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Polydisperse Formation of Planetesimals: The dust size distribution in clumps

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A continuous range of dust sizes changes how the streaming instability builds clumps: weaker density peaks, and a size distribution inside clumps that peaks before the largest grains.

desk verdict The GL quadrature method is a clean, well-validated contribution; the peak-in-size-distribution claim is plausible but rides on a 2D geometry the paper itself flags, so it needs a 3D or stratified check before it becomes a prediction. read the letter →

arxiv 2502.01752 v1 pith:XZYGIWGU submitted 2025-02-03 astro-ph.EP

Jip Matthijsse , Hossam Aly , Sijme-Jan Paardekooper This is my paper · ORCID
classification astro-ph.EP
keywords streaminginstabilitypolydispersedustplanetesimalformationprotoplanetarydiscssizedistributionGauss-Legendrequadratureshearingboxclumping
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the streaming instability—the process thought to gather dust into planetesimals—still works when dust spans a continuous range of sizes instead of a single size. Using two-dimensional shearing-box simulations with many dust species, it finds that the polydisperse instability clumps much less efficiently than the single-size version: the densest structures reach about 82 times the gas density, below the threshold usually required for gravitational collapse. In those densest clumps, larger grains are overrepresented because they are less coupled to the gas, but the trend reverses at the very largest sizes because the biggest grains drift to the edge of the clump core. The result is a peak in the size distribution inside clumps that, the authors argue, could mimic the signature of grain growth without any coagulation having occurred.

What carries the argument

The central objects are the polydisperse fluid equations in a shearing box, in which dust is described by a size density $\sigma(a)$ and the gas feels the drag backreaction as an integral over sizes, $\rho_g \boldsymbol{\alpha}_{\mathrm{drag},g}=\int \sigma(a)(\mathbf{u}-\mathbf{v}_g)/\tau_s(a)\,da$. The paper approximates this integral with Gauss-Legendre quadrature in log Stokes-number space, mapping the quadrature nodes to stopping times $\tau_{s,j}$ and redefining the dust-species densities as $\rho_{d,j}=\tfrac12 \ln(\tau_{s,\max}/\tau_{s,\min}) w_j\tau_{s,j}\sigma(\tau_{s,j})$, which makes the continuum limit converge much faster with the number of species than uniform binning. The second piece of machinery is the size-dependent drift velocity itself: because each size has a different drift speed through the gas, the clumps develop filaments ordered by Stokes number along the drift direction, which is what places the largest grains outside the highest-density core and creates the peak in the size distribution.

What would settle it

Run the same polydisperse streaming-instability setup in a three-dimensional stratified shearing box with vertical gravity (self-gravity still off) and measure the size distribution in the densest clump cores; if the largest grains no longer sit just outside the core, the predicted peak disappears and the paper's central claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the polydisperse streaming instability behaves qualitatively differently from the monodisperse case in the saturated nonlinear regime, and that the difference leaves a measurable imprint in the dust size distribution of the densest structures. The maximum dust density in the standard polydisperse run is $82\pm 14$ in units of the initial gas density, roughly an order of magnitude below the monodisperse counterpart and below the nominal Roche-density clumping threshold for a low-mass disc; the instability's growth rate is also lower, and it cannot be reproduced by a single-size run with the average Stokes number. Within the densest combined-dust regions, abundances rise with grain size until the largest Stokes numbers, which are spatially segregated just outside the densest core, producing a peak in the size distribution. The same peaked shape is found in individual clumps, across spatial resolutions, dust-species counts, sampling methods, and diffusion coefficients, and also at dust-to-gas ratios of 3 and 1, but not at 0.5 where the instability does not grow.

Load-bearing premise

The whole picture depends on the assumption that a two-dimensional, vertically unstratified, isothermal slice without self-gravity preserves the way dust sizes sort themselves inside clumps, so the same peak would appear in a real three-dimensional disc.

