Global regularization method for planar restricted three-body problem

Abstract

In this paper, global regularization method for planar restricted three-body problem is purposed by using the transformation z = x+iy = ? cos n(u+iv), where i = ??1, 0 < ? ? 1 and n is a positive integer. The method is developed analytically and computationally. For the analytical developments, analytical solutions in power series of the pseudotime ? are obtained for positions and velocities (u, v, u', v') and (x, y, x?, y?) in both regularized and physical planes respectively, the physical time t is also obtained as power series in ?. Moreover, relations between the coefficients of the power series are obtained for two consequent values of n. Also, we developed analytical solutions in power series form for the inverse problem of finding ? in terms of t. As typical examples, three symbolic expressions for the coefficients of the power series were developed in terms of initial values. As to the computational developments, the global regularized equations of motion are developed together with their initial values in forms suitable for digital computations using any differential equations solver. On the other hand, for numerical evolutions of power series, an efficient method depending on the continued fraction theory is provided.