Modelling dust coagulation, dynamical drag and turbulent mixing during star and disc formation

Technical Summary: Modelling Dust Coagulation, Dynamical Drag and Turbulent Mixing during Star and Disc Formation

Problem Statement
Dust plays a critical role in both star and planet formation, influencing gas cooling, ionization levels, and the coupling of magnetic fields to the gas. While dust coagulation into larger grains (pebbles) is essential for planetesimal formation and core growth, previous three-dimensional hydrodynamical simulations of star formation have largely neglected the dynamical evolution of dust relative to the gas. Earlier methods, such as those by Bate (2022), treated dust populations as evolving independently within each fluid element (SPH particle), assuming dust was perfectly coupled to the gas. This approximation held for the early, low-density phases of collapse but failed to account for dust migration, settling, and mixing over longer timescales, particularly within protoplanetary discs where densities are higher and grain growth is rapid. Furthermore, existing models often lacked a self-consistent treatment of how turbulence drives both the relative velocities necessary for grain growth and the turbulent diffusion that mixes dust vertically.

Methodology
The authors present a new implementation within the smoothed particle hydrodynamics (SPH) code SPHNG that combines three distinct physical processes:

1. Dust Coagulation: Utilizing the discretised mean-field Smoluchowski equation method (Hirashita & Yan 2009; Bate 2022), the code tracks the evolution of a population of dust grains across $N$ logarithmically spaced size bins for every SPH gas particle. The method considers binary collisions with a sticking probability of unity below a specific coagulation threshold velocity.
2. Dynamical Drag (MULTIGRAIN): The authors integrate the MULTIGRAIN algorithm (Hutchison et al. 2018), which solves the continuum fluid equations for a mixture of gas and multiple dust species in the terminal velocity approximation. This allows for the calculation of bulk differential velocities between dust sizes and the gas due to pressure gradients and gravity.
3. Implicit Integration and Turbulent Diffusion: A key innovation is the use of a recently developed implicit integration method (Elsender & Bate 2024) to solve the dust drag equations. This avoids the restrictive time-step constraints of explicit methods for grains with long stopping times and prevents unphysical negative dust fractions. Additionally, the authors introduce a new implicit method to model turbulent diffusion. They treat the turbulent stirring of dust as a diffusion process ($D_i = \alpha_{SS} c_s H / Sc$), where the Schmidt number ($Sc$) accounts for the coupling efficiency of different grain sizes. This diffusion term is solved using a Gauss-Seidel operator-splitting approach, allowing dust to mix between fluid elements.

The code tracks the evolution of dust from the collapse of a molecular cloud core, through the first hydrostatic core phase, and into the formation and evolution of a protoplanetary disc, utilizing sink particles to follow the system beyond stellar core formation.

Key Contributions

• First 3D Coupled Simulation: This work represents the first three-dimensional hydrodynamical simulation to simultaneously model dust coagulation, dynamical gas drag, and turbulent mixing during the early stages of star and disc formation.
• Implicit Diffusion Scheme: The development of a robust implicit solver for turbulent dust diffusion allows for stable simulations of dust mixing without the severe time-step limitations associated with explicit diffusion schemes.
• Unified Treatment: The code unifies the treatment of relative velocities (driving growth) and turbulent diffusion (driving mixing) within a single framework, using the same turbulence parameter ($\alpha_{SS}$) for both processes in the test cases.

Results
The authors present results from both test calculations (dust settling in a vertical column) and full star formation simulations:

• Test Calculations: The code successfully recovers analytic solutions for the equilibrium between dust settling and turbulent diffusion. Tests confirm that including turbulent diffusion prevents the formation of unrealistically thin dust layers and maintains a finite dust scale height.
• Star Formation Simulations:
  • First Hydrostatic Core: Consistent with previous work (Bate 2022), rapid dust growth occurs within the first hydrostatic core, producing locally mono-disperse populations with sizes exceeding 100 $\mu$m deep inside the core. The inclusion of drag and diffusion had negligible impact on this early phase due to short evolutionary timescales.
  • Protoplanetary Disc Evolution: Once the disc forms, the inclusion of turbulent diffusion significantly alters dust evolution.
    • Enhanced Growth: Turbulent diffusion substantially enhances the rate of dust grain growth. By continuously supplying small and intermediate-sized grains to high-density regions (where large grains are growing), diffusion prevents the local depletion of "food" for coagulation.
    • Turbulence Dependence: The level of turbulence ($\alpha_{SS}$) is critical. High turbulence ($\alpha_{SS} = 10^{-3}$) leads to rapid growth to mono-disperse populations of $\sim30-100$ $\mu$m grains throughout the disc. Lower turbulence ($\alpha_{SS} = 10^{-4}$) results in slower growth and a more vertically stratified dust population, with larger grains confined closer to the mid-plane.
    • Dust Settling: In simulations with weak turbulence, dust settling leads to a significant increase in the mid-plane dust-to-gas ratio (by >50%), while small grains remain depleted near the mid-plane and confined to surface layers.
  • Drag vs. Diffusion: Comparisons of simulations with and without specific processes reveal that dust-gas dynamical drag has little effect on dust distributions on these timescales, as grains remain well-coupled. In contrast, the inclusion of turbulent diffusion has an enormous effect, smoothing dust distributions and accelerating coagulation.

Significance and Claims
The paper claims that this new methodology enables, for the first time, the self-consistent modeling of dust growth, mixing, and migration during the formation of stars and their discs. The authors emphasize that while the inclusion of drag and diffusion does not alter the fundamental findings regarding dust growth in the first hydrostatic core, it is crucial for the subsequent evolution of the protoplanetary disc. Specifically, they claim that turbulent dust diffusion provides a substantial enhancement to the rate of dust grain growth by acting as a source of small grains to regions where large grains are forming.

The authors remain modest regarding the scope of their findings, noting that:

• The results depend on the adopted turbulence model (e.g., $\alpha$-disc vs. MHD turbulence).
• The simulations currently consider coagulation but not fragmentation; different coagulation thresholds could lead to fragmentation and repopulation of small grains.
• The radiative transfer calculations still assume standard interstellar opacities rather than using the evolving local grain size distributions, a step identified as a necessary future extension.

Ultimately, the work provides a tool to compare 3D hydrodynamical simulations with the wealth of new observational data from facilities like ALMA, VLT-SPHERE, and JWST, particularly regarding dust settling and turbulent stirring in edge-on discs.