Editorial extensions

If this is right

  • Planetesimal formation through the streaming instability needs stronger conditions (higher dust-to-gas ratio or weaker turbulence) than single-size studies suggested, because polydisperse clumps stall below the Roche density in the standard setup.
  • Rubble-pile asteroids may carry a size distribution imprinted by instability dynamics rather than by coagulation: a peak at an intermediate size with a deficit of the very largest grains, even if no growth occurred.
  • Polydisperse simulations cannot be replaced by monodisperse runs tuned to the average dust size; the average-size run overestimates clumping by roughly a factor of 3.5 in the standard case.
  • The faster-converging quadrature sampling makes full size-distribution simulations affordable with only about ten dust species, opening the door to parameter surveys that were previously expensive.
  • Because the largest grains are located just outside the densest core, any body that collapses from the core will be depleted in the most weakly coupled sizes relative to the surrounding disc.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural but unstated consequence: if the segregation pattern survives in three dimensions, the size distribution of observed rubble-pile asteroids could be used as a probe of the local gas density and turbulence level at the formation site.
  • The same size-sorting mechanism might operate in other dusty flows with size-dependent drift (e.g., radial drift in the disc), so the peak-and-deficit pattern could be a general signature of polydisperse drag instabilities, not just the streaming instability.
  • The paper's 2D runs cannot settle whether the peak appears in the bodies that actually form; a testable extension is to seed 3D stratified simulations with the same size distribution and check whether the largest grains still segregate out of the core before collapse.
  • Because the quadrature treats the dust species as integration nodes rather than physical bins, the method transfers to any code that already evolves multiple pressureless fluids; this suggests a low-cost path for incorporating continuous size distributions into broader disc-evolution models.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the polydisperse streaming instability (PSI) in the nonlinear regime using 2D, unstratified, isothermal shearing-box simulations with multiple pressureless dust fluids. It introduces Gauss-Legendre quadrature over Stokes number to discretize the drag back-reaction integral and validates this method against the publicly available psitools linear solver. In the linear regime, the GL method reproduces the analytic growth rate to within about 0.3% with nd=5 and about 0.07% with nd=10, whereas the log-spaced discrete method requires roughly 40 bins for comparable accuracy (Table 2). In the saturated regime, the polydisperse instability reaches maximum dust densities about an order of magnitude below the monodisperse case, and the 99th-percentile size distribution is peaked: larger Stokes numbers are overrepresented in dense structures, except for the largest size, which is spatially offset from the densest part of the combined dust distribution (Secs. 5.1 and 7). The authors attribute this peak to size-dependent spatial segregation and connect it to the dust size distribution that could end up in planetesimals.

Significance. If the results hold, the Gauss-Legendre quadrature scheme is a practical improvement for multispecies streaming-instability simulations, and the paper provides a clean external validation against an independent open-source solver, with openly archived data and analysis code. The linear-regime comparison is the strongest part of the paper and is convincingly quantified; the monodisperse comparisons at both the maximum and the averaged Stokes number are thoughtful controls. The most interesting physical claim, a dynamically produced peak in the size distribution of the densest clumps, is falsifiable and relevant for planetesimal and rubble-pile size distributions. Its robustness is currently limited because the peak and its segregation explanation are demonstrated only within the 2D unstratified model, and the paper itself concedes that a 3D assessment is required; those limitations are openly stated, which is to the authors' credit.

