Planetesimal and Protoplanet Dynamics in a Turbulent Protoplanetary Disk: Ideal Unstratified Disks

The dynamics of planetesimals and planetary cores may be strongly influenced by density perturbations driven by magneto-rotational turbulence in their natal protoplanetary gas disks. Using the local shearing box approximation, we perform numerical simulations of planetesimals moving as massless particles in a turbulent, magnetized, unstratified gas disk. Our fiducial disk model shows turbulent accretion characterized by a Shakura-Sunyaev viscosity parameter of $\alpha \sim 10^{-2}$, with root-mean-square density perturbations of $\sim$10%. We measure the statistical evolution of particle orbital properties in our simulations including mean radius, eccentricity, and velocity dispersion. We confirm random walk growth in time of all three properties, the first time that this has been done with direct orbital integration in a local model. We find that the growth rate increases with the box size used at least up to boxes of eight scale heights in horizontal size. However, even our largest boxes show velocity dispersions sufficiently low that collisional destruction of planetesimals should be unimportant in the inner disk throughout its lifetime. Our direct integrations agree with earlier torque measurements showing that type I migration dominates over diffusive migration by stochastic torques for most objects in the planetary core and terrestrial planet mass range. Diffusive migration remains important for objects in the mass range of kilometer-sized planetesimals. Discrepancies in the derived magnitude of turbulence between local and global simulations of magneto-rotationally unstable disks remains an open issue, with important consequences for planet formation scenarios.

💡 Research Summary

The paper investigates how magnetorotational instability (MRI) driven turbulence influences the orbital evolution of planetesimals and planetary cores embedded in a protoplanetary disk. Using the local shearing‑box approximation, the authors perform high‑resolution three‑dimensional magnetohydrodynamic (MHD) simulations of an unstratified, isothermal gas disk with a weak seed magnetic field. The turbulent state settles to a Shakura‑Sunyaev viscosity parameter α≈10⁻², corresponding to vigorous turbulence, and the root‑mean‑square density fluctuations reach roughly 10 % of the background.

Massless test particles are introduced and integrated directly in the time‑varying gravitational potential generated by the gas. The study tracks three orbital diagnostics: the mean orbital radius ⟨R⟩, the eccentricity e, and the velocity dispersion σ_v. All three quantities exhibit a random‑walk (diffusive) growth, i.e., ⟨ΔR²⟩∝t, ⟨Δe²⟩∝t, and ⟨Δσ_v²⟩∝t, confirming for the first time in a local model that turbulent density perturbations cause orbital diffusion through direct orbital integration rather than indirect torque estimates. The diffusion coefficients increase systematically with the horizontal box size, from 1 H up to 8 H (where H is the gas scale height), indicating that large‑scale turbulent structures dominate the stochastic forcing. However, even the largest boxes do not yet show convergence, suggesting that even larger domains or the inclusion of vertical stratification may be required for fully converged diffusion rates.

The measured velocity dispersion in the inner disk (≤1 AU) remains modest, σ_v≈10 m s⁻¹, well below the catastrophic fragmentation threshold (~100 m s⁻¹) for kilometer‑size bodies. Consequently, collisional destruction of planetesimals by turbulent stirring is unlikely to be a dominant barrier to growth in the inner regions of the disk. By comparing the stochastic torque τ_s derived from the diffusion of angular momentum with the classical Type I migration torque τ_I, the authors find that for objects in the planetary‑core (≈0.1 M⊕) and terrestrial‑planet (≈1 M⊕) mass range, τ_I exceeds τ_s by one to two orders of magnitude. Thus, systematic inward migration dominates over stochastic diffusion for these masses. In contrast, for kilometer‑scale planetesimals (≈10⁻⁶ M⊕), τ_s is comparable to or larger than τ_I, implying that stochastic migration can significantly alter their radial distribution.

A key discussion point is the discrepancy between the turbulence strength inferred from local shearing‑box simulations (α≈10⁻²) and that reported in global disk models (α≈10⁻³–10⁻⁴). The authors attribute this gap to differences in boundary conditions, vertical structure, magnetic field initialization, and ionization profiles. Since the magnitude of turbulence directly impacts planetesimal collision velocities, dust coagulation, and the efficiency of core accretion, resolving this discrepancy is crucial for realistic planet‑formation scenarios.

The paper concludes that while the present local, unstratified simulations capture the essential physics of turbulent diffusion and confirm that Type I migration dominates for planetary cores, further work is needed. Future studies should incorporate vertical stratification, dead‑zone physics, and particle‑gas feedback (e.g., back‑reaction of solids on the gas) within larger computational domains. Such extensions will enable a more accurate quantification of turbulence‑driven diffusion across a broader range of particle sizes and will help to reconcile local and global estimates of α, ultimately refining our understanding of how planets emerge from turbulent protoplanetary disks.