The previous definition of the ICO is motivated by our comparison with the results of numerical relativity. Indeed we shall confront the prediction of the standard (Taylorbased) postNewtonian approach with a recent result of numerical relativity by Gourgoulhon, Grandclément, and Bonazzola [123, 126]. These authors computed numerically the energy of binary black holes under the assumptions of conformal flatness for the spatial metric and of exactly circular orbits. The latter restriction is implemented by requiring the existence of an “helical” Killing vector, which is timelike inside the light cylinder associated with the circular motion, and spacelike outside. In the numerical approach [123, 126] there are no gravitational waves, the field is periodic in time, and the gravitational potentials tend to zero at spatial infinity within a restricted model equivalent to solving five out of the ten Einstein field equations (the socalled IsenbergWilsonMathews approximation; see Ref. [114] for a discussion). Considering an evolutionary sequence of equilibrium configurations Refs. [123, 126] obtained numerically the circularorbit energy and looked for the ICO of binary black holes (see also Refs. [52, 124, 154] for related calculations of binary neutron and strange quark stars).
Since the numerical calculation [123, 126] has been performed in the case of corotating black holes, which are spinning with the orbital angular velocity , we must for the comparison include within our postNewtonian formalism the effects of spins appropriate to two Kerr black holes rotating at the orbital rate. The total relativistic mass of the Kerr black hole is given by^{34}
where is the spin, related to the usual Kerr parameter by , and is the irreducible mass given by ( is the hole’s surface area). The angular velocity of the corotating black hole is hence, from Equation (195), Physically this angular velocity is the one of the outgoing photons that remain for ever at the location of the lightlike horizon. Combining Equations (195, 196) we obtain and as functions of and , This is the right thing to do since is the basic variable describing each equilibrium configuration calculated numerically, and because the irreducible masses are the ones which are held constant along the numerical evolutionary sequences in Refs. [123, 126]. In the limit of slow rotation we get where is the moment of inertia of the black hole. Next the total massenergy is which involves, as we see, the usual kinetic energy of the spin.To take into account the spin effects our first task is to replace all the masses entering the energy function (194) by their equivalent expressions in terms of and the two irreducible masses. It is clear that the leading contribution is that of the spin kinetic energy given by Equation (199), and it comes from the replacement of the rest massenergy (where ). From Equation (199) this effect is of order in the case of corotating binaries, which means by comparison with Equation (194) that it is equivalent to an “orbital” effect at the 2PN order (i.e. ). Higherorder corrections in Equation (199), which behave at least like , will correspond to the orbital 5PN order at least and are negligible for the present purpose. In addition there will be a subdominant contribution, of the order of equivalent to 3PN order, which comes from the replacement of the masses into the “Newtonian” part, proportional to , of the energy (see Equation (194)). With the 3PN accuracy we do not need to replace the masses that enter into the postNewtonian corrections in , so in these terms the masses can be considered to be the irreducible ones.
Our second task is to include the specific relativistic effects due to the spins, namely the spinorbit (SO) interaction and the spinspin (SS) one. In the case of spins and aligned parallel to the orbital angular momentum (and righthanded with respect to the sense of motion) the SO energy reads
Here we are employing the formula given by Kidder et al. [146, 144] (based on seminal works of Barker and O’Connell [7, 8]) who have computed the SO contribution and expressed it by means of the orbital frequency . The derivation of Equation (200) in Ref. [146, 144] takes into account the fact that the relation between the orbital separation (in the harmonic coordinate system) and the frequency depends on the spins. We immediately infer from Equation (200) that in the case of corotating black holes the SO effect is equivalent to a 3PN orbital effect and thus must be retained with the present accuracy (with this approximation, the masses in Equation (200) are the irreducible ones). As for the SS interaction (still in the case of spins aligned with the orbital angular momentum) it is given by The SS effect can be neglected here because it is of order 5PN for corotating systems. Summing up all the spin contributions we find that the suplementary energy due to the corotating spins is [23] where . The complete 3PN energy of the corotating binary is finally given by the sum of Equations (194) and (202), in which we must now understand all the masses as being the irreducible ones (we no longer indicate the superscript “irr”), which for the comparison with the numerical calculation must be assumed to stay constant when the binary evolves.

In conclusion, we find that the location of the ICO as computed by numerical relativity, under the helicalsymmetry and conformalflatness approximations, is in good agreement with the postNewtonian prediction. See also Ref. [88] for the results calculated within the effectiveonebody approach method [60, 61] at the 3PN order, which are close to the ones reported in Figure 1. This agreement constitutes an appreciable improvement of the previous situation, because the earlier estimates of the ICO in postNewtonian theory [145] and numerical relativity [180, 9] strongly disagreed with each other, and do not match with the present 3PN results. The numerical calculation of quasiequilibrium configurations has been since then redone and refined by a number of groups, for both corotational and irrotational binaries (see in particular Ref. [74]). These works confirm the previous findings.
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