasize the theory of orbits, which was developed in profound work
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