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On the Mass Budget Problem of Protoplanetary Disks: Streaming Instability and Optically Thick Emission

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Overdense dust filaments produced by the streaming instability are optically thick at (sub-)millimeter wavelengths, so standard flux-based disk mass estimates can undercount the true dust mass by factors of 2–7 in the inner disk.

desk verdict Genuine step forward on how streaming instability plus scattering can bias ALMA mass estimates, but the headline 2–7 range is conditional on the boxes being placed in the strong-clumping regime, and the paper's own 100 au control shows how much depends on that choice. read the letter →

arxiv 2506.10435 v1 pith:PO6A2VIC submitted 2025-06-12 astro-ph.EP

Daniel Godines , Wladimir Lyra , Luca Ricci , Chao-Chin Yang , Jacob B. Simon , Jeonghoon Lim , Daniel Carrera This is my paper · ORCID
classification astro-ph.EP
keywords protoplanetarydisksstreaminginstabilitydustemissionradiativetransferopticallythickmassestimatesexcessmissingproblem
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 argues that the long-standing 'missing mass' problem of protoplanetary disks can be partly explained by a breakdown of the usual observational assumption that (sub-)millimeter emission is optically thin. In shearing-box simulations of the streaming instability with self-gravity, the instability gathers dust into overdense filaments whose effective optical depth at (sub-)millimeter wavelengths reaches $\tau>1$; when the emergent intensity is converted to a dust mass with the standard optically thin formula, the inferred mass falls short of the true dust mass by factors of roughly 2–7 at 10 au and 1.5–2.4 at 30 au. The bias persists even when the fraction of the area that is optically thick is small, and dust scattering roughly doubles to triples the correction. If this holds, flux-based disk mass surveys are lower limits on the solid content available for planet formation.

What carries the argument

The load-bearing pair is the streaming instability acting as the clumping agent and the mass-excess ratio $\Lambda_\nu \equiv m_{\mathrm{true}}/m_{\mathrm{obs},\nu}$ as the measure of observational bias. The radiative transfer uses the effective optical depth $\tau^{\mathrm{eff}}_\nu = \kappa^{\mathrm{eff}}_\nu \Sigma_d$, with a density-weighted dust opacity that includes both absorption and scattering, and the emergent intensity is computed from a plane-parallel, isothermal slab solution with isotropic scattering. Four dust species (0.36–12 mm radii) evolve in a three-dimensional shearing box with self-gravity, and dust that exceeds twice the Hill density is converted into sink particles representing planetesimals.

What would settle it

Take a resolved (sub-)millimeter image of an inner-disk region where the streaming instability is expected to operate and measure the optical depth of individual filaments directly, for example from multi-wavelength flux ratios or from the spectral index. If no structure with $\tau\gtrsim1$ is found in a disk whose independently measured mass already matches the optically thin estimate, the proposed hidden-mass mechanism is absent for that disk. Alternatively, a full radiative-transfer model of a real disk that recovers the same mass as the optically thin flux argument would falsify the claimed bias for that disk.

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Extended reading notes

Core claim

The central claim is that the streaming instability's overdense filaments are optically thick at (sub-)millimeter wavelengths, so interpreting their emission under the optically thin assumption produces a mass excess $\Lambda_\nu = m_{\mathrm{true}}/m_{\mathrm{obs},\nu}$ of $\sim$2–7 in the inner disk. In the 10 au and 30 au runs, clumps and filaments reach effective optical depths $\tau^{\mathrm{eff}}_\nu \gtrsim 1$ at 0.3–3 mm, and even a small filling factor of optically thick columns is enough to bias the inferred mass. Scattering by millimeter-to-centimeter grains suppresses the emergent intensity and increases $\Lambda_\nu$ by factors of $\sim$1.5–3 relative to absorption-only calculations, while in the optically thin, high-albedo 100 au control case the mass excess can drop slightly below unity, meaning the standard estimate overpredicts mass by up to $\sim$15%.

Load-bearing premise

The 10 au and 30 au runs are initialized in the strong-clumping regime of the streaming instability (the dust-to-gas ratio relative to the pressure-gradient parameter, $Z/\Pi$, above the clumping threshold); if real disks at those radii have lower dust-to-gas ratios, different grain-size distributions, or stronger turbulence, the overdense filaments and the 2–7 correction factors may not appear.

