Analytical Solution with Two Time Scales of Circular Restricted Three-Body Problem
Abstract
Analytical solution performs a vital role in a wide variety of deep space exploration missions, especially the periodic solutions which were believed to be the unique avenue to solve three-body problem by Poincaré. As the absence of a general solution for the problem, an approximate analytical solution of the circular restricted three-body problem is addressed by employing multiple scales method in conjunction with some analytical techniques. It is worthwhile to note that the presented solution in three-dimensional space contains two time scales which is significative to improve and perfect the known literature.
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Introduction
An existence of conditional periodic solutions of circular restricted three-body problem (CR3BP) was presented by Gao [1]. However, because of lacking exact analytical solutions for the problem, a considerable amount of researchers were attracted to develop the approximate analytical solutions of the problem. In 1973, Farquhar and Kamel [2] developed an approximation method for this type of orbit. Richardson [3, 4] presented a third-order analytical solution about the collinear libration points of the CR3BP based on the method of successive approximations and a technique similar to the LindstedtPoincaré method. Lu and Zhao [5] put forward a kind of improved third-order approximate analytical solution in 2009, which is more accurate than the classical analytic solution of Richardson. In addition, Nayfeh [6, 7] studied two types of resonance near the planar triangular libration points under a case of small amplitude.
Conclusion
Since the governing equations of the third body is time-dependent in inertial frame, which appear as a high-dimensional nonlinear autonomous system. According to the abundant known literature, this system seems unlikely to be solved via analytical approaches.