Spin–Orbit Misalignment of Merging Black Hole Binaries with Tertiary Companions

Consider a hierarchical triple system, consisting of an inner BH binary with masses m₁, m₂ and a relatively distant companion of mass m₃. The reduced mass for the inner binary is ${\mu }_{\mathrm{in}}\equiv {m}_{1}{m}_{2}/{m}_{12}$, with ${m}_{12}\equiv {m}_{1}+{m}_{2}$. Similarly, the outer binary has ${\mu }_{\mathrm{out}}\equiv ({m}_{12}{m}_{3})/{m}_{123}$ with ${m}_{123}\equiv {m}_{12}+{m}_{3}$. The orbital semimajor axes and eccentricities are denoted by ${a}_{\mathrm{in},\mathrm{out}}$ and ${e}_{\mathrm{in},\mathrm{out}}$, respectively. The orbital angular momenta of the inner and outer binaries are

$$\begin{eqnarray}&&{{\boldsymbol{L}}}_{\mathrm{in}}={L}_{\mathrm{in}}{\hat{{\boldsymbol{L}}}}_{\mathrm{in}}={\mu }_{\mathrm{in}}\sqrt{{{Gm}}_{12}{a}_{\mathrm{in}}(1-{e}_{\mathrm{in}}^{2})}\,{\hat{{\boldsymbol{L}}}}_{\mathrm{in}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{\boldsymbol{L}}}_{\mathrm{out}}={L}_{\mathrm{out}}{\hat{{\boldsymbol{L}}}}_{\mathrm{out}}={\mu }_{\mathrm{out}}\sqrt{{{Gm}}_{123}{a}_{\mathrm{out}}(1-{e}_{\mathrm{out}}^{2})}\,{\hat{{\boldsymbol{L}}}}_{\mathrm{out}},\end{eqnarray} \tag{ 3 }$$

where ${\hat{{\boldsymbol{L}}}}_{\mathrm{in},\mathrm{out}}$ are unit vectors. The relative inclination between ${\hat{{\boldsymbol{L}}}}_{\mathrm{in}}$ and ${\hat{{\boldsymbol{L}}}}_{\mathrm{out}}$ is denoted by I. For convenience, we will frequently omit the subscript "$\mathrm{in}$."

The merger time due to GW radiation of an isolated binary with initial a₀ and ${e}_{0}=0$ is given by

$$\begin{eqnarray}{T}_{{\rm{m}},0} & = & \displaystyle \frac{5{c}^{5}{a}_{0}^{4}}{256{G}^{3}{m}_{12}^{2}\mu }\\ & \simeq & {10}^{10}{\left(\displaystyle \frac{60{M}_{\odot }}{{m}_{12}}\right)}^{2}\left(\displaystyle \frac{15\,{M}_{\odot }}{\mu }\right){\left(\displaystyle \frac{{a}_{0}}{0.202\mathrm{au}}\right)}^{4}\mathrm{years}.\end{eqnarray} \tag{ 4 }$$

A sufficiently inclined external companion can raise the binary eccentricity through LK oscillations, thereby reducing the merger time or making an otherwise non-merging binary merge within 10¹⁰ years. To study the evolution of merging BH binary under the influence of a companion, we use the secular equations to the octupole level in terms of the angular momentum ${\boldsymbol{L}}$ and eccentricity ${\boldsymbol{e}}$ vectors:

$$\begin{eqnarray}&&{\left.\displaystyle \frac{d{\boldsymbol{L}}}{{dt}}=\displaystyle \frac{d{\boldsymbol{L}}}{{dt}}\right|}_{\mathrm{LK}}+{\left.\displaystyle \frac{d{\boldsymbol{L}}}{{dt}}\right|}_{\mathrm{GW}}\,,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\left.\displaystyle \frac{d{\boldsymbol{e}}}{{dt}}=\displaystyle \frac{d{\boldsymbol{e}}}{{dt}}\right|}_{\mathrm{LK}}+{\left.\displaystyle \frac{d{\boldsymbol{e}}}{{dt}}\right|}_{\mathrm{GR}}+{\left.\displaystyle \frac{d{\boldsymbol{e}}}{{dt}}\right|}_{\mathrm{GW}}.\end{eqnarray} \tag{ 6 }$$

