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

TL;DR

We address the problem of obtaining an analytic coordinate-time function $T(u)$ at the $2PN$ order for compact binaries on eccentric and circular orbits. The method decomposes the integrating function $\tau(u)$ into $0$PN, $1$PN, and $2$PN contributions, and to handle the nonlinear $v(u)$ relation rewrites certain terms as finite truncated sums plus an accumulation function to enforce continuity. The resulting $T(u)$ is expressed as $t_{\rm 0PN}+c^{-2}t_{\rm 1PN}+c^{-4}t_{\rm 2PN}$, with the $2PN$ piece taking the form $t_{\rm 2PN}(u)=D_1\bar{\mathcal{A}}_1+D_2\bar{\mathcal{A}}_2+D_3$, where $\bar{\mathcal{A}}_i$ are accumulation-corrected arctangent functions. Applications to six binary neutron-star systems and six binary black-hole configurations demonstrate good agreement with observations or simulations (mean relative errors from $2\times10^{-3}\%$ to about $4\%$). The work provides a practical, fast route to analytical templates and motivates extensions to higher PN orders and spin.

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.

Figures (3)

• Figure 1: Plot of the function $\chi$ versus the eccentric anomaly $u\in[0,2\pi]$ for different values of $m_{\rm max}=1,\dots,15$.
• Figure 2: Three-dimensional plot showing the link among $\Delta(i),r_0,e_0$ for ten different fixed values of $m_{\rm max}$.
• Figure 3: Numerical integration of $T(u)$ (black line) and analytical formula (red dashed line) with $u\in[0,40\pi]$. The following parameter values have been used: $m_1=1.60 M_{\odot}$, $m_2=1.17 M_{\odot}$, $\gamma=0.7$, $R_0=r_0(GM)=100M$, and $\dot{r}(0)=0$. The infinite sums have been truncated at $m_{\rm max}=10$.