On parametric resonance near the libration point L1 of a planar restricted photogravitational three-body problem


DOI: 10.34759/trd-2022-126-03

Аuthors

Avdyushkin A. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: avdyushkin.a.n@ya.ru

Abstract

A planar elliptic restricted photogravitational three-body problem is considered, i.e. it is investigated the motion of a low-mass body under the influence of both gravitational forces and light pressure forces acting from two massive bodies that move along known Keplerian orbits. It is assumed that the all three bodies move in the same plane. There is a particular solution in this problem describing the motion, which the low-mass body is located on the segment between the attracting centers at the so-called collinear libration point L1.

In this paper, we study the problem of the collinear libration point L1 stability in the case of small eccentricity of the massive bodies’ orbits. The system of perturbed motion equations is written in Hamiltonian form. It is established that in this system there are possible both basic and combinational parametric resonances leading to instability L1.

The normal form of the Hamiltonian quadratic part of the perturbed motion equations is obtained in explicit form by the method of a small parameter. This made it possible to reduce the linear stability problem L1 to the equivalent stability problem of a linear autonomous system with a normalized Hamiltonian. The explicit expressions defining the boundaries of the parametric resonance regions were found on the basis of this autonomous system and it was obtained the stability conditions L1 in the linear approximation. Previously, the regions of stability and instability were obtained numerically in [12]. Carried out in that work the numerical analysis results are in good agreement with the results obtained analytically for small values of eccentricity in this article.

Keywords:

Hamiltonian system, double third-order resonance, stability in the linear approximation, periodic motions

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