Gravitational waveforms from unequal-mass binaries with arbitrary spins under leading order spin-orbit coupling

TL;DR

This work develops analytic gravitational-wave templates for spinning binaries with leading-order spin-orbit coupling in quasi-circular orbits and small mass asymmetry. By constructing a conservative 2PN Hamiltonian ${ m H}={ m H}_{ m N}+{ m H}_{ m 1PN}+{ m H}_{ m 2PN}+{ m H}_{ m SO}$ and introducing a rotating spin geometry, the authors derive first-principles evolution equations for three configuration angles and obtain perturbative first-order corrections about the equal-mass case. They compute the orbital motion and provide explicit quadrupolar waveforms $h_+$ and $h_ imes$ in terms of evolving spins and orbital angles, suitable for incorporation into GW templates. The work clarifies how unequal masses and general spin configurations modify precession, orbital phase, and the resulting quadrupolar GW signal, while noting future extensions to eccentricity and radiation reaction.

Abstract

The paper generalizes the structure of gravitational waves from orbiting spinning binaries under leading order spin-orbit coupling, as given in the work by Königsdörffer and Gopakumar [PRD 71, 024039 (2005)] for single-spin and equal-mass binaries, to unequal-mass binaries and arbitrary spin configurations. The orbital motion is taken to be quasi-circular and the fractional mass difference is assumed to be small against one. The emitted gravitational waveforms are given in analytic form.

Figures (2)

• Figure 1: Binary geometry completed by a rotating spin coordinate system. The usual reference frame is ($\boldsymbol{e_X}, \boldsymbol{e_Y}, \boldsymbol{e_Z}$) having chosen $\boldsymbol{e_Z}$ to be aligned with $\boldsymbol{J}$. The vectors $\boldsymbol{L}, \boldsymbol{S}_{1}, \boldsymbol{S}_{2}$ describe the orbital angular momentum and the individual spins, respectively. The angle $\Theta$ denotes the inclination angle of $\boldsymbol{L}$ w.r.t. $\boldsymbol{J}$, which is -- of course -- to be taken as a time dependent quantity. The orbital plane, being perpendicular to $\boldsymbol{L}$ by construction, is spanned by the orthonormal vectors $\boldsymbol{j}$ and $\boldsymbol{i}$, where the latter one intersects the invariable plane at the angle $\Upsilon$ measured from $\boldsymbol{e_X}$. The spin-coordinate system is constructed out of the orbital dreibein ($\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}$) by a rotation of $\alpha_\text{ks}$ around $\boldsymbol{i}$, such that the vector pointing from $\boldsymbol{L}$ to $\boldsymbol{J}$ is the total spin $\boldsymbol{S}_{1}+\boldsymbol{S}_{2}$. The angle $\alpha_{12}$ is measured between $\boldsymbol{S}_{1}$ and $\boldsymbol{S}_{2}$. The spin $\boldsymbol{S}_{1}$, projected into the ($\boldsymbol{j}_s, \boldsymbol{i}_s \equiv \boldsymbol{i}$) plane is rotated by an angle $\phi_s$ from $\boldsymbol{i}$, and $\boldsymbol{S}_{1}$ itself is moving on the circle (with variable radius) embedded in the figure.
• Figure 2: The geometry of the binary, having added the observer related frame $(\boldsymbol{p},\boldsymbol{q},\boldsymbol{N})$ (in dashed and dotted lines) with $\boldsymbol{N}$ as the line--of--sight vector, after removing the angles in the spin frame. The line--of--sight vector is chosen to lie in the $\boldsymbol{e}_{Y}$--$\boldsymbol{e}_{Z}$--plane, and measures an angle $i_0$ (associated with the rotation around $\boldsymbol{e_X}$) from $\boldsymbol{e_Z}$, such that $\boldsymbol{p} = \boldsymbol{e}_{X}$, and this is the point where the orbital plane meets the plane of the sky. Because of this rotation, the angle $i_0$ is also found between the vector $\boldsymbol{q}$, itself positioned in $(\boldsymbol{e_Y}, \boldsymbol{e_Z})$, too, and $\boldsymbol{e_Y}$. The grey area in the graphics completely lies in the orbital plane, spanned by $(\boldsymbol{i}, \boldsymbol{j})$ and $\varphi$ measures the angle between the separation vector $\boldsymbol{r}$ and $\boldsymbol{i}$. 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 $\Theta$. Please note that $\boldsymbol{L}$ does not lie on the unit sphere, only $\boldsymbol{k}$ does!