Third post-Newtonian accurate generalized quasi-Keplerian parametrization for compact binaries in eccentric orbits

TL;DR

The paper develops a 3PN accurate Kepler-type parametrization for eccentric compact binaries, providing a unified framework in both ADM-type and harmonic coordinates. It builds a generalized quasi-Keplerian form with $r=a_r(1-e_r\cos u)$ and a detailed Kepler-like equation for the mean anomaly, incorporating PN corrections through $n$, $\Phi$, and higher harmonics via $v$-dependent terms. Explicit 3PN expressions for orbital elements $(a_r,e_r,n,e_t,\Phi,e_\phi)$ are given as functions of the conserved energy $E$, angular momentum $h$, and mass ratio $\eta$, with gauge-invariant quantities $n$ and $\Phi$ corroborated across gauges. This parametrization enables ready-to-use 3PN templates for eccentric inspirals, provides a robust PN diagnostic tool for simulations, and supports future extensions to radiation-reaction effects and precise pulsar timing analyses.

Abstract

We present Keplerian-type parametrization for the solution of third post-Newtonian (3PN) accurate equations of motion for two non-spinning compact objects moving in an eccentric orbit. The orbital elements of the parametrization are explicitly given in terms of the 3PN accurate conserved orbital energy and angular momentum in both Arnowitt, Deser, and Misner-type and harmonic coordinates. Our representation will be required to construct post-Newtonian accurate `ready to use' search templates for the detection of gravitational waves from compact binaries in inspiralling eccentric orbits. Due to the presence of certain 3PN accurate gauge invariant orbital elements, the parametrization should be useful to analyze the compatibility of general relativistic numerical simulations involving compact binaries with the corresponding post-Newtonian descriptions. If required, the present parametrization will also be needed to compute post-Newtonian corrections to the currently employed `timing formula' for the radio observations of relativistic binary pulsars.

Figures (1)

• Figure 1: The geometrical interpretation of the eccentric and true anomalies, $u$ and $v$, appearing in the Keplerian parametrization. The auxiliary circle of radius $a$ circumscribes the orbital ellipse with semi-major axis $a$. The points $O$ and $F$ stand for the origin and the focus of the ellipse.