Analytical coordinate time at the second post-Newtonian order

TL;DR

The paper derives the analytic expression for the coordinate time $t$ as a function of the eccentric anomaly $u$ at the $2PN$ order for eccentric compact binaries in GR. It uses a quasi-Keplerian parametrization and expands angular relations into an infinite cosine series that is truncated at $m_{\max}$ to control accuracy. The $2PN$ contribution $t_{2PN}(u)$ is expressed as $t_{2PN}(u)=D_1 A_1+D_2 A_2+D_3$ with $A_1$ and $A_2$ defined as arctan expressions of $u$ and eccentricities, and both $A_1$ and $A_2$ are made continuous by a common accumulation function $F(u)$. Numerical validation against direct integration shows a mean relative error around $0.3\%$ for $m_{\max}=10$, indicating practical usefulness for GW template generation and higher PN extensions.

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 $t(u)$. 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.

Figures (1)

• Figure 1: Numerical integration of $t(u)$ (black line) and analytical formula \ref{['eq:Int2']} (red dashed line) with $u\in[0,20\pi]$, where the coordinate time has been scaled by $GM$ (i.e., $t(GM)$). The following parameter values have been used: $m_1=1.60 M_{\odot}$, $m_2=1.17 M_{\odot}$, $\gamma=0.7$, $r_0(GM)=100M$, and $\dot{r}(0)=0$. The infinite sum has been truncated at $m_{\rm max}=10$. We have employed the same values exploited in Fig. 3 of Ref. LetteraDBA for facilitating the comparison with the 1PN accurate formula.