Tate Homology and Powered Flybys

In this paper we show that in the planar circular restricted three body problem there are either infinitely many symmetric consecutive collision orbits or at least one periodic symmetric consecutive collision orbit for all energies below the first critical energy value. Using Levi-Civita regularization allows us to distinguish two different kinds of symmetric consecutive collision orbits and prove the above claim for both of them separately, one corresponding to a solar eclipse and the other to a lunar eclipse. By interpreting the orbits as Hamiltonian chords between two different Lagrangian submanifolds we can use a perturbed version of $G$-equivariant Lagrangian Rabinowitz Floer homology to prove the existence of this kind of consecutive collision orbit. To calculate this homology we show that under certain conditions the $G$-equivariant Lagrangian Rabinowitz Floer homology is equal to the Tate homology of $G$.