Compact binary system dynamics at the second post-Newtonian order: Analytical formula of the coordinate time for eccentric and circular orbits

This work is based on our published letter, where we developed the analytical expression of the coordinate time in terms of the eccentric anomaly at the second post-Newtonian (PN) order in general relativity (GR) for a compact binary system moving on eccentric orbits. The aim of this paper is to provide more details about the performed calculations and to produce other new results. More specifically, we focus on deriving the analytical expression of the coordinate time at the second PN order for circular orbits, and then discuss two astrophysical applications involving binary neutron star and black hole systems.