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arxiv: 2604.26794 · v1 · pith:IXTYCAQ6new · submitted 2026-04-29 · 🌌 astro-ph.EP

The Influence of the Fractal Dimension on Dust Evolution in Protoplanetary Disks

J. E. Sch\"oll , C. P. Dullemond , C. Dominik This is my paper

Pith reviewed 2026-05-07 12:09 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords fractaldustgrowthdimensionporositycaseeffectaggregates
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The pith

Lower fractal dimensions allow dust aggregates to reach higher masses in protoplanetary disks but slow overall growth and do not help trigger streaming instability when collision velocities are assumed independent of porosity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In protoplanetary disks, microscopic dust grains collide and stick to form larger, often porous aggregates. Porosity is quantified by the fractal dimension: lower values mean fluffier structures with more empty space inside. The authors implement this in the DustPy simulation code by making porosity a prescribed function of particle mass, using the fractal dimension as a tunable parameter. They focus on how this affects two main obstacles to growth: the fragmentation barrier, where collisions shatter particles, and the bouncing barrier, where particles rebound instead of sticking. The model accounts for how porosity changes particle motion through the disk gas but deliberately ignores how porosity might change the outcomes of individual collisions. Simulations show that fluffier aggregates (smaller fractal dimension) grow to larger masses, yet the process takes longer. The highest Stokes numbers, which measure how particles drift relative to the gas, stay similar in fragmentation-limited cases but drop in bouncing-limited cases as porosity increases. Under the assumption that the critical velocities for bouncing or fragmentation do not change with porosity, more porous growth does not promote the streaming instability thought to concentrate solids into planetesimals; it can even hinder it in bouncing-limited regimes. Particles still remain far smaller than planetesimals.

Core claim

The dust evolution is strongly influenced by the fractal dimension. Although larger masses are reached for smaller fractal dimensions, the particles are still much smaller than planetesimals. Under the assumption that the bouncing/fragmentation velocity does not depend on the fractal dimension or filling factor, fractal growth is not beneficial for the streaming instability to occur in the case of fragmentation-limited growth and even disadvantageous in the case of bouncing-limited growth.

Load-bearing premise

The bouncing and fragmentation velocities are independent of fractal dimension or filling factor, combined with the decision to neglect the effect of porosity on collision outcomes while only including its effect on dynamics.

read the original abstract

Context: During the first stages of dust coagulation in protoplanetary disks, the dust aggregates are expected to have a high degree of porosity. Most models of dust growth, however, do not take this into account. The reason for this is the technical complexity of this problem. Furthermore, the coagulation/fragmentation kernel for colliding porous or fractal dust aggregates is not well understood. Aims: We wish to explore the effect of aggregate porosity on the evolution of the dust population in protoplanetary disks, with an emphasis on the fragmentation and the bouncing barrier. Methods: We use the DustPy code, and implement porosity as a prescribed function of particle mass with the fractal dimension as a free parameter. In this way, we parameterize the ill-constrained physics of colliding porous/fractal aggregates, and we can explore the effect of different porosity prescriptions. We take into account the effect of porosity on the dust dynamics, while neglecting its effect on the collision outcomes. Results: We find that larger particle masses are reached for lower fractal dimensions. The maximum Stokes numbers that are reached do not depend on the fractal dimension in the case of fragmentation-limited growth and decrease with decreasing fractal dimension in the case of bouncing-limited growth. Furthermore, particle growth is slower for smaller fractal dimensions in our models. Conclusions: The dust evolution is strongly influenced by the fractal dimension. Although larger masses are reached for smaller fractal dimensions, the particles are still much smaller than planetesimals. Under the assumption that the bouncing/fragmentation velocity does not depend on the fractal dimension or filling factor, fractal growth is not beneficial for the streaming instability to occur in the case of fragmentation-limited growth and even disadvantageous in the case of bouncing-limited growth.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper explores the effect of aggregate porosity on dust evolution in protoplanetary disks by implementing porosity in the DustPy code as a prescribed mass-dependent function parameterized by the fractal dimension D_f. This affects dust dynamics (Stokes number, radial drift) but explicitly neglects any effect on collision outcomes. Simulations show that lower D_f yields larger maximum particle masses but slower growth; maximum Stokes numbers are independent of D_f in fragmentation-limited growth and decrease with lower D_f in bouncing-limited growth. Under the assumption that bouncing and fragmentation velocities are independent of D_f or filling factor, the authors conclude that fractal growth is not beneficial for streaming instability in fragmentation-limited cases and disadvantageous in bouncing-limited cases, with particles remaining much smaller than planetesimals.

Significance. If the independence assumption holds, the work provides a systematic parametric exploration of how fractal dimension influences dust growth barriers and streaming instability conditions via numerical forward integration. It usefully isolates the dynamical effects of porosity while highlighting that current models still fall short of planetesimal sizes. The approach of using D_f as a free parameter to probe ill-constrained porous collision physics is a strength for sensitivity studies, though the overall impact is limited by the lack of quantitative validation or tests of the neglected physics.

