Syzygies for periodic orbits in the restricted three-body problem

In this paper we show the existence of syzygies for all periodic orbits inside the bounded Hill's region of the planer circular restricted three-body problem with energy below the second critical value. The proof will follow some ideas of Birkhoff to compute the roots of partial derivatives of the effective potential. Birkhoff's methods are extended to higher energies and a new base case is created and shown to fulfil the requirements. An other step from Birkhoff is scrutinized to continue the statement to all mass ratios. The final step is achieved by integrating over periodic orbits. Applying the same methods to Hill's lunar problem delivers similar results in that setting as well.