Editorial extensions

If this is right

  • Observed (sub-)mm disk masses in the inner about 30 au should be read as lower limits; the true dust mass can be 2–7 times larger where the streaming instability is active.
  • Dust scattering cannot be neglected when converting (sub-)mm fluxes to masses; including it raises the correction factor by 1.5–3.
  • A small optically thick fraction is enough to bias mass estimates: filling factors below about 0.05 can still hide a significant fraction of the mass.
  • At larger radii where the instability does not strongly clump dust, the standard mass estimate can slightly overestimate the dust mass (by up to about 15%) because scattering boosts the emergent intensity.
  • Longer-wavelength observations that see optically thin emission are needed to recover the hidden mass.

Reading between the lines

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

  • Were these simulations representative of real disks, the population-level missing mass would shrink by the same 2–7 factor only where strong clumping occurs; disks with lower dust-to-gas ratios or stronger turbulence would need little or no correction.
  • Because the simulations are evaluated face-on and the paper notes that optical depth grows with inclination, the quoted mass excess is a lower limit for typical inclined disks; extending the radiative transfer to inclined viewing would likely increase the inferred bias.
  • A natural test is to apply the same optically thin inversion to synthetic images of disks with measured substructure: if filaments observed at 0.87 mm have $\tau\ge1$, the multi-wavelength spectral index of those disks should show the flux-deficit pattern predicted here.
  • This result implies that pebble-accretion efficiencies inferred from comparisons of disk dust masses to exoplanet masses could be systematically underestimated; reconciling the mass budget may not require near-unity accretion efficiency if the disk masses themselves are undercounted.
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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

2 major / 5 minor

Summary. The paper investigates whether optically thick (sub-)millimeter emission from dust filaments formed by the streaming instability can bias dust mass estimates that assume optically thin emission. The authors run three local shearing-box simulations at 10, 30, and 100 au with four dust sizes, self-gravity, and sink particles, then perform radiative transfer with DSHARP opacities including scattering. They define the mass excess Lambda = m_true / m_obs from the standard optically thin estimator and report Lambda ~ 2-7 at 10 au, ~ 1.5-2.4 at 30 au, and ~ 1 at 100 au. The paper concludes that such clumping can partially explain the protoplanetary disk mass budget problem.

Significance. The paper provides a clean numerical experiment with a useful control run at 100 au, a public code release (protoRT), and a physically motivated mechanism linking streaming-instability clumping to observational bias. If the result is robust, it strengthens the argument that flux-based disk masses are lower limits and that optically thick filaments can hide substantial mass even at low filling factors. The main limitation is the narrow parameter space: the strong-clumping regime is entered by construction through Z=0.03 and an ad hoc per-species normalization of the clumping criterion, so the generality of the 2-7 factors is not yet demonstrated.

major comments (2)
  1. [Section 2.1.1 and Fig. 2] The placement of the 10 au and 30 au runs inside the strong-clumping regime is determined by the initial solid-to-gas ratio Z=0.03 together with the normalization of the clumping criterion to Z/Pi/4 (Eq. 11 and Fig. 2). The paper does not test the sensitivity of the resulting mass excess to either Z or this normalization; the 100 au run, which lies below the same threshold, produces no filaments and yields Lambda ~ 1. Because the authors themselves note in Section 2.1.1 that the precise clumping threshold is an active area of research, the central claim Lambda ~ 2-7 cannot yet be considered robust. A low-Z control run (e.g., Z=0.01 at 30 au) or a comparison with the unnormalized Z/Pi criterion is needed to determine whether the result is a generic property of streaming-instability clumping or a consequence of the specific parameter choice.
  2. [Eq. (29) and Section 4] The mass excess values are computed for local shearing boxes of size 0.2H, yet Section 4 compares them directly to disk-integrated mass budget discrepancies reported by Manara et al. (2018) and Mulders et al. (2021). The manuscript should either perform a radial integration over a disk model to derive a disk-integrated mass correction factor or explicitly restrict the claim to the modeled patches; the 2-7 range arises from a single 10 au patch and, as the authors state in Section 2.2, the face-on geometry gives a lower limit, so the direct comparison with survey-level mass deficits may overstate the applicability of the quoted factors.
minor comments (5)
  1. [Section 2.1.1 and Fig. 2] The text defines Z in Eq. (11) as the total dust-to-gas ratio, but the y-axis of Fig. 2 and the clumping argument use the per-species value Z/4; the authors should state explicitly that each species is initialized with Z/4 and label the figure accordingly.
  2. [Section 3 and Fig. 8] The text uses 'T' for the orbital period in the description (e.g., 'before t/T ~ 25') while earlier in the paper 'P' is used for the orbital period; the notation should be standardized.
  3. [Section 2.3, Eq. (27)] The quantity f_thick,nu is defined as the ratio of the mean intensity to the blackbody intensity, not as a geometric fraction of optically thick area; the name 'optically thick fraction' is misleading and should be replaced with a term such as 'normalized intensity' or 'emissivity ratio' to avoid confusion with f_fill.
  4. [Section 2.2] The opacity binning procedure is described clearly, but it would be helpful to state the four bin boundaries explicitly (a_min and a_max for each size bin) in the text, rather than only in Fig. 5, to make the opacity averaging reproducible.
  5. [Section 4] The paper compares its results with those of Rucska and Wadsley (2023) and Scardoni et al. (2021), but it does not state quantitatively how the inclusion of scattering changes the comparison; a sentence summarizing the difference after including scattering would clarify the advance over this earlier work.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the 2–7 mass-excess values are emergent from simulated intensities, not imposed by construction; a minor overlapping-author citation (Lim et al. 2025) sets the clumping-regime input but does not determine the target result.