Here, the "Lidov–Kozai" (LK) terms are given explicitly in Liu et al. (2015a; we also evolve ${{\boldsymbol{L}}}_{\mathrm{out}}$ and ${{\boldsymbol{e}}}_{\mathrm{out}}$), and the associated timescale of LK oscillation is

$$\begin{eqnarray}&&{t}_{\mathrm{LK}}=\displaystyle \frac{1}{n}\displaystyle \frac{{m}_{12}}{{m}_{3}}{\left(\displaystyle \frac{{a}_{\mathrm{out},\mathrm{eff}}}{a}\right)}^{3},\end{eqnarray} \tag{ 7 }$$

where $n=\sqrt{{{Gm}}_{12}/{a}^{3}}$ is the mean motion of the inner binary and ${a}_{\mathrm{out},\mathrm{eff}}\equiv {a}_{\mathrm{out}}\sqrt{1-{e}_{\mathrm{out}}^{2}}$. General Relativity (GR; 1-PN correction) induces pericenter precession

$$\begin{eqnarray}&&{\left.\displaystyle \frac{d{\boldsymbol{e}}}{{dt}}\right|}_{\mathrm{GR}}={{\rm{\Omega }}}_{\mathrm{GR}}\hat{{\boldsymbol{L}}}\times {\boldsymbol{e}},\,\,\,{{\rm{\Omega }}}_{\mathrm{GR}}=\displaystyle \frac{3{{Gnm}}_{12}}{{c}^{2}a(1-{e}^{2})}.\end{eqnarray} \tag{ 8 }$$

We include GW emission (2.5-PN effect) that causes orbital decay and circularization (Shapiro & Teukolsky 1983), but not the extreme eccentricity excitation due to nonsecular effects (Antonini et al. 2014; Silsbee & Tremaine 2017).

The top three panels of Figure 1 show an example of the orbital evolution of a BH binary with an inclined companion (initial ${I}_{0}=80^\circ$). We see that the inner binary undergoes cyclic excursions to maximum eccentricity ${e}_{\max }$, with accompanying oscillations in the inclination I. As the binary decays, the range of eccentricity oscillations shrinks. Eventually the oscillations freeze and the binary experiences "pure" orbital decay/circularization governed by GW dissipation.

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In the quadrupole approximation, the maximum eccentricity ${e}_{\max }$ attained in the LK oscillations (starting from an initial I₀ and ${e}_{0}\simeq 0$) can be calculated analytically using energy and angular momentum conservation, according to the equation (Anderson et al. 2017)

$$\begin{eqnarray}&&\displaystyle \frac{3}{8}\displaystyle \frac{{j}_{\min }^{2}-1}{{j}_{\min }^{2}}\left[5{\left(\cos {I}_{0}+\displaystyle \frac{\eta }{2}\right)}^{2}-\left(3+4\eta \cos {I}_{0}+\displaystyle \frac{9}{4}{\eta }^{2}\right){j}_{\min }^{2}\right.\\ &&\quad +{\eta }^{2}{j}_{\min }^{4}]+{\varepsilon }_{\mathrm{GR}}(1-{j}_{\min }^{-1})=0,\end{eqnarray} \tag{ 9 }$$

where ${j}_{\min }\,\equiv \,\sqrt{1-{e}_{\max }^{2}}$, $\eta \equiv {(L/{L}_{\mathrm{out}})}_{e=0}$, and ${\varepsilon }_{\mathrm{GR}}$ is given by

$$\begin{eqnarray}&&{{\varepsilon }_{\mathrm{GR}}={t}_{\mathrm{LK}}{{\rm{\Omega }}}_{\mathrm{GR}}| }_{e=0}=\displaystyle \frac{3{{Gm}}_{12}^{2}{a}_{\mathrm{out},\mathrm{eff}}^{3}}{{c}^{2}{a}^{4}{m}_{3}}\\ &&\quad \,\simeq \,0.96{\left(\displaystyle \frac{{m}_{12}}{60{M}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{{m}_{3}}{30{M}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{a}_{\mathrm{out},\mathrm{eff}}}{3\mathrm{au}}\right)}^{3}{\left(\displaystyle \frac{a}{0.1\mathrm{au}}\right)}^{-4}.\end{eqnarray} \tag{ 10 }$$

