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 the letter Phys. Lett. B, 865, 139484 (2025), where we developed the analytical expression of the coordinate time in terms of the eccentric anomaly at the second post-Newtonian order in General Relativity 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 will focus on deriving the analytical expression of the coordinate time at the second Post-Newtonian order for circular orbits and then discuss two astrophysical applications involving binary neutron star and black hole systems.