Properties of gravitoturbulent accretion disks

We explore the properties of cold gravitoturbulent accretion disks - non-fragmenting disks hovering on the verge of gravitational instability - using a realistic prescription for the effective viscosity caused by gravitational torques. This prescription is based on a direct relationship between the angular momentum transport in a thin accretion disk and the disk cooling in a steady state. Assuming that opacity is dominated by dust we are able to self-consistently derive disk properties for a given $\dot M$ assuming marginal gravitational stability. We also allow external irradiation of the disk and account for a non-zero background viscosity which can be due to the MRI. Spatial transitions between different co-existing disk states (e.g. between irradiated and self-luminous or between gravitoturbulent and viscous) are described and the location of the boundary at which disk must fragment is determined in a variety of situations. We demonstrate in particular that at low enough $\dot M$ external irradiation stabilizes gravitoturbulent disk against fragmentation all the way to infinity thus providing means of steady mass transport to the central object. Implications of our results for the possibility of planet formation by gravitational instability in protoplanetary disks and star formation in the Galactic Center and for the problem of feeding supermassive black holes in galactic nuclei are discussed.

💡 Research Summary

The paper investigates the structure and stability of cold, gravitoturbulent accretion disks—thin, non‑fragmenting disks that sit just above the threshold of gravitational instability (Toomre Q≈1). The authors adopt a physically motivated prescription for the effective viscosity generated by self‑gravity: in a steady‑state, the viscous stress parameter α_eff is directly linked to the cooling time t_cool through α_eff≈(4/9)γ(Ω t_cool)⁻¹, where Ω is the Keplerian angular frequency and γ the adiabatic index. This relation, originally derived from Gammie’s (2001) numerical experiments, captures the idea that gravitational torques are regulated by how fast the disk can radiate away the heat they generate.

Because the disks are assumed to be cold enough that dust dominates the opacity, the authors employ a Rosseland mean opacity κ∝T² (with a weak density dependence) appropriate for sub‑micron silicate grains. Combining this opacity law with the α–t_cool relation yields analytic scalings for temperature T(r), surface density Σ(r), and mid‑plane density ρ(r) as functions of the mass accretion rate \dot M and radius r, under the marginal‑stability condition Q≈1.

Two external agents are incorporated: (1) irradiation from a central star, an active galactic nucleus, or any external luminous source, characterized by a flux F_irr that sets a floor temperature T_irr≈(F_irr/σ)¹⁄⁴; and (2) a background “non‑gravitational” viscosity α_MRI that may arise from magnetorotational turbulence or other microphysical processes. The presence of these agents partitions the disk into four distinct regimes:

1. Self‑luminous gravitoturbulent zone – cooling is internal, α_eff is set by the cooling time, and the disk follows the Q≈1 scaling.
2. Irradiated gravitoturbulent zone – external heating fixes T≈T_irr, forcing Σ to adjust so that Q stays near unity.
3. Viscous (MRI‑dominated) zone – α_MRI≫α_eff, the disk behaves like a standard α‑disk, and Q can be >1.
4. Fragmentation zone – when t_cool becomes short enough that α_eff drops below a critical value (≈0.06), the disk can no longer transport angular momentum without breaking up, and gravitational fragmentation occurs.

The authors solve for the transition radii between these zones as functions of \dot M, F_irr, and α_MRI. A key result is that at sufficiently low accretion rates (≈10⁻⁸ M_⊙ yr⁻¹ for typical parameters), external irradiation can keep the entire disk warm enough that the cooling time remains long, pushing the fragmentation radius to infinity. In this regime the disk can transport mass inward indefinitely without forming clumps, providing a steady supply channel to the central object. Conversely, at higher \dot M (≈10⁻⁶ M_⊙ yr⁻¹), cooling becomes rapid, the fragmentation radius moves inward to tens of astronomical units, and the disk is prone to break up into bound fragments—conditions favorable for direct gravitational instability planet formation or for star formation in dense nuclear disks.

The interplay between α_MRI and the gravitoturbulent α_eff is highly non‑linear. When α_MRI≈10⁻³, the irradiated gravitoturbulent region can be extensive, allowing Q to stay near unity over a large radial span. If α_MRI is larger (≈10⁻²), the gravitational torque becomes negligible and the disk is essentially an MRI‑driven α‑disk throughout. Thus, the level of magnetorotational activity and the strength of external heating are decisive parameters that dictate whether a gravitoturbulent state can exist at all.

The paper connects these theoretical findings to observations. For protoplanetary disks such as HL Tau or TW Hya, the inferred low irradiation and moderate accretion rates place the fragmentation boundary at ~30–50 AU, consistent with the presence of massive rings and gaps seen by ALMA that could be the signatures of early‑stage gravitational fragmentation. In the Galactic Center, intense X‑ray/UV fields from the central black hole can stabilize otherwise unstable nuclear disks, allowing a continuous inflow of gas onto the supermassive black hole without the disk fragmenting into stars.

In summary, the authors present a unified framework that couples gravitational torque, radiative cooling, external irradiation, and background viscosity. By deriving explicit analytic expressions for the disk structure and the locations of regime transitions, they clarify under which astrophysical conditions a gravitoturbulent disk can survive, when it will fragment, and how it can serve as an efficient conduit for mass transport. These results have broad implications for theories of planet formation by gravitational instability, star formation in extreme environments such as the Galactic Center, and the feeding of supermassive black holes in galactic nuclei.