full rationale

The mass-excess chain is self-contained and not circular. The paper defines Λν = m_true/m_obs,ν (Eq. 29), where m_obs,ν is obtained by inserting the simulated emergent intensity into the standard optically thin estimator Σ_d,obs = <I_ν>/(B_ν κ_ν) (Eq. 28). Nothing in that pipeline is fitted to the Λν values: the opacities are taken from the DSHARP model, the radiative transfer uses a published analytic scattering solution (Miyake & Nakagawa 1993), and the sink-particle treatment is described on physical grounds (density ≥ 2ρ_H). The 100 au run serves as a control: the same estimator and the same code yield Λ≈1 at 3 mm and no filaments, showing that the ~2–7 range is a property of the simulated clumped state rather than an artifact of the definitions. The only overlapping-author self-citation with any role in setup is the Lim et al. (2025) clumping criterion used in Sec. 2.1.1 to place the 10 and 30 au runs in the strong-clumping region of the Z/Π plane, with the paper's own Z/Π/4 normalization for four species. This is an input condition, not a fitted parameter and not the target result; the mass excess is computed from the subsequent dust evolution and radiative transfer. The choice of Z=0.03 and of the Z/Π/4 normalization is an important sensitivity/robustness limitation, but it is not a circular reduction: the paper does not claim to predict these mass-excess factors for arbitrary disks, and the claim is explicitly anchored to simulations initialized in the clumping regime. A non-circular but legitimate concern is that the headline factors may not apply if real inner-disk dust-to-gas ratios are below the clumping threshold; this belongs under correctness risk, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a specific disk model, opacity model, and clumping criterion; these are reasonable but not externally anchored, so the numerical mass excess values carry model dependence. No new physical entities are introduced; sink particles are numerical representations of planetesimals within the existing physics of the simulation.

free parameters (4)
  • Initial dust-to-gas ratio Z = 0.03
    Set for all three simulations; controls the dust surface density that determines where the disk becomes optically thick.
  • Sink particle density threshold = 2 times the Hill density
    Chosen by hand in Section 2.1.1 to filter out clumps disrupted by turbulence; affects how much mass is locked into planetesimals.
  • Clumping criterion normalization Z/Pi/4 = Z/Pi divided by 4 species
    Used to place the 10 and 30 au simulations in the strong-clumping regime of Lim et al. (2025); the division by species is an assumption.
  • Dust species grain radii = 12, 3.6, 1.2, and 0.36 mm
    Four discrete sizes chosen from the DSHARP distribution; the absorption and scattering opacities, and thus the mass excess, depend on these sizes.
assumptions (5)
  • domain assumption The Miyake and Nakagawa (1993) analytic solution for the mean intensity assumes an isothermal slab, isotropic scattering, and no incident radiation at the disk surface.
    Used in Equation 23 to compute the source function; real disks with vertical temperature gradients would change the emergent intensity and the inferred bias.
  • domain assumption DSHARP dust opacities are representative of grains in protoplanetary disks.
    Adopted for Equations 15 to 17; the authors note that DIANA opacities would roughly double the mass excess.
  • domain assumption The streaming instability produces strong dust clumping when Z/Pi/4 exceeds the threshold identified by Lim et al. (2025).
    Determines which simulations are expected to form filaments; cited as an active area of research.
  • ad hoc to paper Gravitational collapse into planetesimals occurs when the local dust density reaches twice the Hill density.
    Set by hand in Section 2.1.1 as a practical collapse threshold that accounts for turbulent diffusion.
  • domain assumption The disk is vertically isothermal and observed face-on.
    Used for the temperature profile and 1D radiative transfer; the authors state face-on viewing gives a lower limit on the mass excess.

