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On Schwarzschild periodic solutions in the restricted three body problem Olund, Brian Russel

Abstract

Consider the restricted three body problem in which we have a central body S (the sun), a perturbing planet J (Jupiter) whose mass is small compared to that of the sun, and a planetoid P of negligible mass. We consider the special case in which we have the following restrictions: 1) The perturbing planet J moves in a circle with the sun as center. 2) The orbit of P is an ellipse in the same plane as the orbit of J and with the sun at one focus. 3) If the perturbing influence of J were ignored, the mean motion n₀ of P would be related to the mean motion n of J by n₀/n' = p/q , where p and q are positive relatively prime integers. The period of this system in the unperturbed motion is T₀ = 2πq/n' . We wish to see under what conditions a periodic solution can be found for the perturbed motion. Using the method of a small parameter Schwarzschild has shown that, under certain conditions, if the mass m' of Jupiter is sufficiently small, all three bodies will return to the same relative position as initially after a time T = T₀ (1+τ) except that the entire system will have rotated through a small angle. τ is of the order of m'/m(sun) and vanishes with m’. The paper is divided into two parts. The first part is devoted to a method for calculating the period and the mean values of the orbital elements. The second part is devoted to a method for calculating the period and the initial values of the orbital elements.

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