Circumbinary Disks and Gas Torques

A circumbinary disk does not exert one universal “inspiral torque.” The binary opens a cavity, excites nonaxisymmetric waves, feeds streams and minidisks, and accretes both mass and angular momentum. Negative gravitational torque from the outer disk competes with positive and negative contributions close to each hole and with the angular momentum carried by captured gas. The sign of $\dot a$ follows only after all terms and mass growth are combined in one quasi-steady budget.

Helpful background. Black-Hole Geometry supplies relativistic inner scales; Relativistic Matter and Compact Objects supplies gas closure; Black-Hole Astrophysics supplies accretion and host context; Two-Body Dynamics supplies binary energy and angular momentum; Gravitational-Wave Astronomy supplies the radiation-reaction comparison.

Cavity, streams, and angular-momentum flux

Let the binary have total mass $M=M_1+M_2$, mass ratio $q=M_2/M_1\le1$, semimajor axis $a$, and eccentricity $e$. Resonant and direct gravitational forcing clears a low-density cavity of order a few $a$, but streams cross it and feed circum-single minidisks. The inner disk can become eccentric and precess, so an orbital average may still depend on a longer disk-cycle average.

Separate the binary torque into

$$\dot J_b =\dot J_{\rm grav} +\dot J_{\rm acc},$$

where $\dot J_{\rm grav}$ is the volume-integrated gravitational force of the gas on the two bodies and $\dot J_{\rm acc}$ contains the momentum actually removed with accreted mass. A radial disk budget also includes advective, Reynolds or Maxwell, and viscous fluxes. In a settled calculation their radial changes close against the torque deposited between radii. Failure of that closure usually signals an unequilibrated outer disk, a boundary flux, or inconsistent sink bookkeeping.

For a circular binary,

$$J_b=\mu\sqrt{GMa}, \qquad \mu=\frac{M_1M_2}{M},$$

and therefore

$$\frac{\dot a}{a} =2\frac{\dot J_b}{J_b} -2\frac{\dot M_1}{M_1} -2\frac{\dot M_2}{M_2} +\frac{\dot M}{M}.$$

A positive torque does not automatically imply expansion, because the growing masses change the angular momentum required at fixed $a$. Eccentricity adds the factor $\sqrt{1-e^2}$ and requires a separate $\dot e$ equation.

Why calculations disagree

The torque is sensitive to $q$, $e$, disk aspect ratio $H/R$, viscosity or MHD stress, thermodynamics, magnetic topology, self-gravity, and the sink prescription. Thin disks concentrate mass near the cavity wall and can strengthen negative outer torque. Hotter disks leak more readily into the cavity. Slow sinks retain gas near the components and change the strongest local torque. A fixed circular orbit suppresses the very orbital response one is trying to infer unless work and torque are recorded consistently.

Tang, MacFadyen, and Haiman 2017, §§ 2–4 demonstrates that changing the minidisk sink time can change the magnitude and even the sign of the measured torque in two-dimensional viscous calculations. Duffell et al. 2020, §§ 2–4 maps mass ratio and viscosity and finds positive net torque for part of the near-equal-mass circular parameter space. Neither result licenses extrapolation to a thin, radiative, eccentric, three-dimensional MHD disk.

Long integrations matter. The gas initially placed outside the cavity must establish a quasi-steady mass and angular-momentum flux through the measurement region; the eccentric inner disk must sample its precession cycle; and the outer boundary must not drain into the answer. Report torque as a function of radius and component, not only as one final number.

Results and open questions

The evidence cutoff is 2026-08-18. The migration sign remains unresolved outside the limited formulations that have converged. The parameter regime is part of every claim: $q$, $e$, $H/R$, stress prescription, thermodynamic closure, disk mass, magnetic flux, sink scale, and boundary condition must accompany the torque sign.

