Precession-Averaged Dynamics and Morphologies

Quasicircular spinning binaries exhibit three nested timescales. The orbit is fastest, spin precession is intermediate, and radiation reaction shrinks the binary most slowly. When both separations are large, the precession cycle can itself be averaged, reducing a vector evolution to a one-dimensional nutation problem and a secular inspiral. The resulting “morphology” classifies the relative azimuth of the two spins; it is a property of a stated orbit-averaged PN system, not a permanent label of an exact merger.

Required background. Presymplectic Potentials and Currents supplies the reduced spin phase space.

Helpful background. The Einstein Equation fixes curvature, Weak Metric Perturbations and Gauge Freedom supplies perturbative variables, Black-Hole Geometry and Mechanics supplies the Kerr matching, and Relativistic Two-Body Dynamics supplies multiscale averaging.

Three separated timescales

Consider black holes with constant-magnitude canonical spins, including their Kerr spin-induced quadrupoles. Adopt a Newton–Wigner-type spin gauge, quasicircular orbit-averaged 2PN spin precession, and leading adiabatic radiation reaction. Define $M=m_1+m_2$, $\mu=m_1m_2/M$, $\nu=\mu/M$, and $q=m_1/m_2\geq1$. At fixed mass ratio, the characteristic powers of orbital speed $v$ are

$$T_{\mathrm{orb}}\sim\frac{GM}{v^3}, \qquad T_{\mathrm{prec}}\sim\frac{GM}{v^5}, \qquad T_{\mathrm{rr}}\sim\frac{GM}{\nu v^8}.$$

Thus

$$\frac{T_{\mathrm{orb}}}{T_{\mathrm{prec}}}\sim v^2, \qquad \frac{T_{\mathrm{prec}}}{T_{\mathrm{rr}}}\sim\nu v^3.$$

Both ratios are small in the weak-field inspiral. Orbit averaging removes the fast orbital phase; precession averaging then replaces a full nutation cycle by its secular effect on the slowly decreasing orbital angular momentum.

On a precession timescale, the spin magnitudes $S_1,S_2$, orbital magnitude $L$, total magnitude

$$J=\lvert\mathbf L+\mathbf S_1+\mathbf S_2\rvert,$$

and the dimensionless projected effective spin

$$\xi=\frac{[(1+q^{-1})\mathbf S_1+(1+q)\mathbf S_2] \cdot\hat{\mathbf L}}{GM^2}$$

are constant in the chosen truncation. Its conservation needs the full spin-spin and Kerr quadrupole–monopole terms, as explained on the spin–orbit page. The total-spin magnitude

$$S=\lvert\mathbf S_1+\mathbf S_2\rvert$$

generically oscillates between turning points $S_-$ and $S_+$. It is therefore a convenient coordinate for a precession cycle away from special limits such as equal masses, one vanishing spin, or a zero-amplitude equilibrium. In those cases $S$ can be constant even while the vectors precess, so the integral formulas below must be taken as limits or replaced by the vector equations.

Effective potentials and turning points

Angular-momentum triangles first bound the allowed interval:

$$\max(|J-L|,|S_1-S_2|)\leq S \leq\min(J+L,S_1+S_2).$$

For $L,S\ne0$, define the cosine between $\mathbf L$ and total spin, and the components of each spin along their sum,

$$c_S=\frac{J^2-L^2-S^2}{2LS}, \qquad a_1=\frac{S^2+S_1^2-S_2^2}{2S}, \qquad a_2=S-a_1.$$

The two spins have equal and opposite transverse components of magnitude $b=(S_1^2-a_1^2)^{1/2}$. If $\psi$ is their rotation angle about $\mathbf S$ relative to the $\mathbf S$–$\mathbf L$ plane, then

$$\begin{aligned} \mathbf S_1\cdot\hat{\mathbf L} &=a_1c_S+b\sqrt{1-c_S^2}\cos\psi,\\ \mathbf S_2\cdot\hat{\mathbf L} &=a_2c_S-b\sqrt{1-c_S^2}\cos\psi. \end{aligned}$$

Substitute into $\xi$. Its extrema at fixed $S$ give two effective-potential branches,

$$\xi_\pm(S) =\frac{[(1+q^{-1})a_1+(1+q)a_2]c_S \ \pm(q-q^{-1})b\sqrt{1-c_S^2}}{GM^2}.$$

They bound the allowed values $\xi_-(S)\leq\xi\leq\xi_+(S)$ and correspond to coplanar configurations. Either branch can contain $\Delta\Phi=0$ or $\pi$ segments; its sign does not label one morphology globally. At fixed $\xi$, the endpoints of the allowed $S$ interval give $S_-$ and $S_+$. This construction is equivalent to Gerosa et al. 2015, § II A–C, Eqs. (8)–(20), PDF, with $q$ inverted and $G$ restored.