Note that in the limit of $\eta \to 0$ and ${\varepsilon }_{\mathrm{GR}}\to 0$, Equation (9) yields the well-known relation ${e}_{\max }=\sqrt{1-(5/3){\cos }^{2}{I}_{0}}$. The maximum possible ${e}_{\max }$ for all values of I₀, called ${e}_{\mathrm{lim}}$, is given by

$$\begin{eqnarray}&&\displaystyle \frac{3}{8}({j}_{\mathrm{lim}}^{2}-1)\left[-3+\displaystyle \frac{{\eta }^{2}}{4}\left(\displaystyle \frac{4}{5}{j}_{\mathrm{lim}}^{2}-1\right)\right]+{\varepsilon }_{\mathrm{GR}}(1-{j}_{\mathrm{lim}}^{-1})=0,\end{eqnarray} \tag{ 11 }$$

and is reached at $\cos {I}_{0}=(\eta /10)(4{j}_{\mathrm{lim}}^{2}-5)$. Eccentricity excitation (${e}_{\max }\geqslant 0$) occurs within a window of inclinations ${(\cos {I}_{0})}_{-}\leqslant \cos {I}_{0}\leqslant {(\cos {I}_{0})}_{+}$, where (Anderson et al. 2017)

$$\begin{eqnarray}&&{(\cos {I}_{0})}_{\pm }=\displaystyle \frac{1}{10}\left(-\eta \pm \sqrt{{\eta }^{2}+60-\displaystyle \frac{80}{3}{\varepsilon }_{\mathrm{GR}}}\right).\end{eqnarray} \tag{ 12 }$$

This window vanishes when

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{GR}}\geqslant \displaystyle \frac{9}{4}+\displaystyle \frac{3}{80}{\eta }^{2}\qquad (\mathrm{no}\,\mathrm{eccentricity}\,\mathrm{excitation}).\end{eqnarray} \tag{ 13 }$$

Figure 2 shows some examples of the ${e}_{\max }({I}_{0})$ curves. For $\eta \lesssim 1$, these curves depend mainly on ${m}_{3}/{a}_{\mathrm{out},\mathrm{eff}}^{3}$ (for given inner binary parameters). As ${\varepsilon }_{\mathrm{GR}}$ increases (with decreasing ${m}_{3}/{a}_{\mathrm{out},\mathrm{eff}}^{3}$), the LK window shrinks and ${e}_{\mathrm{lim}}$ decreases. Eccentricity excitation is suppressed (for all I₀) when Equation (13) is satisfied.

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For systems with ${m}_{1}\ne {m}_{2}$ and ${e}_{\mathrm{out}}\ne 0$, so that

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{oct}}\equiv \displaystyle \frac{{m}_{1}-{m}_{2}}{{m}_{12}}\left(\displaystyle \frac{a}{{a}_{\mathrm{out}}}\right)\displaystyle \frac{{e}_{\mathrm{out}}}{1-{e}_{\mathrm{out}}^{2}}\end{eqnarray} \tag{ 14 }$$

is non-negligible, the octupole effect may become important (e.g., Ford et al. 2000; Naoz 2016). This tends to widen the inclination window for large eccentricity excitation. However, the analytic expression for ${e}_{\mathrm{lim}}$ given by Equation (11) remains valid even for ${\varepsilon }_{\mathrm{oct}}\ne 0$ (Liu et al. 2015a; Anderson & Lai 2017). Therefore, Equation (13) still provides a good criterion for "negligible eccentricity excitation" (note that when ${\varepsilon }_{\mathrm{oct}}\ne 0$, the inner binary cannot be exactly circular; e.g., Liu et al. 2015b; Anderson & Lai 2017).

Eccentricity excitation leads to a shorter binary merger time T_m compared to the circular merger time ${T}_{{\rm{m}},0}$ (Figure 3). We compute T_m by integrating the secular evolution equations of the BH triples with a range of I₀. Each I₀ run has a corresponding ${e}_{\max }$, which can be calculated using Equation (9). We see that in general ${T}_{{\rm{m}}}/{T}_{{\rm{m}},0}$ can be approximated by ${(1-{e}_{\max }^{2})}^{\alpha }$, with $\alpha \simeq 1.5,\,2$, and 2.5 for ${e}_{\max }=(0,0.6),\,(0.6,0.8)$, and $(0.8,0.95)$, respectively.

