Coalescing binary systems of compact objects to (post)$^{5/2}-Newtonian order. V. Spin Effects

TL;DR

This work extends the PN treatment of coalescing binaries to include spin-orbit and spin-spin couplings, deriving spin contributions to the radiative multipoles, waveform, and energy/angular momentum losses within the Blanchet-Damour-Iyer formalism. It shows that spins induce orbital-plane precession and direct amplitude corrections, with spin-orbit effects significantly impacting the orbital phase and waveform, especially for systems with large mass ratios or rapidly spinning components; spin-spin effects are typically smaller. The analysis focuses on circular orbits, analyzes polarization states in a precessing frame, and provides detailed results for nonspinning, singly spinning, and doubly spinning binaries, including how detector orientation affects observed modulations. The findings inform gravitational-wave data analysis by clarifying when spin effects are detectable and how they influence parameter estimation, particularly the potential to extract spin information from waveform phase and amplitude modulations.

Abstract

We examine the effects of spin-orbit and spin-spin coupling on the inspiral of a coalescing binary system of spinning compact objects and on the gravitational radiation emitted therefrom. Using a formalism developed by Blanchet, Damour, and Iyer, we calculate the contributions due to the spins of the bodies to the symmetric trace-free radiative multipole moments which are used to calculate the waveform, energy loss, and angular momentum loss from the inspiralling binary. Using equations of motion which include terms due to spin-orbit and spin-spin coupling, we evolve the orbit of a coalescing binary and use the orbit to calculate the emitted gravitational waveform. We find the spins of the bodies affect the waveform in several ways: 1) The spin terms contribute to the orbital decay of the binary, and thus to the accumulated phase of the gravitational waveform. 2) The spins cause the orbital plane to precess, which changes the orientation of the orbital plane with respect to an observer, thus causing the shape of the waveform to be modulated. 3) The spins contribute directly to the amplitude of the waveform. We discuss the size and importance of spin effects for the case of two coalescing neutron stars, and for the case of a neutron star orbiting a rapidly rotating $10M_\odot$ black hole.

Figures (16)

• Figure 1: Amplitude modulation of gravitational waveforms by spin-induced orbital precession, plotted against time to coalescence. System consists of a nonspinning $1 M_\odot$ neutron star and a maximally spinning black hole of $10 M_\odot$. Spin and orbital angular momentum vectors are initially misaligned by $11.3^\circ$. Initial orbital inclination relative to ${\bf J}$ is $i$. The angle $\gamma$ represents the orientation of the detector relative to ${\bf J}$, with the detector located on the x-axis (see Fig. \ref{['source_coordinates']}) such that the source is directly overhead. Curves show envelope of the quadrupole waveform for various detector orientations (The curves for the cases $\gamma = 0$, $i/2$, and $i$ would lie on top of one another, so the first two have been shifted upward for ease of presentation). Gravitational-wave frequency runs from 10 Hz on the right to 300 Hz on the left.
• Figure 2: The source coordinate system. The total angular momentum ${\bf J}$ initially lies along the z-axis. The detector is located in the x-z plane. The spherical angles $(i,\alpha)$ define the direction of the Newtonian angular momentum ${\bf L_N}$ which is perpendicular to the orbital plane. In terms of celestial mechanics, the angle of ascending nodes is $\alpha + \pi/2$.
• Figure 3: A comparison of the magnitudes of the orbital angular momentum and spin angular momentum as the binary inspirals. (a) The equal mass case (assuming the bodies are maximally spinning). (b) The case of a 10:1 mass ratio.
• Figure 4: Gravitational wavefrom plotted against orbital phase for a 10:1.4 mass-ratio system. The smaller body's spin is aligned with the orbital angular momentum ${\bf L}$, while the larger body's spin is tilted by an angle of $30^\circ$ with respect to ${\bf L}$. Plotted is $(D/2\mu)h_+$ for an observer at $\Theta = 90^\circ$. Plots begin at an orbital separation of $15m$ and terminate at $10m$. (a) The total waveform. (b) The quadrupole contribution to the waveform. (c) The first higher-order post-Newtonian correction (${\cal O}(\epsilon^{1/2} )$ beyond the quadrupole). (d) The next post-Newtonian correction. (e) The leading-order spin-orbit contribution to the waveform. (f) The leading-order spin-spin contribution to the waveform (note the different scale). Notice the modulation due to the precession of the orbital plane.
• Figure 5: The wobble of the orbital plane during simple precession (in the absence of gravitational-radiation damping). Plotted are the $x$- and $y$-components of the unit vector ${\bf \hat{L}_N}$ which defines the orbital plane. The total angular momentum ${\bf J}$ is directed out of the page. If the orbital plane were not wobbling as it precessed, the plot would be a circle whose radius depends on the inclination of the orbital plane with respect to ${\bf J}$.