ALIGNMENT OF SUPERMASSIVE BLACK HOLE BINARY ORBITS AND SPINS

Analyses of the Bardeen–Petterson effect indicate that the warp transition radius, which is the innermost radius at which there is significant inclination to the black hole spin axis and therefore dominates the alignment rate, is approximately where the rate at which aligning angular momentum due to the Lense–Thirring torque is transported outward matches the rate at which misaligned angular momentum from the outer disk is transported inward (Nelson & Papaloizou 2000; Sorathia et al. 2013b). Although misaligned angular momentum transport is not well-described by diffusion (Lodato & Price 2010; Sorathia et al. 2013a), we will nonetheless suppose that, at the order of magnitude level, the rate of this process does scale in the way a diffusive process would, and that the radial scale of misaligned angular momentum gradients near radius r is ∼r. Then the rate at which the local misaligned angular momentum changes due to this mixing is ∼fαΩ(h/r)². The factor f encapsulates several uncertainties about the gas alignment process. One, as we have already noted, is the assumption that this process resembles diffusion with radial gradient scales ∼r. Another is the intrinsic rate of inward misaligned angular momentum flow. In the original Bardeen–Petterson paper, f = 1, i.e., radial misaligned angular momentum transport is entirely due to mass accretion. On the other hand, Papaloizou & Pringle (1983) argued that if the stress responsible for accretion acted like an isotropic viscosity, f ∼ α⁻². In smoothed particle hydrodynamic simulations incorporating that assumption, Lodato & Pringle (2007) and Lodato & Price (2010) found that in fact f ≲ 3/α when, as would be expected, the warp is nonlinear. Warp nonlinearity is defined by the criterion $|d \hat{\ell }/d\ln r| < h/r$, for $\hat{\ell }$ a unit vector in the direction of the angular momentum averaged over a radial shell at r. More recently, on the basis of MHD simulations without any sort of phenomenological viscosity prescriptions, Sorathia et al. (2013b) have argued that 1 ≲ f ≲ α⁻¹. In their analysis, $f \sim \alpha ^{-1} {\cal M}^2$ when it is primarily due to radial mixing motions with no net mass inflow, where ${\cal M}$ is the typical Mach number of radial flows induced by the warp. Sorathia et al. (2013a) found that warps generically drive transonic radial flows whose Mach number can be either larger or smaller than unity by factors of several.

For an object located at radius r outside an axisymmetric object of radius R, planetary dynamics theory (e.g., Murray & Dermott 1999) indicates that the precession rate of the line of nodes is

$$\begin{equation} \omega _{\rm bin}={3\over 2}J_2\Omega (R/r)^2\;, \end{equation} \tag{ 1 }$$

where the numerical coefficient

$$\begin{eqnarray} &&J_2={1\over {mR^2}}\int _0^R \, \int _{-1}^{+1} \, d(\cos \theta) \, dr \, \pi r^4 \, (3\cos ^2\theta -1) \rho (r,\cos \theta). \nonumber\\ \end{eqnarray} \tag{ 2 }$$

Here ρ is the mass per unit volume. Note that at a given r, ω_bin is independent of the precise choice of the outer boundary R provided it is large enough to contain all the gravitating mass. The time-averaged mass distribution associated with a circular binary of mass ratio q is two axisymmetric rings; for this configuration, J₂ = −η/2 if one chooses R = a. Thus the line of nodes undergoes retrograde precession, unlike the prograde precession produced by general relativistic frame-dragging. If the orbit is moderately eccentric there is only a modest change in the coefficient because what drives precession is the time-averaged quadrupole moment of the binary.

At the order of magnitude level, the rate at which aligning angular momentum is delivered is ∼ω_binL_⊥, where ω_bin is the precession rate due to the quadrupole moment and L_⊥ is the local misaligned angular momentum (Larwood & Papaloizou 1997). However, Sorathia et al. (2013b) have recently shown that in the context of Lense–Thirring torques, the rate of aligning angular momentum delivery can differ from this estimate by a factor of order unity, which we will call ${\cal I}$. This quantity is composed of two multiplicative factors. One is a dimensionless integral accounting for the fact that the very strong radial dependence of the precession frequency (∼r⁻³ for Lense–Thirring precession, ∼r^−7/2 for classical quadrupolar precession) means that regions of small misalignment at small radius can be disproportionately strong loci of torque; it has the form

$$\begin{equation} \int _0^1 \, dx \, x^{-3/2} \frac{\sin \delta (x)}{\sin \delta _{\rm BP}} \frac{\Sigma (x)}{\Sigma _{\rm BP}}. \end{equation} \tag{ 3 }$$

