N-Body Simulation of Planetesimal Formation through Gravitational Instability and Coagulation. II. Accretion Model

The gravitational instability of a dust layer is one of the scenarios for planetesimal formation. If the density of a dust layer becomes sufficiently high as a result of the sedimentation of dust grains toward the midplane of a protoplanetary disk, the layer becomes gravitationally unstable and spontaneously fragments into planetesimals. Using a shearing box method, we performed local $N$-body simulations of gravitational instability of a dust layer and subsequent coagulation without gas and investigated the basic formation process of planetesimals. In this paper, we adopted the accretion model as a collision model. A gravitationally bound pair of particles is replaced by a single particle with the total mass of the pair. This accretion model enables us to perform long-term and large-scale calculations. We confirmed that the formation process of planetesimals is the same as that in the previous paper with the rubble pile models. The formation process is divided into three stages: the formation of non-axisymmetric structures, the creation of planetesimal seeds, and their collisional growth. We investigated the dependence of the planetesimal mass on the simulation domain size. We found that the mean mass of planetesimals formed in simulations is proportional to $L_y^{3/2}$, where $L_y$ is the size of the computational domain in the direction of rotation. However, the mean mass of planetesimals is independent of $L_x$, where $L_x$ is the size of the computational domain in the radial direction if $L_x$ is sufficiently large. We presented the estimation formula of the planetesimal mass taking into account the simulation domain size.

💡 Research Summary

This paper investigates the formation of planetesimals through gravitational instability (GI) of a dust layer in a protoplanetary disk, using local N‑body simulations that omit gas dynamics. The authors adopt an “accretion model” for collisions: when two particles become gravitationally bound, they are instantly replaced by a single particle whose mass equals the sum of the pair, and whose velocity conserves linear momentum. This approach dramatically reduces particle number during the run, allowing simulations to be extended to many orbital periods and to larger computational domains than previously possible with rubble‑pile (elastic or inelastic rebound) collision models.

The simulations are performed in a shearing‑box framework with periodic boundaries in the radial (x), azimuthal (y), and vertical (z) directions. The initial condition is a thin, vertically settled dust layer composed of equal‑mass particles, with no gas drag or pressure support. The box dimensions (Lx, Ly) are varied to explore size effects.

The evolution proceeds in three distinct stages. First, small density perturbations grow into non‑axisymmetric, spiral‑like structures due to GI. These structures are elongated in the azimuthal direction and are sustained by the Coriolis force and background shear. Second, within the dense filaments, local overdensities become gravitationally bound, forming “seed” planetesimals. The mass of these seeds is found to depend strongly on the azimuthal box size Ly: the mean seed mass scales as Ly^{3/2}. The radial size Lx, provided it exceeds a threshold (≈4–5 scale heights), does not affect the mean seed mass, indicating that the azimuthal extent controls the amount of material that can be gathered into a single bound clump. Third, the seeds undergo mutual collisions and accrete surrounding particles. Because the accretion model instantly merges colliding bodies, the growth phase proceeds more rapidly than in previous rubble‑pile simulations, yet the overall qualitative picture remains the same.

A key quantitative result is an empirical formula for the final planetesimal mass:

 M_planet ≈ C Σ Ly^{3/2}

where Σ is the surface density of the dust layer and C is a dimensionless constant calibrated from the simulations. This relation captures the dependence on the azimuthal domain size and provides a practical way to extrapolate simulation outcomes to realistic disk scales.

The authors emphasize that, even without gas drag, GI alone can produce massive planetesimals, supporting the viability of the “GI‑first” scenario for early planet formation. The accretion model proves to be a computationally efficient surrogate for more detailed collision physics, preserving the essential dynamics of clump formation and growth.

In the discussion, the paper notes several limitations and future directions. The omission of gas means that aerodynamic damping, turbulent stirring, and streaming instability are not represented; incorporating these effects could modify the threshold for GI and the subsequent accretion rates. The particle size distribution is also uniform in the present work; a realistic spectrum might affect the binding probability and the mass spectrum of seeds. Finally, the authors suggest comparing the derived mass–size scaling with observations of dust substructures (e.g., rings and clumps) in ALMA disks to test whether the predicted Ly^{3/2} dependence manifests in real systems.

Overall, the study demonstrates that a simple accretion collision prescription enables long‑term, large‑scale N‑body simulations of dust‑layer GI, confirms the three‑stage formation pathway (structure → seed → growth), and reveals a robust scaling of planetesimal mass with the azimuthal extent of the computational domain. This work provides a valuable benchmark for future, more physically complete models of planetesimal formation.