major comments (3)
  1. [Abstract and Conclusions] Abstract and Conclusions: The headline claim that 'fractal growth is not beneficial for the streaming instability to occur in the case of fragmentation-limited growth and even disadvantageous in the case of bouncing-limited growth' is explicitly conditional on the assumption that bouncing/fragmentation velocities do not depend on fractal dimension or filling factor. No sensitivity analysis or alternative runs are presented to quantify how the maximum masses and Stokes numbers (and thus SI viability) would change if this assumption is relaxed, despite the abstract noting that the coagulation/fragmentation kernel for porous aggregates is ill-constrained; this assumption is load-bearing for the central conclusion.
  2. [Methods] Methods section (porosity implementation): Porosity is included only through its effect on dynamics via the prescribed mass-dependent function with free parameter D_f, while its effect on collision outcomes is neglected. No estimate of the resulting error or justification for the neglect is provided (e.g., via comparison to a model with porosity-dependent velocities), which directly impacts the reported growth rates and barrier locations that underpin the SI assessment.
  3. [Results] Results section: The reported trends in maximum masses, Stokes numbers, and growth timescales emerge from forward integration of the DustPy equations with D_f as an explicit free parameter; no reduction of these quantities to parameter-free expressions or cross-validation against independent models is shown, limiting assessment of whether the D_f dependence is robust beyond the specific numerical setup.
minor comments (2)
  1. [Abstract] The abstract would benefit from explicitly stating the numerical range of fractal dimensions explored and the specific disk parameters (e.g., surface density, turbulence strength) used in the simulations to allow readers to contextualize the results.
  2. [Methods] Notation for the fractal dimension (D_f) and filling factor should be introduced with a clear definition or reference to the functional form in the first methods paragraph for improved readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments on our manuscript. We have carefully considered each point and provide point-by-point responses below. Revisions have been made to improve clarity on assumptions and to add supporting analysis where feasible.

read point-by-point responses

  1. Referee: [Abstract and Conclusions] The headline claim that 'fractal growth is not beneficial for the streaming instability to occur in the case of fragmentation-limited growth and even disadvantageous in the case of bouncing-limited growth' is explicitly conditional on the assumption that bouncing/fragmentation velocities do not depend on fractal dimension or filling factor. No sensitivity analysis or alternative runs are presented to quantify how the maximum masses and Stokes numbers (and thus SI viability) would change if this assumption is relaxed, despite the abstract noting that the coagulation/fragmentation kernel for porous aggregates is ill-constrained; this assumption is load-bearing for the central conclusion.

    Authors: We agree that the conclusions on streaming instability viability rest on the assumption of porosity-independent collision velocities. This choice was made explicitly to isolate dynamical effects of fractal dimension (via Stokes number and radial drift) while using D_f to parameterize the uncertain collision physics, as described in the methods and abstract. A full sensitivity analysis with velocity-dependent models is not possible at present because no established quantitative prescriptions exist for how bouncing and fragmentation thresholds scale with filling factor in the relevant regimes. In the revised manuscript we have added a dedicated paragraph in the discussion section providing a qualitative exploration of possible outcomes if velocities decrease with lower D_f (drawing on literature estimates of porosity effects), and we have strengthened the wording in the abstract and conclusions to foreground the conditional nature of the result. revision: partial

  2. Referee: [Methods] Porosity is included only through its effect on dynamics via the prescribed mass-dependent function with free parameter D_f, while its effect on collision outcomes is neglected. No estimate of the resulting error or justification for the neglect is provided (e.g., via comparison to a model with porosity-dependent velocities), which directly impacts the reported growth rates and barrier locations that underpin the SI assessment.

    Authors: The methods section already states that the effect on collision outcomes is neglected because the coagulation/fragmentation kernel for fractal aggregates remains ill-constrained. We have expanded this justification in the revised version by adding a short discussion of the expected magnitude of the neglected effect, referencing published work indicating that porosity can alter effective collision velocities by factors of a few in certain regimes. A quantitative error budget would require a specific functional form for velocity dependence, which is precisely the physics we are parameterizing rather than resolving; we therefore regard a full error estimate as outside the scope of the present exploratory study. The revised methods now include this additional paragraph. revision: yes

  3. Referee: [Results] The reported trends in maximum masses, Stokes numbers, and growth timescales emerge from forward integration of the DustPy equations with D_f as an explicit free parameter; no reduction of these quantities to parameter-free expressions or cross-validation against independent models is shown, limiting assessment of whether the D_f dependence is robust beyond the specific numerical setup.

    Authors: The trends are obtained from numerical integration because the goal of the work is a systematic parametric survey within the DustPy framework. To provide additional insight we have added an appendix containing a simplified analytic estimate of the fragmentation barrier under the constant-velocity assumption; this shows that the maximum Stokes number scales independently of D_f to leading order when velocities are held fixed, thereby confirming the numerical result in a parameter-free manner. Cross-validation against other coagulation codes is a worthwhile future step but lies beyond the resources available for the current revision; we have noted this explicitly in the conclusions. revision: partial

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on treating fractal dimension as a free parameter that sets porosity versus mass, on the domain assumption that porosity affects only dynamics, and on the ad-hoc choice to hold collision velocities fixed.

free parameters (1)
  • fractal dimension
    Tunable parameter that sets the porosity-mass relation for aggregates.
axioms (2)
  • domain assumption Porosity is implemented as a prescribed function of particle mass
    Allows exploration without solving the full microphysics of porous collisions.
  • ad hoc to paper Effect of porosity on collision outcomes is neglected
    Simplification stated in the methods to focus on dynamical effects.

pith-pipeline@v0.9.0 · 5624 in / 1463 out tokens · 93141 ms · 2026-05-07T12:09:58.228339+00:00 · methodology

discussion (0)

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