Here the radius has been non-dimensionalized in units of R_BP, the Bardeen–Petterson alignment radius. Quantities subscripted "BP" are evaluated at R_BP. These include Σ, the surface density, and δ, the misalignment angle. The other factor accounts for the fact that the direction of the angular momentum carried through the disk to the alignment front is not necessarily exactly opposite the direction of misaligned angular momentum; Sorathia et al. (2013b) estimate that this factor is ≃ 0.5. A very similar formalism should apply to the interaction of a binary with its surrounding disk. The only difference is that the radial coordinate in the dimensionless integral is normalized to R_bin, the radius out to which the circumbinary disk is aligned with the binary orbital plane, and the power of x in the integrand is −2 rather than −3/2.

With this refinement, we may estimate

$$\begin{equation} {R_{\rm bin}\over a}=\left(3 \eta {\cal I}_{\rm bin}\over {4f \alpha }\right)^{1/2}\left(R_{\rm bin}\over h_{\rm bin}\right). \end{equation} \tag{ 4 }$$

Similarly, Lense–Thirring precession produces a transition radius r_BP in each minidisk given by

$$\begin{equation} {r_{\rm BP}\over r_g}\approx \left(2a_* {\cal I}_{\rm BP} \over {f \alpha }\right)^{2/3}\left(r_{\rm BP}\over h_{\rm BP}\right)^{4/3}. \end{equation} \tag{ 5 }$$

We distinguish radii with respect to the center of mass of the system from those within the minidisks by using an upper-case R for the former and a lower-case r for the latter.

The torque on the binary is the radial integral of the precession rate times the misaligned angular momentum i.e.,

$$\begin{equation} N_{\rm bin} = -\eta \pi \frac{{GMa}^2}{R_{\rm bin}} \sin \delta _{\rm bin} {\cal I}_{\rm bin} \Sigma _{\rm bin}. \end{equation} \tag{ 6 }$$

For this estimate, we make several simplifying approximations. We ignore misaligned angular momentum transferred to the binary through accretion, in the expectation that R_bin ≫ a, so that the accreted matter has already been aligned. We also assume that the orientation of the circumbinary material is nearly constant, as is consistent with recent numerical simulations (Maio et al. 2013). If it is not (as in the chaotic accretion scenario of King & Pringle 2006), any changes will only lengthen the orbital plane alignment time T_bin. The Bardeen–Petterson torque is

$$\begin{equation} N_{\rm BP} = 4\pi a_* (r_g c)^2 \left(\frac{r_g}{r_{\rm BP}}\right)^{1/2} \sin \delta _{\rm BP} {\cal I}_{\rm BP} \Sigma _{\rm BP}. \end{equation} \tag{ 7 }$$

For time-steady disks heated only by local accretion, Shakura & Sunyaev (1973) find that when gas pressure exceeds radiation pressure and scattering opacity exceeds free–free opacity (their "middle region"),

$$\begin{equation} \Sigma =2\times 10^7\ {\rm g\ cm}^{-2}\alpha ^{-4/5}({\dot{M}}/{\dot{M}}_{\rm Edd})^{3/5}M_8^{1/5}(R/R_g)^{-3/5}. \end{equation} \tag{ 8 }$$

Here ${\dot{M}}_{\rm Edd}=4\pi {GM}/(\epsilon \kappa c)$ for opacity κ and accretion efficiency $\epsilon \equiv L/({\dot{M}}c^2)$, and M₈ = M/10⁸ M_☉. In the outer Shakura–Sunyaev disk region, where gas pressure exceeds radiation pressure and free–free opacity exceeds scattering opacity,

$$\begin{equation} \Sigma =9\times 10^7\ {\rm g\ cm}^{-2}\alpha ^{-4/5}({\dot{M}}/{\dot{M}}_{\rm Edd})^{7/10}M_8^{1/5}(R/R_g)^{-3/4}. \end{equation} \tag{ 9 }$$

In both these estimates we assume R ≫ R_g.