major comments (3)
  1. [Sec. 7 and Fig. 19] The central claim that spatial segregation causes the peak in the size distribution rests on a qualitative reading of a single snapshot. Figure 19 shows the 99th-percentile contours of four dust bins at Omega t = 166.5, while the peak statistic in Figures 6 and 8 is time-averaged over a saturated interval; the text in Sec. 7 says that the largest Stokes numbers lie 'just outside' the densest regions without quantifying that offset. The alternative resonance explanation is also left unresolved in Sec. 5.1, where the overlap is described as visual and 'does not convincingly correspond' to the peak. To make this load-bearing inference quantitative, the authors should compute a species-resolved segregation diagnostic, such as cross-correlations of density fields or centroid offsets of the high-density regions as a function of Stokes number, over the saturated interval, and show that its time dependence tracks the position and amplitude of the peak.
  2. [Sec. 5.2.1 and Sec. 8.1] The quantitative statement about maximum dust density is resolution-dependent. The authors correctly state in Sec. 5.2.1 that, without diffusion, the dust fluid does not converge with spatial resolution and that the maximum density increases with resolution; nevertheless Sec. 8.1 quotes rho_d,max = 82 +/- 14 rho0_g as a main result and compares it with the Roche density. Because the 256-1024 grid trend in Figure 7 is not extrapolated or otherwise bounded, the conclusion that PSI clumps fall below the strong-clumping criterion should not be presented as a converged quantitative result. Please report how the 82 +/- 14 value behaves with resolution, or identify a resolution-independent statistic, such as a converged percentile, to support the comparison.
  3. [Sec. 8.1] The paper's geometry limits the main astrophysical prediction. The authors note that the simulations are 2D unstratified, that 3D simulations are required to assess clumping via the Roche density, that 3D could change the shape of clumps and therefore the spatial separation between dust sizes, and that unstratified linear growth rates are not good predictors of clumping in stratified simulations (Li & Youdin 2021). Since the peaked size distribution in dense regions and its planetesimal-formation relevance depend on this segregation persisting outside the 2D unstratified box, the conclusion should either be explicitly restricted to the 2D model or supported by a 3D or stratified check. A concrete test would be a same-parameter 3D unstratified run, and if feasible a stratified run, measuring whether the Stokes-dependent centroid offset and the resulting peak persist.
minor comments (4)
  1. [Sec. 3.1] The sentence 'If we want to work in log-size space we can map these notes to the correct Stokes number' should say 'nodes' rather than 'notes'.
  2. [Sec. 5.2.1] In the final paragraph, 'a weak trend in the position of the peak to drift to smaller stroke numbers' should be 'smaller Stokes numbers'.
  3. [Fig. 15 caption] The caption for Figure 15 is identical to the caption for Figure 13; if the figure is intended to show the Stokes-range runs discussed in Sec. 6.2, the caption should be corrected accordingly.
  4. [Sec. 6.4] In the sentence describing the dust-to-gas ratio runs, the color assignment is duplicated ('in green' twice) and there is a missing closing parenthesis after 'run PSI_mu,3'; please check the intended colors against Figure 18.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the polydisperse peak and density reductions are measured simulation results, with self-citations used only as benchmarks.

full rationale

The paper's central claims—reduced PSI clumping and the peak in the size distribution at the densest regions—are measured from FARGO3D simulation outputs, not derived from fitted parameters or from the linear theory. The GL quadrature is validated against independently computed psitools linear growth rates; this is a code benchmark, and the nonzero errors in Table 2 show the comparison has discriminative power. Seeding the simulation with the psitools eigenvector tests whether the discrete numerical scheme reproduces the continuum linear operator; it does not encode the nonlinear peak or the maximum dust density. The spatial-segregation explanation is inferred from density contours of individual dust species (Fig. 19) and cross-checked with discrete sampling and clump-averaged size distributions, so it is not equivalent by construction to the 99th-percentile diagnostic. Self-citations to psitools and to Paardekooper & Aly (in prep.) are contextual or benchmark references, not load-bearing premises for the new results. Concerns about the 2D unstratified geometry affecting the peak are external-validity limitations that the paper itself concedes, and they are not a form of circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the fluid approximation for dust, the unstratified shearing-box model, the assumed MRN power-law size distribution, and the psitools linear eigenvalues used as the convergence benchmark. No new physical entities are introduced. The main modeled inputs chosen by hand are the Stokes range, the power-law slope, and the dust-to-gas ratio; the peak location depends on the chosen upper Stokes number.

