Closed Form Approximations For The Three Body Problem A. B. Mehmood, U. A. Shah and G. Shabbir Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology Topi, Swabi, NWFP, Pakistan Email:
[email protected] Abstract In this paper, an approach is developed to solve the three body problem involving masses which posses spherical symmetry. The problem dates back to the times of Poincare, and is undoubtedly one of the oldest of unsolved problems of classical mechanics. The Poincare’s Dictum comprehensively proves that the problem is truly insolvable as a result of the nature of the instabilities involved. We therefore refute the idea of finding exact solutions. Instead, we develop closed form analytical approximations in place of exact solutions. We will solve the problem for the case when all the masses involved have spherically symmetric mass distributions. The method of solution would include the use of a single mass to replicate the effect of two individual masses on each body. The derivation of solutions will involve the use of the Lambert’s wave function and the solution will comprise of the position vectors expressed as explicit time functions. Introduction Having outlined the nature of our work in the abstract, we go on to provide a more formal definition of the problem that we choose to solve. The problem, by definition is to solve for the position vectors as time functions, of three gravitating masses when they execute free motion under each other’s gravitational influence. The masses form an isolated system in free space, and given the initial position and velocity of each mass, their subsequent free motions are to be examined as accurately as possible. We now state and discuss some of the simplifying assumptions that will be used in solution of the problem. It will be noted that these assumptions will provide us with the luxury of deriving the solutions comfortably on a relative scale, and yet 1 the accuracy of our solutions will be reasonable. Firstly, we will assume that the gravitating bodies involved posses mass distributions that are spherically symmetric. The reason behind this idea is the fact that the three body problem finds most of its applications in celestial mechanics [1-4], involving planets and heavenly bodies as the gravitating masses. Moreover, these planets and heavenly bodies are known to posses mass distributions that are almost always spherically symmetric. As a result it would be impractical, (and perhaps not possible) to solve the problem taking into account any arbitrary mass distributions. Moreover, this assumption would also allow us to treat the gravitating bodies having finite volume, as point masses. This would of course be another simplifying step since the effect o