The similarity in form of these two expressions suggests that we write

$$\begin{equation*} \Sigma =\Sigma _0\alpha ^{-4/5}({\dot{M}}/{\dot{M}}_{\rm Edd})^\gamma M_8^{1/5}(R/R_g)^{-\beta }, \end{equation*}$$

where R = R_bin for the binary torques and R = r_BP for the Bardeen–Petterson torques (and for the Bardeen–Petterson torques ${\dot{M}}\rightarrow {\dot{m}}$ and R_g → r_g). Similarly, M₈ = M/10⁸ M_☉ for the circumbinary disk but it is m₁/10⁸ M_☉ or m₂/10⁸ M_☉ for the minidisks. If, as we shall assume, the transition radius in the circumbinary disk is in the same Shakura–Sunyaev region as the transition radii in the minidisks, then Σ₀, β, and γ are the same for the circumbinary disk as for the minidisks. We note that the few simulations that have been performed in which accretion from the inner edge of a circumbinary disk is distributed between the partners in the binary are consistent with ${\dot{m}}$ being slightly larger for the lower-mass black hole, hence for q significantly smaller than unity it could be that ${\dot{m}}_2/{\dot{m}}_{\rm 2,Edd}$ is considerably larger than ${\dot{m}}_1/{\dot{m}}_{\rm 1,Edd}$.

Rewriting the torques in terms of this notation, we have

$$\begin{eqnarray} N_{\rm bin} &=& {-} \pi \eta (ac)^2 \sin \delta _{\rm bin} {\cal I}_{\rm bin} \Sigma _0 \alpha ^{-4/5} ({\dot{M}}/{\dot{M}}_{\rm Edd})^\gamma M_8^{1/5} \nonumber\\ &&\times(R_{\rm bin}/R_g)^{\beta -1}. \end{eqnarray} \tag{ 10 }$$

Similarly, the Bardeen–Petterson torque on a black hole is

$$\begin{eqnarray} N_{\rm BP} &=& 4\pi a_* (r_g c)^2 \sin \delta _{\rm BP} {\cal I}_{\rm BP} \Sigma _0 \alpha ^{-4/5} ({\dot{m}}/{\dot{m}}_{\rm Edd})^\gamma m_8^{1/5} \nonumber\\ &&\times (r_{\rm BP}/r_g)^{\beta -1/2}. \end{eqnarray} \tag{ 11 }$$

The angular momentum of the binary L_bin = ηM(GMa)^1/2; that of a black hole is L_BH = a_*(G/c)m². After some manipulation, we find that the characteristic time to align the binary is

$$\begin{eqnarray} &&T_{\rm bin} = \frac{1}{\pi } \frac{\alpha ^{4/5}}{ ({\dot{M}}/{\dot{M}}_{\rm Edd})^\gamma M_8^{1/5}}\frac{T_0}{\sin \delta _{\rm bin}{\cal I}_{\rm bin}} \left(\frac{R_{\rm bin}}{R_g}\right)^{1-\beta } \left(\frac{R_g}{a}\right)^{3/2}, \nonumber\\ \end{eqnarray} \tag{ 12 }$$

where T₀ = c/(GΣ₀) = 700 yr for the middle region and 150 yr for the outer region. Likewise, the Bardeen–Petterson alignment time is

$$\begin{equation} T_{\rm BP} = \frac{\alpha ^{4/5}}{ 4\pi ({\dot{m}}/{\dot{m}}_{\rm Edd})^\gamma m_8^{1/5}}\frac{T_0}{\sin \delta _{\rm BP}{\cal I}_{\rm BP}}\left(\frac{r_{\rm BP}}{r_g}\right)^{1/2-\beta } \end{equation} \tag{ 13 }$$

The ratio between the timescales is then

$$\begin{eqnarray} {T_{\rm BP}\over {T_{\rm bin}}} &=& \frac{1}{4} \left(\frac{\dot{M}}{\dot{m}}\right)^\gamma \left(\frac{m}{M}\right)^{\gamma -1/5} \frac{\sin \delta _{\rm bin}{\cal I}_{\rm bin}}{ \sin \delta _{\rm BP}{\cal I}_{\rm BP}} \left(\frac{r_{\rm BP}}{r_g}\right)^{1/2-\beta } \nonumber\\ &&\times \left(\frac{R_{\rm bin}}{a}\right)^{\beta -1} \left(\frac{a}{R_g}\right)^{\beta +1/2}. \end{eqnarray} \tag{ 14 }$$

