Secular Disk Heating & Gap Opening

• Secular disk heating and gap opening are processes where continuous angular momentum transfer from binary or planetary perturbers reshapes and heats astrophysical disks.
• The analytic gap-opening criteria balance tidal torques against viscous replenishment using parameters like disk thickness, sound speed, and enclosed mass to define clear thresholds.
• SPH simulations validate these theories by distinguishing regimes with pronounced gap formation and stalled migration from smooth, rapidly evolving disk profiles.

Secular disk heating and gap opening describe the intertwined processes by which long-term transfers of angular momentum in astrophysical disks—driven by gravitating perturbers such as binaries or planets—simultaneously heat and restructure a disk’s hydrodynamic and thermal properties. These processes are foundational to the evolution of protoplanetary disks, circumbinary disks, and the gaseous environments of supermassive black hole binaries. The dominant mechanisms involve the excitation of global non-axisymmetric perturbations (e.g., ellipsoidal over-densities in the binary case), the subsequent secular gravitational torques these perturbations mediate, and the feedback between gap formation, viscous replenishment, and disk thermal response.

1. Secular Angular Momentum Exchange and Non-Axisymmetric Forcing

Secular disk heating in the context of a massive binary–disk system is driven by the resonant exchange of angular momentum between the binary and the circumbinary disk, but, critically, this exchange for comparable-mass binaries is not dominated by local resonances as in the linear planet–disk regime. Instead, it proceeds via strong, tidally induced, global, non-axisymmetric disk perturbations whose geometry and dynamical response set the torque budget. The binary excites a global overdensity resembling a uniform ellipsoid, with its major axis misaligned with the binary axis by a nearly constant phase angle Δφ. The gravitational interaction between the binary and this ellipsoidal mode produces a secular (orbit-averaged) torque,

$$\tau_{\rm sec} = \langle \boldsymbol{r}_b \times \nabla \Phi_{\rm ellip} \rangle$$,

where $\Phi_{\rm ellip}$ is the potential due to the non-axisymmetric density enhancement in the disk. This torque extracts angular momentum from the binary and deposits it into the gaseous disk, driving both orbital evolution of the binary and heating (i.e., increased velocity dispersion and pressure support) of the disk medium. The key point is that the angular momentum extraction is a secular process—it accumulates over many dynamical times, causing a gradual but persistent rearrangement of disk structure.

2. Analytic Gap-Opening Criteria and Critical Thresholds

Gap opening is controlled by the relative efficiency of outward angular momentum transport (from the binary torque) compared to the inward replenishment of material by disk viscosity. The paper presents two mathematically equivalent analytic criteria:

1. Velocity-based form: $(v/v_{\rm bin})^2 \cdot (c_s/v)^3 \leq 0.095$
2. Structural parameter form: $(h/r)^3 \cdot [1 + 8M(<r)/M_{\rm bin}] \leq 0.095$

Here, $v$ is the characteristic orbital velocity of the circumbinary disk, $v_{\rm bin}$ the binary’s Keplerian velocity, $c_s$ the sound speed (determining the disk’s thermal pressure support), $h$ the disk’s scale height at radius $r = a/2$ (with $a$ the binary separation), and $M(<r)/M_{\rm bin}$ the ratio of disk mass enclosed within $r$ to the total binary mass.

If the product on the left satisfies the inequality, the binary’s secular torque dominates: gas in the co-orbital region cannot be efficiently replenished by viscous diffusion, and a persistent, deep gap forms. If the product exceeds the threshold, viscous or pressure forces efficiently smooth out any attempted evacuation, leaving the gap unformed or quickly refilled. Thinner disks (small $h/r$) and disks with relatively low mass interior to the binary are more susceptible to gap formation.

3. SPH Simulation Regimes: "Opened" versus "Closed" Disks

Three-dimensional SPH simulations validate the theoretical prescription. Simulated systems organize cleanly into two regimes:

Regime	Gap Depth	Binary Migration	Surface Density
Opened	Pronounced	$<10\%$ per orbit	Ring/gap profile; strong pile-up at edge
Closed	Absent/Weak	Rapid	Smooth, low-amplitude undulations

The division between these regimes appears as a sharp threshold in the space of dimensionless variables given by the forms above. Simulations with $h/r$ and $M(<r)/M_{\rm bin}$ below the analytic boundary correspond to persistent gap formation; systems above the threshold refill and remain smooth. Resolution tests confirm the robustness of this bifurcation.

4. Secular Heating, Binary Migration, and Analogies to Planet–Disk Interaction

The disk torque not only carves gaps but incrementally raises the velocity dispersion (thermal and turbulent) within the disk, increasing the "heat" in a secular sense as angular momentum is deposited. The analogy to planet–disk interaction is explicit: for binaries where no gap forms, the system experiences Type I–like migration—rapid and efficient exchange of angular momentum via non-axisymmetric secular perturbations. Once the gap forms, migration stalls to a slower, Type II–like regime, with the binary and inner disk edge locked and migration rate set by the much slower viscous inflow.

However, for binary mass ratios near unity (as opposed to planet–disk systems with extreme mass ratios), the locus of the angular momentum transfer is fundamentally different: it is the global non-axisymmetric density structure, not the accumulation of localized resonant torques, that mediates disk heating and exchange.

5. Impact of Disk Structural Properties and Enclosed Mass

The analytic gap criterion emphasizes two key physical controls: (1) disk thickness (parameterized by $h/r$) and (2) the inner mass budget (the enclosed mass ratio $M(<r)/M_{\rm bin}$). Disks that are "puffy" (high $h/r$), or have large masses within the critical radius, are more resistant: the pressure and inertia in these configurations provide sufficient back-pressure to buffer against gap opening, ensuring that secular torques become insufficient before a clean gap can form. In contrast, thin and/or low-mass disks enable efficient gap excavation and associated dynamical reorganization.

6. Broader Astrophysical Consequences

This unified picture for secular disk heating and gap opening has multiple ramifications:

• Circumbinary migration and massive black hole coalescence: In gas-rich post-merger nuclei, whether or not a circumbinary gap forms sets the regime of SMBH binary inspiral, controlling the hand-off from gas-driven to gravitational-wave-driven coalescence.
• Comparisons with other disk systems: The clean segregation of gap-opening parameter space in both analytic and simulation work allows for direct application to observed circumbinary disk systems, including protoplanetary and circumstellar environments.
• Observational diagnostics: Gas pile-ups, the width and depth of cavities, and the dynamical state of the disk all afford potential diagnostics for system parameters—especially in the era of high-resolution spectral imaging.

7. Limitations and Regime of Validity

The analytic criteria are derived for isothermal disks under controlled conditions; extensions to disks with varying equations of state, strong turbulence, or more complex thermodynamics require recalibration. Furthermore, in astrophysical situations with extremely high viscosity or strong non-ideal MHD effects, the torque balance controlling gap formation will shift.

In summary, secular disk heating and gap opening in the context of a comparable-mass binary and circumbinary disk are governed by the secular exchange of angular momentum through a long-lived, tidally induced, non-axisymmetric perturbation. Analytic gap-opening criteria, validated by SPH simulations, provide clear thresholds in parameter space that delineate regimes of gap formation and persistent disk heating. This framework connects binary migration, the emergence of disk cavities, and long-term disk restructuring, with consequences for the dynamical and observational states of a wide range of accretion disk systems (1209.5988).