I am just beginning a research project on the study of closed orbits, specifically as related to hydrogen, and I wanted a little bit more information on what exactly makes something a "closed orbit". In several of the following questions I will reference a thesis text that I am reading which is the basis for my attempt to computationally model the closed orbits of hydrogen.
$1.$ What model of the atom does closed orbit theory use? My intuition suggests that we should use the Bohr model, with a closed orbit occurring when the electron returns to the Bohr radius around the core. This seems simple and it allows us to remain in cartesian coordinates as there will be no singularity at $r = 0$. However, the thesis I am reading uses a semi-parabolic coordinate system and converts the Hydrogen Hamiltonian to a scaled energy form to eliminate the dependence on the external electric field and coulomb force. The author writes that this new form of scaled energy allows us to disregard the nucleus as the electron has no dependence on it. Why are we allowed to eliminate the proton in this model of hydrogen?
$2.$ I understand that closed orbit theory involves highly excited atoms but I am not sure what principal quantum $n$ is appropriate for the atom to be sufficiently excited. The thesis I am reading uses the value $n = 30$ but I don't understand why.
Thanks for the help! It is greatly appreciated!