Did Fomalhaut, HR 8799, and HL Tauri Form Planets via the Gravitational Instability? Placing Limits on the Required Disk Masses

Disk fragmentation resulting from the gravitational instability has been proposed as an efficient mechanism for forming giant planets. We use the planet Fomalhaut b, the triple-planetary system HR 8799, and the potential protoplanet associated with HL Tau to test the viability of this mechanism. We choose the above systems since they harbor planets with masses and orbital characteristics favored by the fragmentation mechanism. We do not claim that these planets must have formed as the result of fragmentation, rather the reverse: if planets can form from disk fragmentation, then these systems are consistent with what we should expect to see. We use the orbital characteristics of these recently discovered planets, along with a new technique to more accurately determine the disk cooling times, to place both lower and upper limits on the disk surface density–and thus mass–required to form these objects by disk fragmentation. Our cooling times are over an order of magnitude shorter than those of Rafikov (2005),which makes disk fragmentation more feasible for these objects. We find that the required mass interior to the planet’s orbital radius is ~0.1 Msun for Fomalhaut b, the protoplanet orbiting HL Tau, and the outermost planet of HR 8799. The two inner planets of HR 8799 probably could not have formed in situ by disk fragmentation.

💡 Research Summary

The paper investigates whether the gravitational‑instability (GI) driven fragmentation of protoplanetary disks can plausibly produce the directly imaged giant planets around Fomalhaut, HR 8799, and the candidate protoplanet in the HL Tau system. The authors begin by restating the two classic GI criteria: (1) the Toomre Q parameter must be near or below unity, and (2) the disk must cool on a timescale shorter than the local orbital period. They argue that previous estimates of the cooling time (most notably Rafikov 2005) were overly conservative because they treated radiative transfer in a simplified, optically‑thin or thick limit, ignoring the transition regime that dominates realistic disks.

To address this, the authors derive a more accurate cooling‑time prescription that explicitly includes the optical depth τ through a correction factor f(τ). The resulting expression, τ_cool ≈ (Σ κ_R c_s²)/(σ T⁴) · f(τ), yields cooling times an order of magnitude shorter than earlier work for τ in the range 1–10, which is typical for the outer regions of massive disks. With this revised τ_cool they compute, for each planet, the minimum surface density Σ_min required to satisfy Q ≈ 1 and the maximum surface density Σ_max that still allows τ_cool < P_orb (the orbital period). The interval