Spin-orbit resonance and the evolution of compact binary systems

Abstract

Starting with a post-Newtonian description of compact binary systems, we derive a set of equations that describes the evolution of the orbital angular momentum and both spin vectors during inspiral. We find regions of phase space that exhibit resonance behavior, characterized by small librations of the spin vectors around a fixed orientation. Due to the loss of energy and orbital angular momentum through radiation reaction, systems can eventually be captured into these resonance orientations. By investigating the long-term evolution of compact binaries with a variety of initial conditions, we find that the distribution in parameter space can be strongly affected by resonance captures. This has the effect of significantly reducing the size of search space for gravitational wave sources, in turn improving the chances of detecting such sources through methods of template matching. Furthermore, by calculating the expected spin distribution at the end of the inspiral phase, we can predict what are the most likely initial conditions for the plunge phase, a result of great interest for numerical relativity calculations.

Effects of post-Newtonian spin alignment on the distribution of black-hole recoils

“…The gravitationalwave inspiral time is inversely proportional to the binary's symmetric mass ratio η = q/(1 + q) 2 [60], making it hard to preserve accuracy over long PN time evolutions for q < 0.1. However, Schnittman [20] showed that spinorbit resonances only become important inside a "resonance locking" radius r lock /M ≈ [(1 + q 2 )/(1 − q 2 )] 2 . We are therefore justified in treating all binaries with q ≤ 0.1 (for which PN evolutions from r i = 1000M to r f = 10M become problematic) as if the inspiral did not happen: that is, we can compute recoils of binaries with q < 0.1 by simply applying the recoil formula at r = r i .…”

“…Because of these uncertainties, in this paper we simply assume χ 1 = χ 2 = χ for χ = 0.5, 0.75, and 1.0, and we provide predictions for each of these three values. Schnittman [20] showed that the amount of PN spin alignment is insensitive to the spin magnitude for χ 0.5 (see Fig. 11 in his paper).…”

“…At large separations where |L| ≫ |χ i |, Eq. (3.5) of [20] indicates that the constraint for defining a spin-orbit resonance depends only on θ i , not χ i . We therefore expect that PN spin alignment should not be particularly sensitive to our assumption that χ 1 = χ 2 , although additional numerical studies would be required to confirm this expectation.…”

“…Resonance effects are not present for q = 1, but astrophysical binaries are not expected to be precisely equal in mass. For this reason we have performed Monte Carlo PN evolutions of 900 black hole binaries with mass ratio close to but not exactly equal to one (we choose q = 9/11 to facilitate comparisons with Schnittman [20] and our own earlier work [21,22] 1]. We begin at an initial separation r i = 1000M and use the PN equations of motion to evolve the binaries down to a final separation r f = 10M (cf.…”

“…Because of the steep scaling t GR ∝ r 4 mentioned previously, the SBHs decouple from their circumbinary disk and spiral inwards purely under the influence of gravitational radiation at a binary separation r dec ≃ 10 2 − 10 3 M [12], where M = m 1 + m 2 is the total mass of the binary. Schnittman [20] integrated PN equations of motion and spin precession from an initial binary separation r i = 1000M to a final separation r f = 10M (in geometrical units where G = c = 1) and discovered that partially aligned SBH spin distributions during this portion of the inspiral are strongly influenced by the presence of spin-orbit resonances.…”