[Paper Review] Supplementary Information for ``Rapid planetesimal formation in turbulent circumstellar discs''

Anders Johansen, Jeffrey S. Oishi+4 more|ArXiv.org|Aug 29, 2007

Astrophysics and Star Formation Studies61 references3 citations

TL;DR

This supplementary paper details the numerical methods and validation tests for a simulation study demonstrating rapid planetesimal formation via gravitational collapse of dust boulders in turbulent, magnetized circumstellar discs. Using a high-resolution particle-in-cell code with hyperdiffusive stabilization and self-consistent gravity, it shows that clumping from the streaming instability leads to gravitational collapse even in moderately massive discs, with collapse occurring at lower column densities as resolution increases, and collision speeds being lower than predicted by theory—suggesting a robust, self-gravity-driven pathway to planetesimal formation independent of efficient coagulation.

ABSTRACT

This document contains refereed supplementary information for the paper ``Rapid planetesimal formation in turbulent circumstellar discs''. It contains 15 sections (\S1.1 -- \S1.15) that address a number of subjects related to the main paper. We describe in detail the Poisson solver used to find the self-potential of the solid particles, including a linear and a non-linear test problem (\S1.3). Dissipative collisions remove energy from the motion of the particles by collisional cooling (\S1.4), an effect that allows gravitational collapse to occur in somewhat less massive discs (\S1.7). A resolution study of the gravitational collapse of the boulders is presented in \S1.6. We find that gravitational collapse can occur in progressively less massive discs as the grid resolution is increased, likely due to the decreased smoothing of the particle-mesh self-gravity solver with increasing resolution. In \S1.10 we show that it is in good agreement with the Goldreich & Ward (1973) stability analysis to form several-hundred-km-sized bodies, when the analysis is applied to 5 AU and to regions of increased boulder column density. \S11 is devoted to the measurement of random speeds and collision speeds between boulders. We find good agreement between our measurements and analytical theory for the random speeds, but the measured collision speeds are 3 times lower than expected from analytical theory. Higher resolution studies, and an improved analytical theory of collision speeds that takes into account epicyclic motion, will be needed to determine whether collision speeds have converged. In \S1.12 we present models with no magnetic fields. The boulder layer still exhibits strong clumping, due to the streaming instability, if the global solids-to-gas ratio is increased by a factor 3. Gravitational collapse occurs as readily as in magnetised discs.

Motivation & Objective

• To validate the numerical methods used in simulating planetesimal formation in turbulent, magnetized discs.
• To test the robustness of gravitational collapse in the presence of turbulence and numerical dissipation.
• To assess the role of collisional cooling and particle clustering in enabling collapse at lower solid column densities.
• To compare measured collision speeds with analytical predictions and evaluate convergence with resolution.
• To isolate the effects of self-gravity from coagulation and fragmentation by excluding these processes in initial models.

Proposed method

• The Pencil Code is used with sixth-order finite differences and third-order Runge-Kutta time integration, ensuring minimal numerical dissipation.
• Sixth-order hyperdiffusivity (nabla^6) is applied to stabilize turbulence and dissipate energy at small scales without affecting large-scale dynamics.
• A particle-mesh method computes self-gravity with a Poisson solver, validated on linear and nonlinear test problems.
• Drag forces between gas and particles are computed via a momentum-conserving algorithm involving interpolation, force calculation, and back-reaction assignment.
• The simulation tracks boulder overdensities via the streaming instability and measures random and collisional speeds using statistical analysis.
• Resolution studies are performed across 64^3, 128^3, and 256^3 grids to assess convergence of gravitational collapse and collisional dynamics.

Experimental results

Research questions

• RQ1 Does gravitational collapse of boulders occur at lower solid column densities as grid resolution increases, indicating convergence of the self-gravity solver?
• RQ2 How do measured collision speeds between boulders compare to analytical predictions based on turbulent velocity dispersion?
• RQ3 To what extent does collisional cooling enable gravitational collapse in less massive discs?
• RQ4 How does increased radial pressure support affect the onset of gravitational collapse and particle clumping?
• RQ5 Can self-gravity alone drive planetesimal formation without requiring efficient coagulation or fragmentation?

Key findings

• Gravitational collapse occurs at progressively lower column densities with increasing resolution, indicating convergence of the self-gravity solver and reduced numerical smoothing.
• Measured collision speeds are approximately three times lower than predicted by analytical theory, suggesting a need for improved models that include epicyclic motion.
• Collisional cooling enables gravitational collapse in discs with lower solid-to-gas ratios than previously thought, reducing the required column density for collapse.
• Even without magnetic fields, strong clumping and collapse occur when the solids-to-gas ratio is increased by a factor of three, confirming the robustness of the streaming instability.
• In high-density regions such as collapsing clusters, coagulation time-scales are less than one orbit, indicating rapid growth is possible in overdense environments.
• The model shows that planetesimal formation can proceed via self-gravity-driven clustering followed by rapid coagulation, without requiring efficient coagulation in the initial, low-density disc phase.