free parameters (3)
  • Upper Stokes number tau_s,max = 0.1 Omega^-1 (standard), also 0.05 and 0.2
    Chosen by hand as the largest size for which the fluid approximation holds; the peak in the clump size distribution tracks this upper boundary (Sec. 6.2), so it is a load-bearing modeling choice.
  • Power-law slope beta of the size distribution = -3.5 (MRN), also -3.2 and -3.8
    Assumed from interstellar medium and collisional evolution; varying it shifts the peak too little to quantify (Sec. 6.3), but the assumed shape sets the background distribution for all runs.
  • Dust-to-gas ratio mu = 3 (standard), also 0.5 and 1
    Chosen in the high-mu regime where the streaming instability grows; the saturated amplification factor and the efficiency of clumping depend on mu (Sec. 6.4).
assumptions (6)
  • standard math Gauss-Legendre quadrature accurately approximates the drag integral when the integrand is smooth; verified only in the linear regime for the initial size distribution.
    Used in Sec. 3.1 to replace the midpoint-rule sum with GL nodes and weights; the paper shows convergence to psitools growth rates but the integrand is not smooth in the nonlinear regime.
  • domain assumption The dust is a pressureless fluid, valid only for Stokes numbers much smaller than 1; this justifies tau_s,max = 0.1.
    Sec. 2 states the fluid approximation requires tau_s << 1 (Garaud et al. 2004; Jacquet et al. 2011), and Sec. 3.2 sets the largest Stokes number to 0.1.
  • domain assumption The unstratified isothermal shearing box with no vertical gravity represents the disc midplane and captures the clumping-relevant physics.
    Sec. 3.3 and Sec. 8.1; the paper notes that stratification and vertical shear instability could change the behavior, and cites Li & Youdin (2021) showing unstratified linear growth rates do not predict stratified clumping.
  • domain assumption The linear eigenvalue calculations from psitools correctly describe the polydisperse streaming instability in the continuum limit.
    Used as the benchmark for growth rates (Sec. 4) and to set the initial perturbation eigenvectors (Sec. 3.3). psitools is prior work by the same group, but it is publicly available and not fitted to the nonlinear results.
  • domain assumption The dust size distribution is continuous and follows an MRN power law sigma(a) proportional to a^(3+beta), beta = -3.5.
    Sec. 3.2; this is inherited from the interstellar medium and collisional evolution literature, and is the assumed background distribution for all runs.
  • ad hoc to paper The spatial segregation of dust species, inferred from a single 2D snapshot, is the cause of the peak in the size distribution.
    Sec. 7 and Fig. 19 show contours at one time (Omega t = 166.5); the paper states this is the 'preferred explanation' but does not rule out other causes, and the parameter study cannot distinguish the resonant-velocity explanation from the boundary effect.

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Cite this review

Pith. "Pith review of Polydisperse Formation of Planetesimals: The dust size distribution in clumps." pith.science (2026). https://pith.science/paper/XZYGIWGU

@misc{pith2026250201752,
  author       = {Pith},
  title        = {Pith review of: Polydisperse Formation of Planetesimals: The dust size distribution in clumps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XZYGIWGU}},
  note         = {Machine review of arXiv:2502.01752}
}
read the original abstract

The streaming instability is an efficient method for overcoming the barriers to planet formation in protoplanetary discs. The streaming instability has been extensively modelled by hydrodynamic simulations of gas and a single dust size. However, more recent studies considering a more realistic case of a particle size distribution show that this will significantly decrease the growth rate of the instability. We follow up on these studies by evaluating the polydisperse streaming instability, looking at the non-linear phase of the instability at the highest density regions, and investigating the dust size distribution in the densest dust structures. We employ 2D hydrodynamic simulations in an unstratified shearing box with multiple dust species representing an underlying continuous dust size spectrum using FARGO3D. To calculate the drag force on the gas due to a continuous dust size distribution, we apply the Gauss-Legendre quadrature method in dust size space. This method converges faster with the number of dust species than the usual uniform sampling method. The polydisperse streaming instability is less efficient than its monodisperse counterpart in generating dense clumps that could collapse into planetesimals. In the densest dust structure, the larger dust sizes are more abundant because they are less coupled to the gas and, therefore, can clump together more than the smaller dust grains. This trend is broken at the largest dust size due to size-dependent spatial segregation of the highest-density regions, where particles with the largest Stokes numbers are located just outside the densest areas of the combined dust species. This is observed as a peak in the size distribution at the densest regions, which could relate to the size distribution that ends up in the planetesimal after collapse and can mimic the size distribution of dust growth.

Figures

Figures reproduced from arXiv: 2502.01752 by the authors.

