The Innermost Stable Circular Orbit of Binary Black Holes

TL;DR

This work addresses locating the innermost stable circular orbit (ISCO) for binary black holes by constructing quasicircular initial data that combine Cook's conformal-imaging framework with Brandt-Brügmann's puncture method in a three-sheeted topology. The approach solves the momentum constraint analytically using Bowen-York extrinsic curvature and the Hamiltonian constraint numerically via the puncture ansatz, focusing on equal-mass, non-spinning binaries and minimizing the binding energy $E_b=E-2M$ along sequences of constant horizon area to identify ISCO. The main finding is that ISCO parameters agree with Cook's results within a few percent, indicating the underlying manifold structure has only a small effect on the ISCO and that the method is simpler while enabling potential extensions beyond conformal flatness. This provides a practical framework for generating binary black hole initial data and assessing ISCO sensitivity to background geometry, with implications for gravitational-wave modeling and numerical relativity.

Abstract

We introduce a new method to construct solutions to the constraint equations of general relativity describing binary black holes in quasicircular orbit. Black hole pairs with arbitrary momenta can be constructed with a simple method recently suggested by Brandt and Bruegmann, and quasicircular orbits can then be found by locating a minimum in the binding energy along sequences of constant horizon area. This approach produces binary black holes in a "three-sheeted" manifold structure, as opposed to the "two-sheeted" structure in the conformal-imaging approach adopted earlier by Cook. We focus on locating the innermost stable circular orbit and compare with earlier calculations. Our results confirm those of Cook and imply that the underlying manifold structure has a very small effect on the location of the innermost stable circular orbit.

Figures (3)

• Figure 1: The binding energy $\bar{E}_b$ for $\bar{C} = 2.5$ and $\bar{J} = 3.0$ for different locations of the outer boundary $\bar{Z}_{\rm out}$ (at constant grid resolution $\bar{h} = 0.03125$, top panel) and different grid resolutions $\bar{h}$ (at constant outer boundary $\bar{Z}_{\rm out} = 6$, bottom panel). The dashed line only includes contributions to the ADM mass from inside the computational grid, and the solid line denotes the corrected value (see text).
• Figure 2: Contours of the conformal factor $\psi$ for a configuration close to the innermost stable circular orbit ($\bar{C} = 2.25$ and $\bar{J} = 2.95$). The contours (solid lines) logarithmically span the interval $\psi = 1$ and $\psi = 9.2$. Note that the apparent horizons, marked by the thick dashed lines, are not concentric with the contours of the conformal factor. Instead, they are dragged along by the black holes and lag slightly behind in their (counter-clockwise) orbit.
• Figure 3: The effective potential $\bar{E}_b$ as a function of proper separation $\bar{l}$ for the following values of the angular momentum $\bar{J}$: 2.9, 2.92, 2.94, 2.95, 2.96, 2.97, 2.98, 3.00, 3.02, 3.06, 3.10, 3.15, 3.20, 3.25 (from bottom to top). Quasicircular orbits correspond to minima in the effective potential. The bold line connects these minima and represents a sequence of quasicircular orbits. This sequence terminates at the innermost stable circular orbit.