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

TL;DR

This paper derives analytical formulas for the coordinate time at the second post-Newtonian order for eccentric and circular orbits in compact binary systems, aiding astrophysical modeling of neutron star and black hole binaries.

Contribution

It provides detailed derivations of the coordinate time expressions at 2PN order for both eccentric and circular orbits, extending previous work and enabling precise astrophysical applications.

Findings

Analytical formula for coordinate time in eccentric orbits at 2PN order.

Analytical formula for coordinate time in circular orbits at 2PN order.

Application to binary neutron star and black hole systems.

Abstract

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.