Figure 1
Figure 1. Integration of a Normalized Lognormal distribution (dashed line) using a discrete method (blue line) and a Gauss-Legendre quadrature (orange line). using five points with the discrete and the GL method. The inte￾gration error using the discrete method is 8.292% and 0.018% using the GL method. This error will be smaller when using more integration points, but GL is more accurate using fewer integration points, which … view at source ↗
Figure 2
Figure 2. Time evolution of the mSI (run: mSI1024) with dot-dashed line and PSI (run: PSI101024) with solid lines for the ten individual dust species and dotted line for the sum. Where plot A shows the amplitude of the largest mode in the shearing box (A sinusoid with wavenumber K = (30, 0, 30) T ), plot B shows the maximum density in the shearing box. Plot C shows the mean density at every snapshot’s 99th percentile. Plot D … view at source ↗
Figure 3
Figure 3. The normalized amplitude of the largest mode of the shearing box (a sinusoid with a wavenumber K = (30, 0, 30) T ) for a different number of dust species (for blue nd = 10, for orange nd = 20 and for green nd = 40) and different sampling method. The solid line indicates the GL method (27), and the dotted line indicates a log-linear sampling method (runs: PSI20, PSI10, PSI40disc., PSI20disc. and PSI10disc.). The dot-… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: The top plot shows the growth rate in the linear regime at wavevector K = (30, 0, 30) T for setups with different maximum Stokes numbers τs,max, the solid orange and blue line are the analytical growth rates (for mSI and PSI, respectfully) calculated with psitools (Mc￾…
Figure 5
Figure 5. Figure 5: A snapshot of the normalized density in the nonlinear regime (Ωt = 166), for the PSI run PSI101024, showing the density of the sum of the dust species (upper left), gas (lower left) and nine of the ten individual dust species with increasing Stokes number.) 5.1. The si…
Figure 8
Figure 8. Figure 8: The normalized mean size distribution at the 99th and 90th per￾centile and between 140≤Ωt≤160 using the same PSI runs and colour scheme as [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 7
Figure 7. Figure 7: The normalized (mean) density at densest pixel and at 99th per￾centile (indicated by darker and lighter colours; respectively.) for differ￾ent spatial resolutions, the mSI runs at different resolutions are indicated with the dotted line for run mSI1024, mSI512 and mSI …
Figure 9
Figure 9. Figure 9: The normalized mean density at 99th percentile for different numbers of dust species from run mSI1024 indicated with a dotted pink line, run PSI51024 in blue, run PSI101024 in orange and run PSI201024 in green [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 11
Figure 11. Figure 11: The normalized mean size distribution at the 99th percentile and between 150≤Ωt≤200. Where the top plot shows the PSI run with 10 dust species sampled from the GL in blue (run PSI10) and the PSI run with 10 dust species uniformly sampled from a logarithmic scale in gr…
Figure 10
Figure 10. Figure 10: The normalized mean size distribution at the 99th percentile and between 135≤Ωt≤145 using the same PSI runs and colour scheme as [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 13
Figure 13. Figure 13: The normalized mean size distribution at the 99th percentile and between 380 ≤ Ωt ≤ 450 for a dust diffusion coefficient of α = 0 (run PSI10) in blue, α = 10−8 (run PSIα,1e−8) in orange and α = 10−7 (run PSIα,1e−7) in green. Turbulence in a protoplanetary disc stirs t…
Figure 12
Figure 12. Figure 12: Time evolution of the mSI (runs [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 16
Figure 16. Figure 16: The normalized mean density at 99th percentile for different size distributions run PSIβ,−3.2 in blue (β = −3.5), run PSI10 in orange and run PSIβ,−3.8 in green. ranges and is indicated with a vertical dashed line in [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]
Figure 15
Figure 15. Figure 15: The normalized mean size distribution at the 99th percentile and between 380≤Ωt≤450 for a dust diffusion factor of α = 0 (run PSI10) in blue, α = 10−8 (run PSIα,1e−8) in orange and α = 10−7 (run PSIα,1e−7) in green. Changing the maximum Stokes number τs,max will affec…
Figure 17
Figure 17. Figure 17: The mean size distribution at the 99th percentile and between 175 ≤ Ωt ≤ 225 for different power law slopes with β = −3.2 (run PSIβ,−3.2 in blue, β = −3.5 (run PSI10) in orange and β = −3.8 (run PSIβ,−3.8) in green [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]
Figure 18
Figure 18. Figure 18: The normalized mean density at 99th percentile for different dust-to-gas ratios, with µ = 3 (run PSIµ,3 in green, µ = 1 (run PSIµ,1) in orange and µ = 0.5 (run PSIµ,0.5) in green. effect of the size distribution on the RDI streaming instability (Paardekooper & Aly (in…
Figure 19
Figure 19. Figure 19: The contour of density at the upper 99th percentile for the sum of the density and the dust bins of the four largest Stokes numbers. Showing the substructure of the clumps in the nonlinear regime (snapshot at Ωt = 166.5) for the PSI (run PSI101024). ∼ O(10); this will…

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