For time evolution, use the orbit-averaged precession vectors

$$\begin{aligned} \boldsymbol\Omega_1 &=\frac{G}{2r^3}\left[ \left(4+\frac3q\right)\mathbf L -\frac{3GM^2\xi}{1+q}\hat{\mathbf L}+\mathbf S_2\right],\\ \boldsymbol\Omega_2 &=\frac{G}{2r^3}\left[ (4+3q)\mathbf L -\frac{3qGM^2\xi}{1+q}\hat{\mathbf L}+\mathbf S_1\right], \end{aligned}$$

with $L=\mu\sqrt{GMr}$ at this order. Evolve $\dot{\mathbf S}_A=\boldsymbol\Omega_A\times\mathbf S_A$ and $\dot{\mathbf L}=-\dot{\mathbf S}_1-\dot{\mathbf S}_2$. Differentiating $S^2$ gives a scalar equation,

$$\dot S =\frac{(\boldsymbol\Omega_1-\boldsymbol\Omega_2) \cdot(\mathbf S_1\times\mathbf S_2)}{S}.$$

The triangle relations and fixed $\xi$ determine its magnitude as a function of $S$; its sign reverses at each turning point. The nutation period is

$$\tau_{\mathrm{nut}} =2\int_{S_-}^{S_+}\frac{dS}{\lvert dS/dt\rvert}.$$

A precession average of any quantity $X(S)$ is

$$\langle X\rangle_{\mathrm{prec}} =\frac{2}{\tau_{\mathrm{nut}}} \int_{S_-}^{S_+} \frac{X(S)\,dS}{\lvert dS/dt\rvert}.$$

This weighted integral, not a uniform average over $S$, is what enters the slow radiation-reaction equations.

The effective-potential construction and its analytic checks are developed in Kesden et al. 2015, pp. 1–5; precession-averaged inspiral is developed in Gerosa et al. 2015, §§II–IV.

The three morphologies

Project both spins into the instantaneous orbital plane and let $\Delta\Phi$ be their relative azimuth. Over one nutation cycle it can:

• circulate, passing through all angles;
• librate about $0$, oscillating without reaching $\pi$;
• librate about $\pi$, oscillating without reaching $0$.

At a morphology boundary, one spin becomes aligned or antialigned with $\mathbf L$ at a turning point. Its projection into the orbital plane then vanishes, so $\Delta\Phi$ is undefined there even though the spin vectors remain regular. Slow radiation reaction moves $(L,J)$ through this boundary and changes the azimuthal classification Gerosa et al. 2015, § II C, Eqs. (20)–(24), PDF. Such a transition is not generically a dynamical separatrix with a divergent period. The distinction between azimuthal libration and a true resonant separatrix is discussed in Correia 2016, §§ 3–5, Fig. 1.

The labels are useful because they compress an entire precession cycle and can retain formation information. They depend, however, on the PN order, chosen spin variables, circular averaging, and adequate timescale separation. Near merger or transitional precession, direct higher-fidelity evolution is required.

Direct-integration check and breakdown

For a proposed precession-averaged trajectory:

1. hold $L$ fixed and integrate the full orbit-averaged vector equations for several nutation cycles;
2. verify conservation of $J$, $\xi$, and $S_A$;
3. measure $S_\pm$ and the range of $\Delta\Phi$;
4. compare them with the effective-potential roots and morphology;
5. restore radiation reaction and reduce its strength, checking convergence toward the precession-averaged drift.

The approximation fails when $T_{\mathrm{orb}}\not\ll T_{\mathrm{prec}}$, when $T_{\mathrm{prec}}\not\ll T_{\mathrm{rr}}$, near an actual instability or degeneracy that makes the measured cycle too slow, or when eccentricity and higher multipoles introduce unmodeled frequencies. Ordinary morphology boundaries alone do not establish failure. Numerical drift in spin magnitude can also imitate a morphology transition, so invariant residuals must accompany the classification.

Exercises

Why is a uniform average over $S\in[S_-,S_+]$ incorrect?

Solution

The system spends more time where $\lvert dS/dt\rvert$ is small, especially near a turning point. Time averaging uses $dt=dS/\lvert dS/dt\rvert$ on each half-cycle, producing the displayed weight. A uniform average would underweight the turning regions and give the wrong secular torque.

State a direct criterion that distinguishes circulation from libration.

Solution

Unwrap $\Delta\Phi(t)$ over a converged nutation cycle. Circulation has a net change of $\pm2\pi$ and crosses both coplanar angles. Libration has zero winding number and remains in an interval around either $0$ or $\pi$. The result should agree with the effective-potential turning-point classification.

References

• Correia, Alexandre C. M. “Cassini States for Black Hole Binaries.” Monthly Notices of the Royal Astronomical Society: Letters 457, L49–L53 (2016). DOI. Open PDF.
• Gerosa, Davide, Michael Kesden, Ulrich Sperhake, Emanuele Berti, and Richard O’Shaughnessy. “Multi-Timescale Analysis of Phase Transitions in Precessing Black-Hole Binaries.” Physical Review D 92 (2015): 064016. DOI.
• Kesden, Michael, Davide Gerosa, Richard O’Shaughnessy, Emanuele Berti, and Ulrich Sperhake. “Effective Potentials and Morphological Transitions for Binary Black-Hole Spin Precession.” Physical Review Letters 114 (2015): 081103. DOI.