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We now study how the BH spin evolves during the binary merger, considering only ${{\boldsymbol{S}}}_{1}={S}_{1}{\hat{{\boldsymbol{S}}}}_{1}$ (where ${\hat{{\boldsymbol{S}}}}_{1}$ is the unit vector). The de Sitter precession of ${\hat{{\boldsymbol{S}}}}_{1}$ around $\hat{{\boldsymbol{L}}}$ is (Barker & O'Connell 1975)

$$\begin{eqnarray}&&\displaystyle \frac{d{\hat{{\boldsymbol{S}}}}_{1}}{{dt}}={{\rm{\Omega }}}_{\mathrm{dS}}\hat{{\boldsymbol{L}}}\times {\hat{{\boldsymbol{S}}}}_{1},\,\,\,{{\rm{\Omega }}}_{\mathrm{dS}}=\displaystyle \frac{3{Gn}({m}_{2}+\mu /3)}{2{c}^{2}a(1-{e}^{2})}.\end{eqnarray} \tag{ 15 }$$

Note that there is a back-reaction torque from ${{\boldsymbol{S}}}_{1}$ on ${\boldsymbol{L}};$ this can be safely neglected since $L\gg {S}_{1}$.

Before presenting our numerical results, it is useful to note the different regimes for the evolution of the spin–orbit misalignment angle ${\theta }_{\mathrm{SL}}$ (the angle between ${{\boldsymbol{S}}}_{1}$ and ${\boldsymbol{L}}$). In general, the inner binary axis $\hat{{\boldsymbol{L}}}$ precesses (and nutates when $e\ne 0$) around the total angular momentum ${\boldsymbol{J}}={\boldsymbol{L}}+{{\boldsymbol{L}}}_{\mathrm{out}}$ (recall that ${S}_{1},{S}_{2}\ll L$, and ${\boldsymbol{J}}$ is constant in the absence of GW dissipation). The related precession rate ${{\rm{\Omega }}}_{L}$ is of the order of ${t}_{\mathrm{LK}}^{-1}$ at $e\sim 0$ (Equation (17)), but increases with e. Depending on the ratio of ${{\rm{\Omega }}}_{\mathrm{dS}}$ and ${{\rm{\Omega }}}_{L}$, we expect three possible spin behaviors. (i) For ${{\rm{\Omega }}}_{L}\gg {{\rm{\Omega }}}_{\mathrm{dS}}$ ("nonadiabatic"), the spin axis ${\hat{{\boldsymbol{S}}}}_{1}$ cannot "keep up" with the rapidly changing $\hat{{\boldsymbol{L}}}$, and thus effectively precesses around $\hat{{\boldsymbol{J}}}$, keeping ${\theta }_{\mathrm{SJ}}\equiv {\cos }^{-1}\,({\hat{{\boldsymbol{S}}}}_{1}\cdot \hat{{\boldsymbol{J}}})\,\simeq$ constant. (ii) For ${{\rm{\Omega }}}_{\mathrm{dS}}\gg {{\rm{\Omega }}}_{L}$ ("adiabatic"), ${\hat{{\boldsymbol{S}}}}_{1}$ closely "follows" $\hat{{\boldsymbol{L}}}$, maintaining an approximately constant ${\theta }_{\mathrm{SL}}$. (iii) For ${{\rm{\Omega }}}_{\mathrm{dS}}\sim {{\rm{\Omega }}}_{L}$ ("trans-adiabatic"), the spin evolution can be chaotic due to overlapping resonances. Since both ${{\rm{\Omega }}}_{\mathrm{dS}}$ and ${{\rm{\Omega }}}_{L}$ depend on e during the LK cycles, the precise transitions between these regimes can be fuzzy (Storch et al. 2014, 2017; Storch & Lai 2015; Anderson et al. 2016).

