Title: THE LIBRATION CASE OF THE STELLAR PROBLEM OF THREE BODIES
Abstract: The methods of nonlinear mechanics are introduced to facilitate the study of the libration case of the stellar problem of three bodies. To illustrate the method, the problem of a lunar satellite perturbed by the earth and under the influence of the 12 and 13 terms of the moon's gravitational field is considered. By means of the energy integral and the equation of motion of the argument of pericenter, a closed curve is obtained in phase space. Analysis of the geometry of the variables in phase space yield the period of libration and expansions of the argument of pericenter and eccentricity in terms of time. In 1936 Ernest W. Brown wrote a series of four papers on the 'Stellar Problem of Three Bodies' with applications to satellite theory (Brown, 1936). The stellar problem is defined by a pair of celestial objects relatively close to each other perturbed by a third body at a considerable distance away. The motion of such a system is similar in many respects to the fundamental problem of the long period motion of a lunar satellite. Thus the moon and its satellite correspond to the close pair perturbed by the earth whose distance to the moon is large compared to the moon's distance from its satellite. In this paper the problem is expanded to include the effects of the J 2 andJ 3 harmonics of the moon. In order to solve the problems he considered, Brown developed a semi-analytic method, based on harmonic analysis, which was valid for arbitrary eccentricities and inclinations. His solution gave a great deal of insight into the nature of the perturba tions and avoided a great deal oftedious computations. In the problem considered by Brown the nature of the problem was such that the resulting motion was of the circulation type. In this paper, the initial conditions are such that libration of the argument of pericenter occurs. The differences between the two types of motion are illustrated by a pendulum which is free to rotate about a horizontal axis. When the pendulum makes complete revolutions about the axis the motion is known as circulation. However when the pendulum swings to and fro like the pendulum of a clock we say libration occurs. This concept applied to satellites says that when the argument of pericenter takes on values over the entire interval between 0 and 2n as well as integral multiples of this range circulation occurs. How ever, when the argument of pericenter is restricted to a smaller interval than between o and 2n, libration occurs. The method described by Brown cannot be carried over to the case when libration occurs. In this paper an entirely different semi-analytic approach is developed which is a direct consequence of geometrical considerations of the problem in Poincare phase space. Discussions of the geometry in phase space is given in books on nonlinear mechanics such as Stoker (1950).
Publication Year: 1970
Publication Date: 1970-01-01
Language: en
Type: article
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