Existence of the Broucke periodic orbit and its linear stability

In this paper, we study the existence and linear stability of the Broucke periodic orbit in the planar three-body problem. In each period of this orbit, there are two binary collisions (or BC for short) between the outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. A surprising simple existence proof of this orbit is given. The initial condition of this orbit is shown to be a supremum of some well-chosen set. The linear stability is then analyzed by Robertsʼ symmetry reduction method. It is shown that the Broucke periodic orbit with equal masses is linearly stable.