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9.5 The innermost circular orbit (ICO)

Having in hand the circular-orbit energy, we define the innermost circular orbit (ICO) as the minimum, when it exists, of the energy function E(x). Notice that we do not define the ICO as a point of dynamical general-relativistic unstability. Hence, we prefer to call this point the ICO rather than, strictly speaking, an innermost stable circular orbit or ISCO. A study of the dynamical stability of circular binary orbits in the post-Newtonian approximation of general relativity can be found in Ref. [43].

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 (Taylor-based) post-Newtonian approach with a recent result of numerical relativity by Gourgoulhon, Grandclément, and Bonazzola [123Jump To The Next Citation Point126Jump To The Next Citation Point]. 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 time-like inside the light cylinder associated with the circular motion, and space-like outside. In the numerical approach [123Jump To The Next Citation Point126Jump To The Next Citation Point] 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 so-called Isenberg-Wilson-Mathews approximation; see Ref. [114] for a discussion). Considering an evolutionary sequence of equilibrium configurations Refs. [123Jump To The Next Citation Point126Jump To The Next Citation Point] obtained numerically the circular-orbit energy E(w) and looked for the ICO of binary black holes (see also Refs. [52124154] for related calculations of binary neutron and strange quark stars).

Since the numerical calculation [123Jump To The Next Citation Point126Jump To The Next Citation Point] has been performed in the case of corotating black holes, which are spinning with the orbital angular velocity w, we must for the comparison include within our post-Newtonian 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 by34

2 2 S2 M = M irr +----2-, (195) 4M irr
where S is the spin, related to the usual Kerr parameter by S = M a, and Mirr is the irreducible mass given by V~ -- Mirr = A/(4p) (A is the hole’s surface area). The angular velocity of the corotating black hole is w = @M/@S hence, from Equation (195View Equation),
S w = -----[---- V~ -------]. (196) 2M 3 1 + 1 - S24 M
Physically this angular velocity is the one of the outgoing photons that remain for ever at the location of the light-like horizon. Combining Equations (195View Equation, 196View Equation) we obtain M and S as functions of Mirr and w,
M = V~ ---Mirr------, 1- 4M 2irrw2 (197) ----4M-i3rrw----- S = V~ -------2---2. 1- 4M irrw
This is the right thing to do since w 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. [123Jump To The Next Citation Point126Jump To The Next Citation Point]. In the limit of slow rotation we get
( 3) S = I w + O w , (198)
where I = 4M 3irr is the moment of inertia of the black hole. Next the total mass-energy is
1 2 ( 4) M = Mirr + 2I w + O w , (199)
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 (194View Equation) by their equivalent expressions in terms of w and the two irreducible masses. It is clear that the leading contribution is that of the spin kinetic energy given by Equation (199View Equation), and it comes from the replacement of the rest mass-energy 2 m c (where m = M1 + M2). From Equation (199View Equation) this effect is of order w2 in the case of corotating binaries, which means by comparison with Equation (194View Equation) that it is equivalent to an “orbital” effect at the 2PN order (i.e. oc x2). Higher-order corrections in Equation (199View Equation), which behave at least like w4, 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 8/3 w equivalent to 3PN order, which comes from the replacement of the masses into the “Newtonian” part, proportional to x oc w2/3, of the energy E (see Equation (194View Equation)). With the 3PN accuracy we do not need to replace the masses that enter into the post-Newtonian corrections in E, 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 spin-orbit (SO) interaction and the spin-spin (SS) one. In the case of spins S 1 and S 2 aligned parallel to the orbital angular momentum (and right-handed with respect to the sense of motion) the SO energy reads

[( 2 ) ( 2 ) ] 5/3 4-M-1- S1-- 4-M-2- -S2- ESO = - m(mw) 3 m2 + n M 21 + 3 m2 + n M 22 . (200)
Here we are employing the formula given by Kidder et al. [146Jump To The Next Citation Point144Jump To The Next Citation Point] (based on seminal works of Barker and O’Connell [78]) who have computed the SO contribution and expressed it by means of the orbital frequency w. The derivation of Equation (200View Equation) in Ref. [146Jump To The Next Citation Point144Jump To The Next Citation Point] takes into account the fact that the relation between the orbital separation r (in the harmonic coordinate system) and the frequency w depends on the spins. We immediately infer from Equation (200View Equation) 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 (200View Equation) 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
2 S1 S2 ESS = mn (mw) ---2--2. (201) M 1 M 2
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 [23Jump To The Next Citation Point]
corot 2 { 2 ( 2) 3 4} DE = m c x (2- 6n)x + - 6n + 13n x + O(x ) , (202)
where x = (mw)2/3. The complete 3PN energy of the corotating binary is finally given by the sum of Equations (194View Equation) and (202View Equation), 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.
View Image

Figure 1: Results for the binding energy EICO versus wICO in the equal-mass case (n = 1/4). The asterisk marks the result calculated by numerical relativity. The points indicated by 1PN, 2PN, and 3PN are computed from the minimum of Equation (194View Equation), and correspond to irrotational binaries. The points denoted by corot 1PN, corot 2PN, and corot 3PN come from the minimum of the sum of Equations (194View Equation) and (202View Equation), and describe corotational binaries.
The Figure 1View Image (issued from Ref. [23Jump To The Next Citation Point]) presents our results for E ICO in the case of irrotational and corotational binaries. Since corot DE, given by Equation (202View Equation), is at least of order 2PN, the result for corot 1PN is the same as for 1PN in the irrotational case; then, obviously, corot 2PN takes into account only the leading 2PN corotation effect (i.e. the spin kinetic energy given by Equation (199View Equation)), while 3PNcorot involves also, in particular, the corotational SO coupling at the 3PN order. In addition we present in Figure 1View Image the numerical point obtained by numerical relativity under the assumptions of conformal flatness and of helical symmetry [123126]. As we can see the 3PN points, and even the 2PN ones, are rather close to the numerical value. The fact that the 2PN and 3PN values are so close to each other is a good sign of the convergence of the expansion; we shall further comment this point in Section 9.6. In fact one might say that the role of the 3PN approximation is merely to “confirm” the value already given by the 2PN one (but of course, had we not computed the 3PN term, we would not be able to trust very much the 2PN value). As expected, the best agreement we obtain is for the 3PN approximation and in the case of corotation, i.e. the point 3PNcorot. However, the 1PN approximation is clearly not precise enough, but this is not surprising in the highly relativistic regime of the ICO.

In conclusion, we find that the location of the ICO as computed by numerical relativity, under the helical-symmetry and conformal-flatness approximations, is in good agreement with the post-Newtonian prediction. See also Ref. [88] for the results calculated within the effective-one-body approach method [60Jump To The Next Citation Point61Jump To The Next Citation Point] at the 3PN order, which are close to the ones reported in Figure 1View Image. This agreement constitutes an appreciable improvement of the previous situation, because the earlier estimates of the ICO in post-Newtonian theory [145] and numerical relativity [1809] strongly disagreed with each other, and do not match with the present 3PN results. The numerical calculation of quasi-equilibrium 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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