From our previous expressions we have

$$\begin{equation} \left(r_{\rm BP}\over r_g\right)^{1/2-\beta }=\left({2a_* {\cal I}_{\rm BP}\over f \alpha }\right)^{(1-2\beta)/3}\left({r_{\rm BP}\over h_{\rm BP}}\right)^{2(1-2\beta)/3} \end{equation} \tag{ 15 }$$

and

$$\begin{equation} \left({R_{\rm bin}\over a}\right)^{\beta -1}=\left({3\eta {\cal I}_{\rm bin} \over 4f \alpha }\right)^{(\beta -1)/2} \left({R_{\rm bin}\over h_{\rm bin}}\right)^{\beta -1}. \end{equation} \tag{ 16 }$$

In the Shakura–Sunyaev middle region

$$\begin{equation} h/r\approx 2\times 10^{-3}\alpha ^{-1/10}({\dot{M}}/{\dot{M}}_{\rm Edd})^{1/5}M_8^{-1/10}(r/R_g)^{1/20} \end{equation} \tag{ 17 }$$

and in the outer region

$$\begin{equation} h/r\approx 8.7\times 10^{-4}\alpha ^{-1/10}({\dot{M}}/{\dot{M}}_{\rm Edd})^{3/20}M_8^{-1/10}(r/R_g)^{1/8}. \end{equation} \tag{ 18 }$$

The extremely weak dependences on all parameters in a given disk region mean that we can assume r_BP/h_BP ∼ R_bin/h_bin ∼ 10³.

Applying this result to the spin alignment time, we find that

$$\begin{eqnarray} T_{\rm BP} &\simeq& \frac{10^{2(1-2\beta)}}{4\pi } T_0 \frac{\alpha ^{4/5}}{(\dot{m}/\dot{m}_{\rm Edd})^\gamma m_8^{1/5} \sin \delta _{\rm BP}} \nonumber\\ &&\times \left(\frac{2a_*}{f\alpha }\right)^{(1-2\beta)/3} {\cal I}_{\rm BP}^{-2\beta /3}. \end{eqnarray} \tag{ 19 }$$

For example, in the Shakura–Sunyaev middle region, where γ = −β = 3/5 and T₀ = 700 yr,

$$\begin{eqnarray} &&T_{\rm BP} \simeq 1.4 \times 10^6 \frac{\alpha ^{4/5}}{(\dot{m}/\dot{m}_{\rm Edd})^{3/5} m_8^{1/5} \sin \delta _{\rm BP}} \left(\frac{2a_*}{f\alpha }\right)^{11/15} {\cal I}_{\rm BP}^{2/5}\hbox{ yr}. \nonumber\\ \end{eqnarray} \tag{ 20 }$$

If typical values of $\dot{m}/\dot{m}_{\rm Edd}$ and α are ∼0.1, the spin alignment time is ∼1 × 10⁶ yr.

Similarly, in the same disk zone we find that the binary orbital plane alignment time

$$\begin{eqnarray} T_{\rm bin} & \simeq & 1.4 \times 10^7 \frac{\alpha ^{4/5}}{(\dot{M}/\dot{M}_{\rm Edd})^{3/5} M_8^{1/5}\sin \delta _{\rm bin}} \nonumber\\ &&\times \left(\frac{a}{R_g}\right)^{-1/10} \left(\frac{3\eta }{4f\alpha }\right)^{4/5} {\cal I}_{\rm bin}^{-1/5}\hbox{ yr}. \end{eqnarray} \tag{ 21 }$$

In other words, the orbital plane alignment time is nearly independent of the size of the binary in gravitational units, a/R_g, and in this case the dependence on α cancels identically.

Setting all disk aspect ratios to 10⁻³ also leads to the result that

$$\begin{equation} \left(r_{\rm BP}\over {r_g}\right)^{1/2-\beta } \! \left(R_{\rm bin}\over a\right)^{\beta -1} \!=\! 10^{-(1+\beta)} \frac{ (2a_* {\cal I}_{\rm BP})^{(1-2\beta)/3}}{(\eta {\cal I}_{\rm bin}/2)^{(1-\beta)/2}} (f\alpha)^{(1+\beta)/6}. \end{equation} \tag{ 22 }$$

