Figure-Eight Orbit with Unequal Masses in the Planar Three-Body Problem

This paper examines the existence of a figure-eight solution to the plane three-body problem with gravitational interaction for unequal masses. A Hamiltonian formulation of the problem is used in the space of forms obtained by factorization with respect to translations and rotations, which allows the system dynamics to be described in terms of a triangular potential. Solutions are found by numerically integrating the equations of motion in the space of forms using the angular variable as an independent variable. Initial conditions are selected using Newton’s method so that the trajectory sequentially passes through Euler configurations. Figure-eight solutions are found for values of the third body mass close to unity, and the corresponding initial data are presented. A numerical stability analysis of the resulting trajectories is performed based on perturbations of the initial conditions, demonstrating the dependence of the dynamic behavior of the solutions on the magnitude of the third body mass deviation from the case of equal masses.