The evolution of the flux-size relationship in protoplanetary discs by viscous evolution and radial pebble drift

Anders Johansen; Jes J{\o}rgensen; Johan Appelgren; John Tobin; Michiel Lambrechts; Nagayoshi Ohashi; Nienke van der Marel

REVIEW 4 major objections 4 minor 1 cited by

The evolution of the flux-size relationship in protoplanetary discs by viscous evolution and radial pebble drift

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that observed class 0/I disc fluxes and radii are best matched by a model where cloud-core angular momentum gives a 40 au centrifugal radius and viscous heating is weak, while class II discs tolerate both 40 au and 10 au…

desk verdict A plausible but visually-staked model-data comparison that deserves review, with the central class 0/I preference stronger than the evidence supports. read the letter →

arxiv 2501.04411 v1 pith:JDBQEKOT submitted 2025-01-08 astro-ph.EP

Johan Appelgren , Anders Johansen , Michiel Lambrechts , Jes J{o}rgensen , Nienke van der Marel , Nagayoshi Ohashi , John Tobin This is my paper · ORCID

classification astro-ph.EP

keywords protoplanetarydiscsflux-sizerelationviscousdiscevolutionradialdriftdustmassestimatesclass0/IIIpopulationsynthesis

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

This paper asks whether the observed relation between millimetre flux and disc size can tell apart different assumptions about how protoplanetary discs form and evolve. It argues that embedded class 0/I discs, the youngest objects, are best matched by a model in which the parent cloud core carries enough angular momentum to place material at a centrifugal radius of 40 au around a solar-like star, and in which viscous heating is only 10% as efficient as the angular-momentum transport coefficient. Older class II discs are reproduced about equally well by 40 au and 10 au cores and by both heating efficiencies, so the embedded phase is the discriminating one. The paper also claims that discs older than about 0.5 Myr are optically thin at 1.3 mm, so standard optically-thin dust mass estimates are accurate to within a factor of about two, and that a population of such discs reproduces the observed decline of dust mass with cluster age. If these claims hold, the flux-size plane becomes a direct probe of the angular momentum budget and heating physics of protoplanetary discs.

What carries the argument

The central object is the flux-size relation: the 1.3 mm continuum flux $F_\nu$ plotted against the flux radius, defined as the radius containing 68% or 95% of the total integrated intensity. The machinery is a one-dimensional model in which a Bonnor-Ebert sphere collapses to form a viscous $\alpha$-disc, with angular-momentum transport coefficient $\alpha_\nu = 10^{-2}$ and a separate viscous-heating coefficient $\alpha_{\rm vh}$, while dust grows to the fragmentation limit and drifts radially inward under the Weidenschilling drift prescription. Emission is computed from the Birnstiel et al. (2018) opacity tables using $I_\nu = B_\nu(1-e^{-\tau_\nu})$, so the flux radius is a direct prediction. The 95% flux radius is the discriminating quantity because it traces the faint outer disc, hence the physical size set by the core's angular momentum, while the total flux carries the temperature and mass information affected by viscous heating.

What would settle it

Measure a sample of class 0/I disc sizes with radiative-transfer modelling instead of single-Gaussian fitting; if the resulting 95% flux radii are systematically about a factor of two smaller than the Gaussian radii, the claimed preference for the 40 au, weak-viscous-heating model would weaken or vanish.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the flux-radius plane separates disc formation physics: a disc evolving from a high-angular-momentum core with a centrifugal radius $R_1 = 40$ au and inefficient viscous heating, $\alpha_{\rm vh} = 10^{-3}$, passes through the observed fluxes and 95% flux radii of the eDisk and VANDAM class 0/I samples during its embedded buildup, whereas a low-angular-momentum core with $R_1 = 10$ au and full viscous heating, $\alpha_{\rm vh} = 10^{-2}$, produces discs that are too small and too bright at that stage. For class II discs both models are compatible, so the claim is specifically that the youngest discs carry the information. A second discovery is that radial drift makes discs optically thin at 1.3 mm for most of their lifetime, so the standard optically-thin mass formula traces true dust mass within a factor of about two after 0.5 Myr, with a factor of about three underestimate during the embedded phase.

Load-bearing premise

The comparison assumes that the Gaussian-fit sizes of embedded class 0/I discs can stand in for the model's 95% flux radius, although the paper cites evidence that such fits can overestimate sizes by up to a factor of two.

Editorial extensions

If this is right

The flux-radius plane can discriminate between formation models: the embedded class 0/I phase rejects low-angular-momentum cores with efficient viscous heating.
For discs older than about 0.5 Myr, standard 1.3 mm dust masses trace true masses to within roughly a factor of two, so cluster-age dust-mass trends can be interpreted as radial-drift depletion rather than opacity artefacts.
Very young and massive embedded discs can have their 1.3 mm dust masses underestimated by up to a factor of about three, making those estimates lower limits.
A population model with 40 au centrifugal radii and weak viscous heating reproduces the observed cumulative dust-mass decline across clusters aged about 0.5 to 5 Myr.
With photoevaporation, discs retain a narrow dusty ring outside the gap, keeping flux radii large while total fluxes drop as the inner disc is cleared.

