Innermost circular orbit of binary black holes at the third post-Newtonian approximation

Abstract

The equations of motion of two point masses have recently been derived at the 3PN approximation of general relativity. From that work we determine the location of the innermost circular orbit or ICO, defined by the minimum of the binary's 3PN energy as a function of the orbital frequency for circular orbits. We find that the post-Newtonian series converges well for equal masses. Spin effects appropriate to corotational black-hole binaries are included. We compare the result with a recent numerical calculation of the ICO in the case of two black holes moving on exactly circular orbits (helical symmetry). The agreement is remarkably good, indicating that the 3PN approximation is adequate to locate the ICO of two black holes with comparable masses. This conclusion is reached with the post-Newtonian expansion expressed in the standard Taylor form, without using resummation techniques such as Padé approximants and/or effective-one-body methods.

Figures (5)

• Figure 1: The possible solutions as a function of the regularization constant $\lambda$. There is no solution when $\lambda<\lambda_0(\nu)$, two possible solutions when $\lambda_0(\nu)\leq\lambda<\lambda_1(\nu)$ [which become degenerate at $\lambda=\lambda_0(\nu)$], and a unique solution when $\lambda_1(\nu)\leq\lambda$. The upper branch, existing between $\lambda_0(\nu)$ and the vertical asymptote at $\lambda=\lambda_1(\nu)$, is actually a maximum of the energy.
• Figure 2: The 3PN energy function $E(\Omega)$ for equal-mass binaries and $\omega_{\rm static}=0$.
• Figure 3: Results for $E_{\rm ICO}$ versus $\Omega_{\rm ICO}$ in the equal-mass case. The asterisk marks the result calculated by numerical relativity. The points indicated by 1PN, 2PN and 3PN are computed from Eq. (\ref{['2']}), and correspond to irrotational binaries. The points denoted by 1PN$^{\rm corot}$, 2PN$^{\rm corot}$ and 3PN$^{\rm corot}$ come from the sum of Eqs. (\ref{['2']}) and (\ref{['10']}), and describe corotational binaries. Both 3PN are 3PN$^{\rm corot}$ are shown for $\omega_{\rm static}=0$.
• Figure 4: Results for $E_{\rm ICO}$ in terms of $\Omega_{\rm ICO}$ in the equal-mass case. The $e$ and $j$-methods are Padé approximants at the 3PN order. EOB refers to the effective-one-body approach at the 3PN order. The points marked by 2PN and 3PN correspond to the standard Taylor post-Newtonian series (this work). The results for Padé, EOB and Taylor are for irrotational binaries.
• Figure 5: Same as FIG. \ref{['fig4']} but for the angular-momentum $J_{\rm ICO}$.