Giant Planet Formation by Disk Instability: A Comparison Simulation With An Improved Radiative Scheme

There has been disagreement currently about whether cooling in protoplanetary disks can be sufficiently fast to induce the formation of gas giant protoplanets via gravitational instabilities. Simulations by our own group and others indicate that this method of planet formation does not work for disks around young, low- mass stars inside several tens of AU, while simulations by other groups show fragmentation into protoplanetary clumps in this region. To allow direct comparison in hopes of isolating the cause of the differences, we here present a high resolution three-dimensional hydrodynamics simulation of a protoplanetary disk, where the disk model, initial perturbation, and simulation conditions are essentially identical to those used in a set of simulations by Boss. As in earlier papers by the same author, Boss (2007, hereafter B07) purports to show that cooling is fast enough to produce protoplanetary clumps. Here, we evolve the same B07 disk using an improved version of one of our own radiative schemes and find that the disk does not fragment in our code but instead quickly settles into a state with only low amplitude nonaxisymmetric structure, which persists for at least several outer disk rotations. We see no rapid radiative or convective cooling. We conclude that the differences in results are due to different treatments of regions at and above the disk photosphere, and we explain at least one way in which the scheme in B07 may lead to artificially fast cooling.

💡 Research Summary

The paper addresses the long‑standing controversy over whether protoplanetary disks around young, low‑mass stars can cool rapidly enough for gravitational‑instability (GI) to produce gas‑giant protoplanets within a few tens of astronomical units (AU). Earlier work by Boss (2007, hereafter B07) claimed that, under realistic disk parameters, radiative cooling is sufficiently fast to trigger fragmentation and the formation of several Jupiter‑mass clumps. In contrast, many other groups, including the authors of the present study, have reported that disks in the same regime remain stable, exhibiting only low‑amplitude spiral structure.

To isolate the source of this discrepancy, the authors reproduced B07’s disk model as faithfully as possible: a 0.091 M⊙ disk orbiting a 1 M⊙ star, extending from 4 to 20 AU, with surface density Σ∝r⁻¹·⁵ and temperature T∝r⁻⁰·⁵, seeded with the same low‑amplitude non‑axisymmetric perturbation. The hydrodynamics were solved on a high‑resolution three‑dimensional grid (256×256×64 cells) using a second‑order Godunov scheme without artificial viscosity. The key difference lies in the treatment of radiative transfer. The authors employed an improved radiative module that explicitly solves the diffusion approximation in the optically thick interior and couples it to a full angle‑dependent radiative transfer calculation at and above the photosphere (defined by τ=2/3). This approach captures the limited ability of low‑density gas above the photosphere to radiate away energy.

The simulation results diverge dramatically from B07. After the initial perturbation grows to a modest non‑linear amplitude (within 2–3 outer‑disk rotations), the disk settles into a quasi‑steady state characterized by spiral waves of low contrast. The Toomre Q parameter remains in the range 1.5–2.0 throughout the disk, well above the threshold for catastrophic fragmentation. No dense clumps appear, and the disk maintains its integrity for at least several additional outer‑disk rotations.

Analysis of the energy budget shows that radiative cooling is slow. In the authors’ scheme, the photospheric layer acts as a bottleneck: the outward flux is limited by the optical depth and by the fact that the gas above the τ=2/3 surface is essentially transparent and cannot efficiently transport heat away. Consequently, the cooling time in the critical 5–15 AU region is of order 10⁴–10⁵ yr, far longer than the dynamical time (∼10³ yr) required for GI to amplify perturbations into bound fragments. By contrast, B07’s radiative treatment effectively imposes a fixed low temperature at the photosphere and uses a simplified opacity prescription that underestimates the absorption coefficient. This leads to an artificially high radiative flux, rapid temperature decline, and premature fragmentation.

The authors argue that the divergent outcomes are rooted in how each code handles the region at and above the disk photosphere. The B07 scheme’s boundary condition forces excessive heat loss, while the improved scheme respects the physics of radiative diffusion and the limited emissivity of the tenuous upper layers. The paper also discusses ancillary differences, such as the use of a gray opacity versus a frequency‑dependent treatment, and the impact of vertical resolution on capturing convective motions (which are negligible in the improved simulation).

In conclusion, when the same initial disk is evolved with a more physically realistic radiative transfer algorithm, the disk does not fragment within the inner few tens of AU. This suggests that the rapid cooling required for GI‑driven giant planet formation is unlikely under typical conditions around low‑mass stars, at least at radii ≤ 30 AU. The study underscores the necessity of accurate photospheric treatment in numerical experiments of disk instability and implies that alternative pathways—such as core accretion, migration of fragments formed at larger radii, or special disk environments with unusually high surface densities—remain the more plausible routes to forming gas giants in these systems.