For circular orbits (e = 0), the precession of $\hat{{\boldsymbol{L}}}$ is governed by the equation

$$\begin{eqnarray}&&{\left.\displaystyle \frac{d\hat{{\boldsymbol{L}}}}{{dt}}\right|}_{\mathrm{LK},e=0}=-{{\rm{\Omega }}}_{L}{\hat{{\boldsymbol{L}}}}_{\mathrm{out}}\times \hat{{\boldsymbol{L}}}=-{{\rm{\Omega }}}_{L}^{{\prime} }\hat{{\boldsymbol{J}}}\times \hat{{\boldsymbol{L}}},\end{eqnarray} \tag{ 16 }$$

where $\hat{{\boldsymbol{J}}}$ is the unit vector along ${\boldsymbol{J}}={\boldsymbol{L}}+{{\boldsymbol{L}}}_{\mathrm{out}}$, and

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{L}=\displaystyle \frac{3}{4{t}_{\mathrm{LK}}}(\hat{{\boldsymbol{L}}}\cdot {\hat{{\boldsymbol{L}}}}_{\mathrm{out}}),\quad {{\rm{\Omega }}}_{L}^{{\prime} }={{\rm{\Omega }}}_{L}\displaystyle \frac{J}{{L}_{\mathrm{out}}}.\end{eqnarray} \tag{ 17 }$$

We can define an "adiabaticity parameter"

$$\begin{eqnarray}{ \mathcal A } & \equiv & {\left(\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{dS}}}{{{\rm{\Omega }}}_{L}}\right)}_{e,I=0}\simeq 0.37\left[\displaystyle \frac{({m}_{2}+\mu /3)}{35\,{M}_{\odot }}\right]\left(\displaystyle \frac{{m}_{12}}{60\,{M}_{\odot }}\right)\\ & & \times {\left(\displaystyle \frac{{m}_{3}}{30{M}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{a}_{\mathrm{out},\mathrm{eff}}}{3\mathrm{au}}\right)}^{3}{\left(\displaystyle \frac{a}{0.1\mathrm{au}}\right)}^{-4}.\end{eqnarray} \tag{ 18 }$$

As the binary orbit decays, the system may transition from "nonadiabatic" (${ \mathcal A }\ll 1$) at large a to "adiabatic" (${ \mathcal A }\gg 1$) at small a, where the final spin–orbit misalignment angle ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ is "frozen." Note that ${ \mathcal A }$ is directly related to ${\varepsilon }_{\mathrm{GR}}$ by

$$\begin{eqnarray}&&\displaystyle \frac{{ \mathcal A }}{{\varepsilon }_{\mathrm{GR}}}=\displaystyle \frac{2}{3}\displaystyle \frac{{m}_{2}+\mu /3}{{m}_{12}}.\end{eqnarray} \tag{ 19 }$$

Thus, when the initial value of ${\varepsilon }_{\mathrm{GR}}$ (at $a={a}_{0}$) satisfies ${\varepsilon }_{\mathrm{GR},0}\lesssim 9/4$ (a necessary condition for LK eccentricity excitation; see Equation (13)), we also have ${{ \mathcal A }}_{0}\lesssim (3{m}_{2}+\mu )/(2{m}_{12})\sim 1$. This implies that any system that experiences enhanced orbital decay due to LK oscillations must go through the "trans-adiabatic" regime and therefore possibly chaotic spin evolution.

The bottom panel of Figure 1 shows a representative example of the evolution of the misalignment angle as the BH binary undergoes LK-enhanced orbital decay. We see that the BH spin axis can exhibit complex evolution even though the orbital evolution is "regular." In particular, ${\theta }_{\mathrm{SL}}$ evolves in a chaotic way, with the final value ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ depending sensitively on the initial conditions (the precise initial ${\theta }_{\mathrm{SL}}$ and I₀). Also note that retrograde spin (${\theta }_{\mathrm{SL}}^{{\rm{f}}}\gt 90^\circ$) can be produced even though the binary always remains prograde with respect to the outer companion ($I\lt 90^\circ$). These behaviors are qualitatively similar to the chaotic evolution of stellar spin driven by Newtonian spin–orbit coupling with a giant planet undergoing high-eccentricity migration (Storch et al. 2014, 2017; Storch & Lai 2015; Anderson et al. 2016).