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Cite this review

Pith. "Pith review of On the Mass Budget Problem of Protoplanetary Disks: Streaming Instability and Optically Thick Emission." pith.science (2026). https://pith.science/paper/PO6A2VIC

@misc{pith2026250610435,
  author       = {Pith},
  title        = {Pith review of: On the Mass Budget Problem of Protoplanetary Disks: Streaming Instability and Optically Thick Emission},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PO6A2VIC}},
  note         = {Machine review of arXiv:2506.10435}
}
abstract

Statistical studies of protoplanetary disks and exoplanet populations often exhibit a "missing mass" problem, where observed dust masses in (sub-)millimeter surveys are significantly lower than expected when compared to the mass of evolved exoplanetary systems. We investigate how the streaming instability and subsequent planetesimal formation in protoplanetary disks might solve this missing mass problem when (sub-)millimeter observations are interpreted under the assumption of optically thin emission. We conduct hydrodynamical simulations of the streaming instability with self-gravity after which radiative transfer calculations with dust scattering are performed to measure the (sub-)millimeter intensity. The measured intensity is then used to estimate the disk mass under the assumption of optically thin emission and compared to the true mass in the simulation to calculate the observational bias via the mass excess. We find that the emission from overdense filaments that emerge due to the streaming instability are optically thick at (sub-)millimeter wavelengths, leading to mass excess factors of $\sim 2-7$, even when the optically thick fraction is low.

Figures

Figures reproduced from arXiv: 2506.10435 by the authors.

Figure 1
Figure 1. Radial profiles of key physical properties in our proto￾planetary disk model. We model a disk orbiting a solar-mass star with a total mass of 𝑀disk = 0.02 𝑀⊙, composed of dust grains with an internal density of 𝜌• = 1.675 (g cm−3 ). The vertical red dashed lines indicate the radial locations of our polydisperse, self-gravitat￾ing simulations. The simulations were set up with Stokes numbers corresponding to the locat… view at source ↗
Figure 2
Figure 2. Regime in which the streaming instability induces strong particle clumping. The normalized solid-to-gas ratio of each simu￾lation is shown on the y-axis and is demarcated by distinct dashed lines. The corresponding stars represent the four species in each simulation, which are placed within the clumping boundary accord￾ing to the respective Stokes number. marked by red dashed lines in all panels. The following sub￾s… view at source ↗
Figure 3
Figure 3. The temporal evolution of three polydisperse streaming instability simulations with self-gravity, shown column-wise. The top row presents the maximum dust density over time, alongside the modified Hill density criterion for planetesimal formation. Four lines represent the evolution of the different species, with the second row showing the quantity of each species that is present over time (i.e., not accreted by plan… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Dust opacity and albedo across ALMA bands. The top panel shows the absorption (blue) and scattering (red) opacities from the DSHARP dust model for four different ALMA bands. The second panel presents the corresponding albedos. Using Equation 15, we calculate the absorp…
Figure 5
Figure 5. Figure 5: Grain size binning for multi-species analysis. The top panel shows the absorption (blue) and scattering (red) opacities for ALMA Band 7. Dashed lines represent the full grain size distri￾bution, while colored markers indicate the binned values used in our radiative tra…
Figure 6
Figure 6. Figure 6: Time evolution of the dust distribution. The panels show the vertically and azimuthally averaged dust density as a function of time for each simulation and radial location. The red dashed line indicates the orbital period of maximum dust density [PITH_FULL_IMAGE:figur…
Figure 7
Figure 7. Figure 7: Radiative transfer results at the 0.87 mm wavelength. Each column corresponds to a different simulation and radial location in the disk. The top and bottom rows show the optical depth and the emergent intensity at the output plane (𝑧 = 𝐿𝑧/2), respectively, at the time …
Figure 8
Figure 8. Figure 8: Observational diagnostics from our three streaming instability simulations, presented column-wise. The top plots display the mass excess over time and as function of both radial location and observed (sub-)mm wavelengths. The corresponding filling factors, average opti…
Figure 9
Figure 9. Figure 9: Absorption-only case. The results from the radiative transfer calculations from our three streaming instability simulations are presented column-wise. The top plots display the mass excess over time and as function of both radial location and observed (sub-)mm waveleng…

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