This form leads to

$$\begin{eqnarray} {T_{\rm BP}\over {T_{\rm bin}}} \! &=&\! \frac{1}{4} 10^{-(1+\beta)} \left(\frac{\dot{M}}{\dot{m}}\right)^\gamma \left(\frac{m}{M}\right)^{\gamma -1/5} \nonumber\\ &&\times \frac{\sin \delta _{\rm bin}}{ \sin \delta _{\rm BP}} \frac{{\cal I}_{\rm bin}^{(1+\beta)/2}}{{\cal I}_{\rm BP}^{2(1+\beta)/3}} \frac{(2a_*)^{(1-2\beta)/3}}{(3\eta /4)^{(1-\beta)/2}} (f\alpha)^{(1+\beta)/6} (a/R_g)^{1/2+\beta }. \nonumber\\ \end{eqnarray} \tag{ 23 }$$

In the middle disk zone, the ratio of spin alignment to orbital plane alignment time is then

$$\begin{eqnarray} {T_{\rm BP}\over T_{\rm bin}} &\simeq& 0.2 \frac{(1+q)^{6/5}}{q^{2/5}} \left(\frac{\dot{M}}{\dot{m}}\right)^{3/5} \nonumber\\ &&\times \frac{\sin \delta _{\rm bin}}{\sin \delta _{\rm BP}} a_*^{11/15} \frac{{\cal I}_{\rm bin}^{1/5}}{{\cal I}_{\rm BP}^{4/15}} (f\alpha)^{1/15} (a/R_g)^{-1/10}, \end{eqnarray} \tag{ 24 }$$

while in the Shakura–Sunyaev outer region the ratio of times is

$$\begin{eqnarray} {T_{\rm BP}\over T_{\rm bin}} &\simeq& 0.3 \frac{(1+q)^{5/4}}{q^{3/8}} \left(\frac{\dot{M}}{\dot{m}}\right)^{7/10} \frac{\sin \delta _{\rm bin}}{\sin \delta _{\rm BP}} a_*^{5/6} \frac{{\cal I}_{\rm bin}^{1/8}}{{\cal I}_{\rm BP}^{1/6}} \nonumber\\ &&\times (f\alpha)^{1/24} (a/R_g)^{-1/4}. \end{eqnarray} \tag{ 25 }$$

In both cases, q in these last two expressions should be interpreted as the ratio between the mass of the black hole whose spin is aligning and the mass of the other black hole. Remarkably, nearly all the parameters in these expressions enter only with very small exponents. Their only significant dependences are on a_* and q. They also depend on $\dot{M}/\dot{m}$, but this ratio is likely always to be ∼O(1). The reason T_BP/T_bin depends so weakly on parameters is that quadrupolar and Lense–Thirring torques act in very similar ways: they are both purely precessional torques, and both precession frequencies scale with radius in almost the same fashion. In the quadrupolar case, ω_p∝r^−7/2, whereas in the relativistic case, ω_p∝r⁻³; both are proportional to a single "strength" parameter (η for the quadrupole, a_* for Lense–Thirring). The only contrast is in the characteristic inner scale of the radial power-law, a for the quadrupole, r_g for Lense–Thirring—and that ratio enters to at most the 1/4 power, and sometimes to only the 1/10 power.

To gain some perspective on this timescale ratio, consider first the situation of near-equal masses, so that m ≃ M/2 and $\dot{M} \sim 2 \dot{m}$. In that case, the only remaining parameter with any significant influence on the ratio T_BP/T_bin is a_*. In the middle region, $T_{\rm BP}/T_{\rm bin} \simeq 0.7a_*^{11/15}(a/R_g)^{-1/10}$, which is ∼0.3 for a ∼ 10⁴R_g when a_* ∼ 1 and smaller when a_* ≪ 1. In the outer region, the ratio changes only slightly, to ${\simeq } a_*^{5/6}(a/R_g)^{-1/4}$, which is ∼0.1 for a ∼ 10⁴R_g. Thus, over a wide range of potentially interesting separations (a ≲ 10⁴R_g), the black hole spins in an equal-mass binary align with the binary orbital plane roughly an order of magnitude faster than the binary orbital plane aligns with the outer circumbinary disk.

When the mass ratio is far from unity, we need to consider the primary and the secondary black holes separately. The time to align either spin increases with black hole mass, but relatively slowly, ∝m^{γ − 1/5}. The secondary's alignment time is therefore shorter than the primary's by a ratio ∼(m₂/m₁)^s, with 0.4 ⩽ s ⩽ 0.5, depending on which accretion regime applies. On the other hand, for fixed accretion rate and total binary mass, the orbital plane alignment time is ∝η^4/5. In other words, unequal-mass ratios permit more rapid orbital alignment because there is less angular momentum whose direction must be changed. Moreover, because T_bin's scaling with η ≡ q/(1 + q)² is stronger than T_BP's scaling with q/(1 + q), the spin alignment time for either black hole, but especially that of the primary, can become longer than the orbital plane alignment time when the mass ratio is extreme.

