Analysis linear orbital stability of periodic motions in the planar circular restricted four-body problem


Аuthors

Volkov E. V.

Mechanical Engineering Research Institute of the Russian Academy of Sciences, 4, M. Khariton'evskii per., Moscow, 101990, Russia

e-mail: evvolkov94@mail.ru

Abstract

In this work we consider the planar restricted circular four-body problem. A small body of negligible mass moves under the action of gravitational attraction forces of three attracting bodies interacting with each other according to the law of universal gravitation. Attracting bodies are located at triangular libration points, i.e. they move in circular orbits, form an equilateral triangle. The motion of all four bodies occurs in the same plane. It is assumed that the sufficient condition for the linear stability of libration points (the Routh’s stability condition) is satisfied. The masses of the two attracting bodies are equal. In this formulation, the restricted four-body problem admits particular solutions that describe the relative equilibrium positions of a small body in a coordinate system rotating together with the attracting bodies. Periodic motions of a small body are possible in the vicinity of stable positions of relative equilibrium.
In this work considers the problem of orbital stability of periodic motions of a small body of negligible mass arising from a stable position of relative equilibrium. Under the assumption of small amplitude of these periodic motions, an analytical study of their orbital stability in a linear approximation was carried out. Using the small parameter method, explicit asymptotic expressions for the boundaries of the parametric resonance region are constructed. Explicit asymptotic expressions for the boundaries of the parametric resonance region are constructed by using method small parameter. The results of the analytical study are in good agreement with the results of the numerical study conducted in [9].

Keywords:

four-body problem, periodic motions, orbital stability

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