Analytical coordinate time at the second post-Newtonian order

Abstract

We derive the analytical expression of the coordinate time t in terms of the eccentric anomaly u at the second post-Newtonian order in General Relativity for a compact binary system moving on eccentric orbits. The parametrization of t with u permits to reduce at the minimum the presence of discontinuous trigonometric functions. This is helpful as they must be properly connected via accumulation functions to finally have a smooth coordinate time $t\left(u\right)$. Another difficulty relies on the presence of an infinite sum, about which we derive a compact form. This effort reveals to be extremely useful for application purposes. Indeed, we need to truncate the aforementioned sum to a certain finite threshold, which strongly depends on the selected parameter values and the accuracy error we would like to achieve. Thanks to our work, this analysis can be easily carried out.