The analysis in the previous section treated the outer boundary of the minidisks as free. In reality, the binary exerts torques on the minidisks (e.g., Katz et al. 1982; Terquem 1998; Martin et al. 2009), and if those torques are sufficient to maintain spin-disk misalignment to smaller radii than the normal Bardeen–Petterson transition radius, the spins will align even faster than we found above. This effect was also analyzed by Martin et al. (2009) with the assumptions that (using our notation) f = 1/α² and the Bardeen–Petterson transition radius is always well inside the radius at which torques from the binary are important.

As a first step, we argue that despite considerable uncertainty in the exact nature of accretion onto black holes in binary black hole systems, it is likely that both minidisks will extend to their tidal truncation radii. Consider an initial situation in which the holes do not have minidisks. If the streams from the circumbinary disk have circularization radii at least a few times the radius of the innermost stable circular orbit (ISCO), then in a steady state in which the accretion rate onto the holes equals the rate at which matter is added to the disk, the total angular momentum of the disk (which by assumption has constant mass) increases because the angular momentum added at the circularization radius exceeds the angular momentum drained into the holes from the ISCO. This increase can only be terminated when the outer part of the disk is far enough away that tidal torque due to the companion can remove angular momentum from the minidisk and transfer it to the binary orbit.

If either minidisk is misaligned with both the orbital plane and its black hole's spin, torques that tend to align the minidisk with the orbital plane compete with Bardeen–Petterson torques that tend to align the minidisk with the black hole spin. Previous analyses of the forced precession of an annulus of gas by a binary (Katz et al. 1982; Terquem 1998; Lubow & Ogilvie 2000) have found that the precession rate of an annulus of gas at radius r < a around the primary due to the torque by the secondary is given by

$$\begin{equation} {\omega _p\over {\Omega _{\rm bin}}}=-{3\over 4}\frac{q}{(1+q)^{1/2}} \left(r/a\right)^{3/2} \cos i, \end{equation} \tag{ 26 }$$

where i is the inclination of the annulus relative to the orbital plane. If we set this precession rate equal to the Lense–Thirring precession rate

$$\begin{equation} \omega _{\rm prec,L-T}={2a_*\over {(r/r_g)^3}}{c\over r_g}\;, \end{equation} \tag{ 27 }$$

we find that the two are comparable at a radius

$$\begin{equation} r_*/r_g \approx \left(\frac{8 a_*}{3q\cos i}\right)^{2/9} (1 + q)^{2/3}(a/R_g)^{2/3}, \end{equation} \tag{ 28 }$$

where R_g is in terms of M.

The ratio of this characteristic radius to the usual Bardeen–Petterson radius (Equation (5)) is

$$\begin{eqnarray} r_*/r_{\rm BP} &\approx& 1 \times 10^{-4} (\cos i)^{-2/9} a_*^{-4/9} (f\alpha /{\cal I}_{\rm BP})^{2/3} \nonumber\\ &&\times (1+q)^{2/3} q^{-2/9} (a/R_g)^{2/3}, \end{eqnarray} \tag{ 29 }$$

where have set h_BP/r_BP = 10⁻³, our fiducial value. If we suppose that $a_* \sim q \sim f\alpha /{\cal I}_{\rm BP} \sim 1$, r_* < r_BP so long as a ≲ 10⁶R_g. This supposition implies that f is as large as it could plausibly be (∼α⁻¹); a smaller value would strengthen this conclusion. However, the maximum value of a/R_g for which r_* < r_BP diminishes if a_* or q are substantially smaller than unity. A larger value of h_BP/r_BP would act in the same direction to greater effect. To evaluate this ratio for the disk around the secondary as it is torqued by the primary, q should be replaced by 1/q', where q' = m₁/m₂. The mass ratio scaling is then ∝(1 + q')^2/3q'^−4/9.

