A fast method for quantitatively measuring stability in a three-body dynamical system

Abstract

The three body problem can be used to investigate many types of dynamical systems when they possess an attraction for each other. The circular restricted three-body problem is a special case of the general three-body problem. The circular restricted three-body problem has no closed form solution and must be numerically integrated to obtain a trajectory of the motion of the third body. A quantitative stability investigation usually requires a tremendous amount of computation due to numerically integrating each individual trajectory. By reducting the amount of numerical integration time, more time can be directed toward numerically investigating the stability relationship of the trajectory sets for the various mass ratios. In this paper a closed form measure of stability for a bounded region in the three-body system is found and used to perform a global (examination of all mass ratio combinations of the three bodies) stability investigation. The developed stability criteria is a function of the initial conditions of the third body and the strength of the potential field. In this study the potential field influencing the bodies is considered to be a gravitational field and it is shown that there are certain mass ratio combinations of the three bodies that result in an optimally stable system.