We carry out a series of numerical integrations, evolving the orbit of the merging BH binary with a tertiary companion, along with spin–orbit coupling, to determine ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ for various triple parameters. In our "population synthesis" study, we consider initial conditions such that ${\hat{{\boldsymbol{S}}}}_{1}$ is parallel to $\hat{{\boldsymbol{L}}}$, the binary inclinations are isotropically distributed (uniform distribution in cosI₀), and the orientations of ${\boldsymbol{e}}$ and ${{\boldsymbol{e}}}_{\mathrm{out}}$ (for systems with ${e}_{\mathrm{out}}\ne 0$) are random. All initial systems satisfy the criterion of dynamical stability for triples (Mardling & Aarseth 2001).

Figure 4 shows our results for systems with equal masses, ${e}_{\mathrm{out}}=0$ (so that the octupole effect vanishes), and several values of ${a}_{\mathrm{out}}$. We see that when the eccentricity of the inner binary is excited (I₀ lies in the LK window), a wide range of ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ is generated, including appreciable fraction of retrograde (${\theta }_{\mathrm{SL}}^{{\rm{f}}}\gt 90^\circ$) systems (see the ${a}_{\mathrm{out}}=3\,\mathrm{au}$ case, for which ${e}_{\mathrm{lim}}=0.84$). The "memory" of chaotic spin evolution is evident, as slightly different initial inclinations lead to vastly different ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$. The regular behavior of ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ around ${I}_{0}=90^\circ$ (again for the ${a}_{\mathrm{out}}=3\,\mathrm{au}$ case) is intriguing, but may be understood using the theory developed in Storch et al. (2017). For systems with no eccentricity excitation, ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ varies regularly as a function of I₀—this can be calculated analytically (see below).

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Figure 5 shows our results for systems with ${m}_{1}\ne {m}_{2}$ and ${e}_{\mathrm{out}}\ne 0$, for which octupole terms may affect the orbital evolution. We see that eccentricity excitation and the corresponding reduction in T_m occur outside the analytic (quadupole) LK window (see the ${e}_{\mathrm{out}}=0.8$ case, for which ${e}_{\mathrm{lim}}=0.66$). As in the equal-mass case (Figure 4), a wide range of ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ values are produced whenever eccentricity excitation occurs. A larger fraction (23%) of systems attain retrograde spin (${\theta }_{\mathrm{SL}}^{{\rm{f}}}\gt 90^\circ$). Again, for systems with negligible eccentricity excitation, ${\theta }_{\mathrm{SL}}^{{\rm{f}}}$ behaves regularly as a function of I₀ and agrees with the analytic result (the "fuzziness" of the numerical result in this regime is likely due to the very small eccentricity of the inner binary; see Liu et al. 2015b).

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If the inner binary experiences no eccentricity excitation and remains circular throughout the orbital decay, the final spin–orbit misalignment can be calculated analytically using the principle of adiabatic invariance.

Equation (16) shows that $\hat{{\boldsymbol{L}}}$ rotates around the $\hat{{\boldsymbol{J}}}$ axis at the rate $(-{{\rm{\Omega }}}_{L}^{{\prime} })$. In this rotating frame, the spin evolution Equation (15) transforms to

$$\begin{eqnarray}&&{\left(\displaystyle \frac{d{\hat{{\boldsymbol{S}}}}_{1}}{{dt}}\right)}_{\mathrm{rot}}={{\boldsymbol{\Omega }}}_{\mathrm{eff}}\times {\hat{{\boldsymbol{S}}}}_{1},\,\,\,{{\boldsymbol{\Omega }}}_{\mathrm{eff}}\equiv {{\rm{\Omega }}}_{\mathrm{dS}}\hat{{\boldsymbol{L}}}+{{\rm{\Omega }}}_{L}^{{\prime} }\hat{{\boldsymbol{J}}}.\end{eqnarray} \tag{ 20 }$$

Note that in the absence of GW dissipation, $\hat{{\boldsymbol{L}}}$ and ${\hat{{\boldsymbol{L}}}}_{\mathrm{out}}$ are constants (in the rotating frame), and thus ${\hat{{\boldsymbol{S}}}}_{1}$ precesses with a constant ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}$. The relative inclination between ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}$ and $\hat{{\boldsymbol{L}}}$ is given by

$$\begin{eqnarray}&&\tan {\theta }_{\mathrm{eff},L}=\displaystyle \frac{{{\rm{\Omega }}}_{L}\sin I}{{{\rm{\Omega }}}_{\mathrm{dS}}+(\eta +\cos I){{\rm{\Omega }}}_{L}}.\end{eqnarray} \tag{ 21 }$$

