Aligned Spins: Orbital Elements, Decaying Orbits, and Last Stable Circular Orbit to high post-Newtonian Orders

TL;DR

The paper advances gravitational two-body dynamics by extending the quasi-Keplerian parameterisation to aligned-spin binaries, incorporating high-order conservative spin terms (3PN point-mass, NNLO spin-orbit, NNLO spin(1)-spin(2), and NLO spin-squared) and leading-order spin-dependent radiation losses. It presents explicit, gauge-consistent expressions for the orbital elements, Keplerian and orbital-phase parameters, and the energy/angular-momentum decay, all within a unified aligned-spin framework and tied to the gauge-invariant variable $x$. An important result is the LSO expression derived from a tuned Kerr-like binding energy, bridging Schwarzschild, Kerr, and PN regimes; the work also provides detailed transformations between harmonic and ADM formalisms and between covariant and canonical spin variables. Together, these contributions yield high-fidelity, analytic gravitational-wave templates for aligned-spin binaries, laying groundwork for future general-orbit and misaligned-spin generalisations and improving waveform modeling for data analysis. The approach underpins precision modeling of late-inspiral dynamics and enhances the interpretability of GW signals from black-hole and neutron-star binaries with aligned spins.

Abstract

In this article the quasi-Keplerian parameterisation for the case that spins and orbital angular momentum in a compact binary system are aligned or anti-aligned with the orbital angular momentum vector is extended to 3PN point-mass, next-to-next-to-leading order spin-orbit, next-to-next-to-leading order spin(1)-spin(2), and next-to-leading order spin-squared dynamics in the conservative regime. In a further step, we use the expressions for the radiative multipole moments with spin to leading order linear and quadratic in both spins to compute radiation losses of the orbital binding energy and angular momentum. Orbital averaged expressions for the decay of energy and eccentricity are provided. An expression for the last stable circular orbit is given in terms of the angular velocity type variable $x$.

Figures (2)

• Figure 1: Last stable circular orbit for $S=0.1$ plotted for different symmetric mass ratios $\eta$ and Kerr spins $a$. (From black to light grey $a=-1.0$ to $a=0.84$. The two uppermost plots contain the cases $a=0.8$ and $a=0.84$ respectively. The difference in Kerr spin between all other plots is $\Delta a = 0.2$.) The black square ($\blacksquare$) denotes the last stable circular orbit of a testmass orbiting a Schwarzschild black hole. Also notice that continuation to large Kerr spins $a$ is invalid because the last stable circular orbit will be of the order of magnitude of the Schwarzschild radius which violates the post-Newtonian approximation (wide separation). The reader should be reminded that the frequency-type quantity $x$ increases as the radius of a circular orbit decreases.
• Figure 2: The motion of the reduced mass (black dot) on an ellipse in the Newtonian case. $O$ denotes the origin and $F$ is one focus of the ellipse. Note that $v$ is not identical to the phase $\phi$ in the post-Newtonian case and loses its meaning as the angle between ${{{\mathbf{n}}}_{12}}{}$ and ${\mathbf{e}}_x$. The area enclosed by the ellipse, the $x$-axis and the vector ${\mathbf{r}}$ in the first quadrant equals the quantity ${\cal N}(t-t_0)$. This figure is taken from Memmesheimer:Gopakumar:Schafer:2004 and modified appropriately.