When r_* < r_BP, binary torques are able to keep the spin and the disk misaligned to smaller radii than would be the case without the torques. The spins of the individual black holes are then aligned more quickly than we found before because there is material misaligned with respect to the spin at smaller radii than would be found without the binary torques, and the Lense–Thirring effects are much stronger at those small radii. We previously found that the Bardeen–Petterson torque scales with radius as $R_{\rm T}^{-1/2+\beta }$. Thus if β = −3/5 or β = −3/4, a factor of 10 reduction in R_T (corresponding, for example to a/r_g ∼ 3 × 10⁴) would increase the rate of spin alignment by a factor of 10–20. At still smaller binary separations relative to R_g, the relative speed-up would be even greater. In Figure 1 we display the final ratio T_BP/T_bin including extra alignment from binary torques, for three different values of the mass ratio, and for both the middle and outer regions, setting the other factors to unity. From this figure it is clear that typically the individual black hole spins will be aligned much more rapidly with the orbital axis than the binary orbital axis will be with the circumbinary disk.

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We can now address the question of the effect of binary eccentricity on the relative alignment times of the orbit and of the individual black hole spins. As we discussed after Equation (2), for the moderate eccentricities e ∼ 0.6 reached in simulations (Artymowicz et al. 1991; Goldreich & Sari 2003; Armitage & Natarajan 2005; Cuadra et al. 2009; Roedig et al. 2011; Roedig & Sesana 2012) the torque of the circumbinary disk on the binary orbit will be comparable to what it is for a circular binary. However, for a given semimajor axis the angular momentum of the binary is less by a factor of (1 − e²)^1/2 than it is for a circular orbit, hence we expect the binary orbital alignment time to be somewhat shorter. For e = 0.6 this factor is 0.8, meaning that the orbital alignment time is not affected much by moderate eccentricities.

The alignment rate of the individual black hole spins could be increased if the outer radii of the minidisks are smaller than r_* and r_BP, because then the characteristic radius is reduced and the torque is increased. This is, however, unlikely to play a significant role for moderate eccentricities. To see this, note that Sepinsky et al. (2007) showed that under most circumstances the effective Roche lobe radius for a binary of semimajor axis a and eccentricity e is within 20% of (1 − e) times the standard Eggleton (1983) value

$$\begin{equation} r_{\rm Roche}/a={0.49q^{2/3}\over {0.6q^{2/3}+\ln (1+q^{1/3})}}. \end{equation} \tag{ 30 }$$

Using our previous expressions, this implies that over the entire range 0 < q < ∞ the ratio of the Roche radius to r_* is within ∼30% of

$$\begin{equation} r_{\rm Roche}/r_*\approx 0.6q^{-4/9}(1-e)\left(a\over R_g\right)^{1/3}\left(3\cos i\over {8a_*}\right)^{2/9}. \end{equation} \tag{ 31 }$$

As we show in the next section, for a < few × 10³R_g gravitational radiation dominates, hence for q < 10 and e < 0.6 we expect that r_Roche  ≳  r_* and the minidisk alignment rate is not affected much. Therefore, the eccentricities likely to be reached in these systems have only a small effect on our conclusions.

Even if the minidisks align with the orbital plane much more rapidly than the orbital plane aligns with distant gas, it is still possible that initially misaligned disks could stay misaligned if the binary orbital evolution time is much shorter than the alignment time. In this section we therefore first demonstrate that when orbital evolution is driven by circumbinary gas, alignment happens on a shorter timescale than the orbit evolves, and hence we expect the black hole spins to be well-aligned with the orbit. We then explore the opposite extreme, when gravitational radiation dominates orbital evolution, which is relevant at smaller binary separations.

At large separations in our gas-driven scenario, the contraction of the binary as well as its alignment by torques from the circumbinary disk are both driven by the gas. There have not been sufficient numerical studies to determine how the rate at which the semimajor axis decreases depends on the mass ratio and binary eccentricity, but for a circular equal-mass binary the results of Shi et al. (2012) imply

$$\begin{equation} {\dot{a}\over a}=-0.8{\dot{M}\over M}, \end{equation} \tag{ 32 }$$

where as before M is the total mass of the binary. Thus the binary shrinks on a characteristic time that is comparable to the time needed to increase the mass of the binary. This time is ∼10⁸ yr or more for an e-folding of mass if accretion proceeds at tens of percent of the Eddington rate. More quantitatively, if the disks are in the Shakura–Sunyaev middle region,

$$\begin{eqnarray} {T_{\rm bin} \over a/{\dot{a}}} &=& 0.25 \frac{\alpha ^{4/5}}{\sin \delta _{\rm bin}{\cal I}_{\rm bin}^{1/5}} (3\eta /4f)^{4/5} (\dot{M}/\dot{M}_{\rm Edd})^{2/5} M_8^{-1/5} \nonumber\\ &&\times (a/R_g)^{-1/10}. \end{eqnarray} \tag{ 33 }$$

Because η ⩽ 0.25, α < 1, and 1 ≲ f ≲ α⁻¹, the orbital plane alignment time is at least 1–2 orders of magnitude shorter than the orbital evolution time. The spin alignment time, even for the primary, is at most comparable to the orbital plane alignment time, so the spin alignment time is even shorter. Thus, in order for the binary orbital plane and spins to break alignment with the circumbinary disk, the orientation of the circumbinary disk must change on timescales substantially shorter than $M/\dot{M} \sim a/\dot{a}$.