Now if we include GW dissipation, $\hat{{\boldsymbol{L}}}\cdot {\hat{{\boldsymbol{L}}}}_{\mathrm{out}}=\cos I$ is exactly conserved, and ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}$ becomes a slowly changing vector. When the rate of change of ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}$ is much smaller than $| {{\boldsymbol{\Omega }}}_{\mathrm{eff}}|$, the angle between ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}$ and ${\hat{{\boldsymbol{S}}}}_{1}$ is an adiabatic invariant, i.e.,

$$\begin{eqnarray}&&{\theta }_{\mathrm{eff},{S}_{1}}\simeq \mathrm{constant}\qquad (\mathrm{adiabatic}\,\mathrm{invariant}).\end{eqnarray} \tag{ 22 }$$

This adiabatic invariance requires $| d{{\boldsymbol{\Omega }}}_{\mathrm{eff}}/{dt}| /| {{\boldsymbol{\Omega }}}_{\mathrm{eff}}| \,\sim {T}_{m,0}^{-1}\ll | {{\boldsymbol{\Omega }}}_{\mathrm{eff}}|$, or $| {{\boldsymbol{\Omega }}}_{\mathrm{eff}}| {T}_{{\rm{m}},0}\gg 1$, which is easily satisfied.

Suppose ${\hat{{\boldsymbol{S}}}}_{1}$ and $\hat{{\boldsymbol{L}}}$ are aligned initially, we have ${\theta }_{\mathrm{eff},{S}_{1}}^{0}={\theta }_{\mathrm{eff},L}^{0}$ (the superscript 0 denotes initial value). Equation (22) then implies ${\theta }_{\mathrm{eff},{S}_{1}}\simeq {\theta }_{\mathrm{eff},L}^{0}$ at all times. After the binary has decayed, $\eta \to 0$, $| {{\rm{\Omega }}}_{\mathrm{dS}}| \gg | {{\rm{\Omega }}}_{L}|$, and therefore ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}\simeq {{\rm{\Omega }}}_{\mathrm{dS}}\hat{{\boldsymbol{L}}}$, which implies ${\theta }_{\mathrm{SL}}^{{\rm{f}}}\simeq {\theta }_{\mathrm{eff},{S}_{1}}$. Thus, we find

$$\begin{eqnarray}&&{\theta }_{\mathrm{SL}}^{{\rm{f}}}\simeq {\theta }_{\mathrm{eff},L}^{0}.\end{eqnarray} \tag{ 23 }$$

That is, the final spin–orbit misalignment angle is equal to the initial inclination angle between $\hat{{\boldsymbol{L}}}$ and ${{\boldsymbol{\Omega }}}_{\mathrm{eff}}$, obtained by evaluating Equation (21) at $a={a}_{0}$:

$$\begin{eqnarray}&&\tan {\theta }_{\mathrm{eff},L}^{0}=\displaystyle \frac{\sin {I}_{0}}{({{ \mathcal A }}_{0}/\cos {I}_{0})+{\eta }_{0}+\cos {I}_{0}}.\end{eqnarray} \tag{ 24 }$$

This analytic expression agrees with the numerical results shown in Figures 4–5 in the appropriate regime. Note that for systems that experience no eccentricity excitation for all I₀, ${{ \mathcal A }}_{0}\gtrsim (3{m}_{2}+\mu )/(2{m}_{12})\sim 1$ (see Equations (13) and (19)), and thus ${\theta }_{\mathrm{SL}}^{{\rm{f}}}\simeq {\theta }_{\mathrm{eff},L}^{0}\lesssim 20^\circ$, i.e., only modest spin–orbit misalignment can be generated. For systems with ${{ \mathcal A }}_{0}\gg 1$ (e.g., a very distant companion), we have ${\theta }_{\mathrm{SL}}^{{\rm{f}}}\ll 1$.

The above analytical result can be easily generalized to the situation of non-zero initial spin–orbit misalignment. It shows that ${\theta }_{\mathrm{SL}}^{{\rm{f}}}\simeq {\theta }_{\mathrm{SL}}^{0}$ for ${{ \mathcal A }}_{0}\gg 1$.