Filling-factor dependence of dust-aggregate collision threshold velocities and its consequence for fractal growth and the streaming instability in protoplanetary disks

Planetesimals are widely believed to form in two steps: dust aggregates in protoplanetary disks grow by mutual collisions until they reach 'pebble' sizes, after which the streaming instability can concentrate them gravitationally into planetesimals. Growth is limited by collision-outcome barriers whose threshold velocities — the bouncing threshold and the fragmentation threshold — are treated in essentially all global dust-evolution models as independent of aggregate porosity, even though early growth is expected to produce highly porous, fractal aggregates. Schöll, Dullemond and Dominik (2026) showed, with a fractal-dimension-parameterized Smoluchowski model, that under this porosity-independence assumption fractal growth does not help (and in the bouncing-limited case actively hinders) reaching the Stokes numbers and dust-to-gas ratios required for the streaming instability, but their conclusion is explicitly conditional: numerical studies suggest bouncing may vanish below a critical filling factor and the fragmentation velocity may decrease with porosity, either of which could reverse it. The open problem is to determine, quantitatively, how the bouncing and fragmentation threshold velocities depend on the filling factor (and fractal dimension) across the porosity range traversed by fractal growth, and to establish whether fractal growth is beneficial, neutral, or disadvantageous for triggering the streaming instability once these dependencies are included. What remains open: the bouncing threshold has been characterized only for filling factors 0.4–0.5, the numerics-versus-laboratory discrepancy on the critical filling factor is unresolved, the fragmentation-threshold porosity dependence has not been revisited since 2016, and no global model combines a fractal-dimension scan with porosity-dependent thresholds.

Background

In protoplanetary disks — the gas- and dust-rich disks around young stars from which planetary systems form — the building blocks of planets begin as micrometer-sized solid grains. These grains collide, stick, and build up increasingly massive dust aggregates. An aggregate's internal structure is characterized by its volume filling factor $\phi$ (the fraction of its volume actually occupied by solid material, equal to the bulk density divided by the monomer material density) and, during the early 'hit-and-stick' phase in which collisions restructure nothing, by its fractal dimension $D$, the exponent in the mass–radius relation $a \propto m^{1/D}$; compact aggregates have $D = 3$, while highly porous, fractal ones have $D$ approaching 2. Because collision energies grow with aggregate mass, hit-and-stick growth eventually gives way to restructuring, compaction, and destructive collisions. Laboratory experiments and numerical simulations therefore classify each collision by its outcome — sticking, bouncing (the aggregates rebound without mass transfer), or fragmentation (collisional breakup) — with the outcome controlled by threshold velocities: collisions above the bouncing threshold $v_{\mathrm{b}}$ (but below the fragmentation velocity $v_{\mathrm{f}}$) rebound, and collisions above $v_{\mathrm{f}}$ fragment. The classic synthesis of laboratory collision experiments by Güttler et al. (2010) and Zsom et al. (2010) established mass-dependent threshold prescriptions and identified the 'bouncing barrier', the size at which collisions begin to rebound and growth stalls, alongside the fragmentation barrier (Blum & Wurm 2008). Global models that evolve the full aggregate mass distribution — Smoluchowski coagulation codes, which solve the mean-field rate equations for the number density of aggregates of each mass (e.g. the publicly available DustPy code), or moments/representative-particle methods (Estrada, Cuzzi & Umurhan 2022; Michoulier, Gonzalez & Price 2024) — then track whether growth can reach sizes at which another process can take over: the streaming instability (Youdin & Goodman 2005; Johansen et al. 2007), a hydrodynamic instability of the particle–gas mixture that concentrates aerodynamically coupled solids into gravitationally collapsing clumps. Simulations of the streaming instability indicate it requires particles with Stokes number (the dimensionless aerodynamic stopping time measuring how strongly a particle couples to the gas) above roughly $10^{-3}$ and a dust-to-gas surface-density ratio of order 0.005–0.01 or above (Carrera, Johansen & Davies 2015; Yang, Johansen & Carrera 2017; Li & Youdin 2021). Because fractal aggregates of a given mass have larger cross sections and lower bulk densities than compact ones, porosity changes both the collision physics and the aerodynamics; whether porous growth helps or hurts the path to the streaming instability depends on details that remain poorly constrained. In particular, distinct-element-method (DEM) simulations, which represent an aggregate as a network of individual monomers (the micrometer-sized constituent grains) connected by cohesive contacts (pioneered by Dominik & Tielens 1997), find that bouncing occurs only for relatively compact aggregates — Wada et al. (2011) and Seizinger & Kley (2013) find no bouncing when the filling factor falls below roughly 0.3–0.4 — whereas laboratory experiments (Blum & Münch 1993; Langkowski, Teiser & Blum 2008; Schräpler, Landeck & Blum 2022) still observe bouncing at lower filling factors, a long-standing discrepancy that Arakawa et al. (2023) proposed to explain through a size dependence of the threshold. For fragmentation, Wada et al. (2009) found the collisional-growth threshold to decrease by about a factor of two between compact ($D = 3$) and fractal ($D = 2$) aggregates, and Gunkelmann, Ringl & Urbassek (2016) found the fragmentation velocity to decrease with decreasing filling factor below $\phi \approx 0.1$. Nevertheless, essentially every global dust-evolution model adopts threshold velocities that depend at most on mass and material, not on porosity. Schöll, Dullemond & Dominik (2026) quantified the consequence of this gap: implementing porosity in DustPy as a prescribed function of mass with the fractal dimension as a free parameter ($D = 2$, 2.5, 3), and accounting for porosity in the aerodynamics (cross sections, Stokes numbers, relative velocities) while neglecting its effect on collision outcomes, they found that lower fractal dimensions reach larger masses but smaller maximum Stokes numbers in bouncing-limited growth, and concluded that, under the porosity-independence assumption, fractal growth is not beneficial for the streaming instability in fragmentation-limited growth and even disadvantageous in bouncing-limited growth. Their conclusion is explicitly conditional, and their discussion identifies precisely what would be needed to lift the condition: threshold-velocity relations $v_{\mathrm{b}}(\phi)$ and $v_{\mathrm{f}}(\phi, D)$ covering the porosity range ($\phi$ down to $10^{-5}$) that fractal growth traverses, including whether bouncing ceases entirely below a critical filling factor.

Problem Statement

Determine, quantitatively, how the collision-outcome threshold velocities of dust aggregates depend on aggregate porosity — specifically the bouncing threshold velocity $v_{\mathrm{b}}(\phi, m)$, equivalently characterizable by a sticking probability, and the fragmentation threshold velocity $v_{\mathrm{f}}(\phi, D, m)$ — over the porosity range traversed by fractal growth in protoplanetary disks, $\phi$ from unity down to values of order $10^{-5}$, including whether bouncing ceases entirely below a critical filling factor and where that critical value lies. Then incorporate these dependencies into a population-level dust-evolution model that treats porosity or fractal dimension as a modeled property, and establish whether fractal (highly porous) growth is beneficial, neutral, or disadvantageous for reaching the conditions for planetesimal formation by the streaming instability — maximum Stokes numbers of order $10^{-3}$ or above together with dust-to-gas surface-density ratios of order 0.005–0.01 or above — thereby testing, and potentially reversing, the conditional conclusion of Schöll et al. (2026) that fractal growth does not help and may hinder the streaming instability. Both parts are jointly required: the threshold-velocity relations and the critical filling factor fix the input physics, but the verdict on the streaming instability is reached only by carrying them through the population-level model.

References

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