Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: The leading order spin-orbit interaction

TL;DR

The paper tackles the problem of obtaining an analytic, post-Newtonian, Keplerian-type parametrization for the conservative dynamics of spinning compact binaries in eccentric orbits, including leading order spin–orbit coupling. It develops a consistent framework by combining a 3PN nonspinning generalized quasi-Keplerian solution with the leading-order spin–orbit terms, yielding explicit parametric expressions for orbital elements and the time/angle evolution in two physically relevant cases: equal-mass binaries with two spins and single-spin binaries with arbitrary masses. The main contributions are explicit expressions for the radial and angular variables, the precession-enabled triad formalism, and the leading quadrupolar gravitational-wave polarizations, all expressed in ADM-type coordinates; these form the foundation for ready-to-use waveform templates and potential 2PN timing formulas for relativistic binaries. The work advances analytical control over spinning, eccentric PN dynamics, enabling efficient template construction, non-chaoticity assessments, and groundwork for extending to higher PN orders and spin-spin interactions with practical astrophysical applications such as gravitational-wave astronomy and binary pulsar timing.

Abstract

We derive Keplerian-type parametrization for the solution of post-Newtonian (PN) accurate conservative dynamics of spinning compact binaries moving in eccentric orbits. The PN accurate dynamics that we consider consists of the third post-Newtonian accurate conservative orbital dynamics influenced by the leading order spin effects, namely the leading order spin-orbit interactions. The orbital elements of the representation are explicitly given in terms of the conserved orbital energy, angular momentum and a quantity that characterizes the leading order spin-orbit interactions in Arnowitt, Deser, and Misner-type coordinates. Our parametric solution is applicable in the following two distinct cases: (i) the binary consists of equal mass compact objects, having two arbitrary spins, and (ii) the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin. As an application of our parametrization, we present gravitational wave polarizations, whose amplitudes are restricted to the leading quadrupolar order, suitable to describe gravitational radiation from spinning compact binaries moving in eccentric orbits. The present parametrization will be required to construct `ready to use' reference templates for gravitational waves from spinning compact binaries in inspiralling eccentric orbits. Our parametric solution for the post-Newtonian accurate conservative dynamics of spinning compact binaries clearly indicates, for the cases considered, the absence of chaos in these systems. Finally, we note that our parametrization provides the first step in deriving a fully second post-Newtonian accurate `timing formula', that may be useful for the radio observations of relativistic binary pulsars like J0737-3039.

Figures (2)

• Figure 1: The binary geometry and interpretations of various angles appearing in this section. Our reference frame is ($\boldsymbol{e}_{X}, \boldsymbol{e}_{Y},\boldsymbol{e}_{Z}$), where the basic vector $\boldsymbol{e}_{Z}$ is aligned with the fixed total angular momentum vector $\boldsymbol{J} = \boldsymbol{L} + \boldsymbol{S}$, such that $\boldsymbol{J} = J \boldsymbol{e}_Z$. The invariable plane ($\boldsymbol{e}_{X}$, $\boldsymbol{e}_{Y}$) is perpendicular to $\boldsymbol{J}$. Important for the observation is the line--of--sight unit vector $\boldsymbol{N}$ from the observer to the source (compact binary). We may have, by a clever choice of $\boldsymbol{e}_{X}$ and $\boldsymbol{e}_{Y}$, the line--of--sight unit vector $\boldsymbol{N}$ in the $\boldsymbol{e}_{Y}$--$\boldsymbol{e}_{Z}$--plane. $\boldsymbol{k} = \boldsymbol{L}/L$ is the unit vector in the direction of the orbital angular momentum $\boldsymbol{L}$, which is perpendicular to the orbital plane. The constant inclination of the orbital plane with respect to the invariable plane is $\Theta$, which is also the precession cone angle of $\boldsymbol{L}$ around $\boldsymbol{J}$. The orbital plane intersects the invariable plane at the line of nodes $\boldsymbol{i}$, with the longitude $\Upsilon$ measured in the invariable plane from $\boldsymbol{e}_{X}$. $\Upsilon$ is also the phase of the orbital plane precession. The orbital plane is spanned by the basic vectors ($\boldsymbol{i}, \boldsymbol{j}$), where $\boldsymbol{j} = \boldsymbol{k} \times \boldsymbol{i}$.
• Figure 2: The convention we adopted to link the orbital frame $(\boldsymbol{i},\boldsymbol{j},\boldsymbol{k})$, the invariable frame $(\boldsymbol{e}_{X},\boldsymbol{e}_{Y},\boldsymbol{e}_{Z})$ and the frame $(\boldsymbol{p},\boldsymbol{q},\boldsymbol{N})$ associated with the observer. Since we choose the line--of--sight unit vector $\boldsymbol{N}$ to lie in the $\boldsymbol{e}_{Y}$--$\boldsymbol{e}_{Z}$--plane, we may align the polarization vector $\boldsymbol{p}$ along $\boldsymbol{e}_{X}$, where the plane of the sky meets the invariable plane. This implies that $\boldsymbol{q}$ also lies in the $\boldsymbol{e}_{Y}$--$\boldsymbol{e}_{Z}$--plane. Therefore the frames $(\boldsymbol{p},\boldsymbol{q},\boldsymbol{N})$ and $(\boldsymbol{e}_{X},\boldsymbol{e}_{Y},\boldsymbol{e}_{Z})$ are connected by a constant angle $i_0$, which is the constant inclination angle between $\boldsymbol{N}$ and $\boldsymbol{J}$. The polarization vectors $\boldsymbol{p}$ and $\boldsymbol{q}$ span the plane of the sky. The inclination of this plane with respect to the orbital plane is the orbital inclination $i$. The inclination of the orbital plane with respect to the invariable plane is denoted by the constant angle $\Theta$.