If instead the binary is close enough that gravitational radiation is important, then binary shrinkage is decoupled from orbital alignment. From Peters (1964), the characteristic timescale of inspiral of a binary due to gravitational radiation is

$$\begin{equation} T_{\rm GW}=a^4(1-e^2)^{7/2} /\left[{256\over 5}{G^3\eta M^3\over c^5}\right], \end{equation} \tag{ 34 }$$

where e is the binary eccentricity. Numerically, this is

$$\begin{eqnarray} T_{\rm GW} &=& 1.3\times 10^6\; {\rm yr}\; (a/10^3 M)^4(0.25/\eta) \nonumber\\ &&\times (M/10^8 \,M_\odot)(1-e^2)^{7/2}, \end{eqnarray} \tag{ 35 }$$

where we have scaled to the symmetric mass ratio for equal masses. Because the time needed to shrink the binary due to gas accretion is ∼10⁸ yr, independent of a, the semimajor axis at which binary shrinking by gas gives way to shrinking by gravitational radiation is a ∼ few × 10³R_g.

Once the binary begins to evolve more rapidly by gravitational wave emission than by interaction with surrounding gas, we expect the orientation of the binary plane to become fixed. The reason is that, so long as R_bin > 10³a (cf. Equation (4)), R_bin will continue to fall well outside the inner edge of the circumbinary disk the entire time until black hole merger. Our estimate (Equation (12)) for the binary orbital alignment time should remain valid. Because T_bin∝a^−3/2, the time for the binary orbital plane to respond to changes in the circumbinary disk orientation becomes longer and longer.

At the same time, even if accretion continues, as the results of Noble et al. (2012) suggest it will, alignment of the black hole spins also becomes slower. The effective coupling radius becomes limited by the tidal truncation radius of the disks r_t as the binary becomes tighter. In addition, if the accretion rate is more than a small fraction of Eddington, when r_t/r_g ≲ 10², the disk is radiation dominated, so that its surface density is described by the Shakura–Sunyaev inner region solution,

$$\begin{equation} \Sigma \simeq {8/3 \over \alpha } (\dot{m}/\dot{m}_{\rm Edd})^{-1} (r/r_g)^{3/2} \kappa ^{-1}. \end{equation} \tag{ 36 }$$

In these conditions, the nominal black hole spin alignment time becomes

$$\begin{equation} T_{\rm BP} = \frac{3}{32\pi } \alpha (\dot{m}/\dot{m}_{\rm Edd}) [ (r_t/a)^{1/2} (a/r_g)]^{-1} \frac{\kappa c/G}{\langle \sin \delta \rangle }, \end{equation} \tag{ 37 }$$

where 〈sin δ〉 is the inclination angle averaged over radius out to the disk's tidal truncation radius r_t. In a circular binary, the primary's r_t ≃ 0.4a while the secondary's r_t ∼ 0.4qa (Artymowicz & Lubow 1994). Relative to the gravitational wave evolution time,

$$\begin{eqnarray} \frac{T_{\rm BP}}{T_{\rm GW}} &\simeq& 0.1 \alpha (\eta /0.25) (\dot{m}/\dot{m}_{\rm Edd}) (r_t/a)^{-1/2} M_8^{-1} \langle \sin \delta \rangle ^{-1} \nonumber\\ &&\times (a/10^3 R_g)^{-5}. \end{eqnarray} \tag{ 38 }$$

In other words, if the disk were suddenly misaligned from the black hole's spin at about the time that gravitational wave emission begins to dominate the binary's evolution, Bardeen–Petterson spin alignment would be completely ineffective once a ≲ 10³R_g. Conversely, whatever mutual relation exists between black hole spins and binary orbit at the time when a ∼ 10³R_g would be preserved until the black holes merge if gas interactions are the only mechanism of reorientation.