Formulation and regime	Evidence or reported result	Discriminating check	Limitation or open status
One-dimensional viscous migration analogy	An outer disk that absorbs binary angular momentum suggests inspiral tied to disk transport	Close total angular momentum and compare binary residence with viscous supply	Cannot represent streams, minidisks, eccentric cavities, or local positive torque
Two-dimensional locally isothermal, circular, near-equal-mass disks	Several long calculations find enough accretion and inner-cavity torque for expansion	Repeat across sink time, radial domain, viscosity, and $H/R$	Sign changes with numerical and thermodynamic choices; radiation and MHD are absent
Thin or eccentric viscous disks	Denser cavity walls or $e\gtrsim0.4$ commonly strengthen negative torque and inspiral	Evolve $a$ and $e$ together with a converged flux decomposition	Thin-disk duration and resolution are costly; eccentric attractors depend on $q$ and closure
Three-dimensional MHD, radiative, or self-gravitating disks	Magnetic stress, vertical structure, cooling, and fragmentation can alter mass supply and torque	Cross-code energy/angular-momentum closure over inflow and precession times	Parameter coverage is sparse; no universal sign has been established

Lai and Muñoz 2023, pp. 517–560 synthesizes the simulations and concludes that circumbinary accretion does not always shrink a binary; secular evolution depends on binary and gas properties. The credible positions are therefore regime-dependent inspiral, regime-dependent expansion, and transitional or stalled behavior. A population model must integrate across this map rather than assign one torque coefficient to all binaries.

A decisive numerical comparison uses common initial and boundary data in at least two independent codes, measures gravitational and accretion torques with the same control surface, and evolves long enough for inflow equilibrium. It refines cells or particles, sink radius and time, domain, and duration; closes mass, energy, and angular momentum; and holds out at least one $(q,e,H/R)$ point as a prediction.

The interpretation could change after a long-duration three-dimensional radiation-MHD survey spans thin and thick disks across mass ratio and eccentricity; after independent codes reproduce the same sign with common sink and boundary definitions; or after an observational binary supplies a calibrated, nondegenerate gas-torque measurement.

Implications and boundaries

Gas affects more than $a$. Preferential accretion can drive $q$ toward unity; torques can excite or damp $e$; minidisk cycles can modulate emission; and disk warps can change spin alignment. These coupled variables alter the later gravitational-wave residence time and source demographics.

The calculation here concerns the gas control volume and its conditional secular torque. Two-Body Dynamics develops the vacuum orbital mechanics, and Gravitational-Wave Astronomy explains how waveforms constrain the source.

Common pitfalls

Reporting gravitational torque alone. Accreted momentum and changing component masses are required before converting $\dot J_b$ to $\dot a$.

Calling a finite disk quasi-steady too early. Initial cavity relaxation can mimic a lasting torque sign before the supply front and eccentric cycle equilibrate.

Treating sink independence at one resolution as convergence. Sink radius, removal time, grid scale, and minidisk viscous time must be varied together.

Convert torque into migration

Consider a circular equal-mass binary that remains equal mass while accreting. Let $\dot M/M=\gamma$ and let the net specific angular momentum delivered to the binary be

$$\ell_0=\frac{\dot J_b}{\dot M} =\lambda\sqrt{GMa}.$$

Derive $\dot a/a$. Evaluate it for $\lambda=0.60$ and $\gamma=0.01\,{\rm Myr^{-1}}$, then for $\lambda=0.20$.

Solution

For equal masses, $\mu=M/4$, so

$$J_b=\frac{M}{4}\sqrt{GMa}, \qquad \frac{\dot J_b}{J_b} =4\lambda\frac{\dot M}{M}=4\lambda\gamma.$$

Both components grow at the same fractional rate $\gamma$. Substitution into the mass-varying angular-momentum identity gives

$$\frac{\dot a}{a} =2(4\lambda\gamma)-2\gamma-2\gamma+\gamma =(8\lambda-3)\gamma.$$

For $\lambda=0.60$, $\dot a/a=1.8\gamma=0.018\,{\rm Myr^{-1}}$: the binary expands by about $1.8\%$ per Myr at that instant. For $\lambda=0.20$, $\dot a/a=-1.4\gamma=-0.014\,{\rm Myr^{-1}}$: it contracts. The sign threshold is $\lambda=3/8$ for this circular, equal-growth case.

References

• Duffell, Paul C., et al. “Circumbinary Disks: Accretion and Torque as a Function of Mass Ratio and Disk Viscosity.” Astrophysical Journal 901 (2020): 25. DOI.
• Lai, Dong, and Diego J. Muñoz. “Circumbinary Accretion: From Binary Stars to Massive Binary Black Holes.” Annual Review of Astronomy and Astrophysics 61 (2023): 517–560. DOI.
• Tang, Yike, Andrew MacFadyen, and Zoltán Haiman. “On the Orbital Evolution of Supermassive Black Hole Binaries with Circumbinary Accretion Discs.” Monthly Notices of the Royal Astronomical Society 469 (2017): 4258–4267. DOI.