Reading between the lines

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

If Gaussian fitting overestimates embedded disc sizes by the factor of two cited in the paper, then the claimed preference for the 40 au, weak-heating model would probably become a weaker preference for an intermediate or 10 au model; radiative-transfer size measurements of the same targets would settle this directly.
The model omits dust diffusion, which the paper notes can enlarge 90% flux radii by up to a factor of two; including diffusion could let lower-angular-momentum cores reproduce the same observed sizes, making the 40 au requirement an upper bound on core angular momentum.
If angular momentum transport is dominated by MHD winds rather than viscosity, discs may not viscously expand, so reproducing large class 0/I radii may require even higher initial angular momentum or an additional expansion mechanism.
The model's spectral index rises to about 3.25 at 1 Myr, above typical observed values, suggesting that fragmentation-limited growth makes mm-sized grains too scarce; allowing grain growth past the fragmentation barrier or dust trapping would raise the mm opacity and could change the inferred dust masses.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this paper uses a disc formation/evolution model with dust drift to compute flux-size tracks and compares them to eDisk/VANDAM class 0/I and class II samples. The headline result is that class 0/I discs favour high angular momentum (R1=40 au) and weak viscous heating (alpha_vh=1e-3); class II discs are consistent with both variants. The supporting claim that discs are optically thin at 1.3 mm after 0.5 Myr, so standard mass estimates are good to factor ~2, is the better-established part.

What is new: the optical-depth-corrected population synthesis, the explicit flux-size tracks for two parameter combos, and the separate examination of angular momentum and heating effects. The paper is honest: it openly notes the size-estimator problem in Sec. 5.3 and includes an appendix correcting a parameter error in A23, which is the kind of transparency you want.

Soft spots: the central discrimination between the two models rests on a visual overlay of model 95% flux radii on Gaussian-fitted observed sizes. The paper itself concedes that a factor-of-2 size overestimate would make the preference unclear. There is no statistical comparison, no error bars, and no parameter exploration beyond the two hand-picked combinations. Since R1 and alpha_vh were chosen to reproduce the data, the agreement in Sec. 3.2 is partly a fit, not an independent prediction. The population synthesis check is useful but uses the same model. Also, the observed sample excludes upper limits, which may bias the comparison toward larger sizes.

Bottom line: this is a readable, useful paper for people comparing disc evolution models to mm continuum surveys. It deserves peer review, but the authors should be pushed to quantify the model-data comparison and, ideally, to test the size estimator directly. The optically-thin mass claim and the class II consistency will likely survive; the class 0/I preference should be treated as tentative.

Referee Report

4 major / 4 minor

Summary. The paper presents a one-dimensional model of protoplanetary disc formation and viscous evolution, coupled to dust growth and radial drift, and uses it to compute the millimeter flux, flux radius, and optically thin dust mass of young discs. Two model variants are compared: one with high angular momentum (R1=40 au) and weak viscous heating (alpha_vh=1e-3), and one with low angular momentum (R1=10 au) and strong viscous heating (alpha_vh=1e-2). The authors report that the first variant best matches the observed fluxes and radii of class 0/I discs, that class II discs are consistent with both models, and that after about 0.5 Myr discs are largely optically thin at 1.3 mm, so standard dust mass estimates are accurate to within a factor of about two. A population synthesis with the preferred parameters reproduces the cumulative dust mass distributions of young star-forming clusters.

Significance. If the conclusions hold, the paper would establish the flux-size plane as a practical diagnostic for the angular momentum budget and viscous heating efficiency of protoplanetary discs, and would strengthen the case that viscous heating is inefficient in embedded discs. The model is clearly described, with explicit equations for the intensity and flux radius and a transparent discussion of parameter choices. The paper is also commendably candid about its limitations, especially the factor-of-two uncertainty in observed disc sizes and the simplifications of the model. The second claim, that discs are optically thin at 1.3 mm for most of their evolution, is a robust output of the model and is useful for the interpretation of dust mass surveys. However, the central model-preference claim is currently supported only by a visual overlay of model tracks on observed data, without a quantitative goodness-of-fit measure or treatment of upper limits, and the paper itself concedes that the factor-of-two size uncertainty could erase the apparent preference. These issues need to be addressed before the central claim can be considered established.

major comments (4)

[Sec. 3.2, Figs. 4-6] The abstract and Sec. 3.2 state that the model with R1=40 au and alpha_vh=1e-3 is 'best able to match' the observed class 0/I flux and radius data. However, this conclusion is drawn by eye from overlaying model tracks on the observed points in Figs. 4-6; no goodness-of-fit statistic, confidence interval, or treatment of measurement errors is provided. The comparison also excludes discs with upper limits on their size, which biases the observed sample toward larger radii and may favor the high-angular-momentum model. The authors should either (i) quantify the agreement (e.g., the fraction of observed points within a given distance of the model track, or a likelihood-based comparison) and show the sensitivity to the excluded upper limits, or (ii) soften the claim to state that the model is 'consistent with' rather than 'best able to match' the data.
[Sec. 2.2, Eq. (6) vs Sec. 5.3] The model flux radius defined in Eq. (6) is the radius containing a given fraction of the integrated model intensity. The observed class 0/I sizes from Ohashi et al. (2023) and Tobin et al. (2020) are derived from single-Gaussian fits to the continuum emission. These are different estimators, and the paper acknowledges in Sec. 5.3 that Gaussian fitting of embedded objects can overestimate true sizes by up to a factor of two (Tung et al. 2024). This is in tension with the statement in Sec. 2.2 that the differences between the two methods 'should be small'. Because the discrimination between the two models in Sec. 3.2 rests on matching the observed 95% radii, the factor-of-two systematic directly affects the central claim. The authors should either compute synthetic observations from the model (including appropriate resolution and fitting procedures) or, at minimum, propagate the factor-of-two uncertainty into the stated model preference.
[Sec. 2.1, Table 1] The preferred values R1=40 au and alpha_vh=1e-3 are not derived from independent constraints; they are selected because the resulting model tracks pass through the observed class 0/I points in the flux-radius plane. Only two values of each parameter are explored (Table 1 and Fig. 6), and no systematic parameter search or quantitative selection metric is presented. The abstract's claim to be 'best able to match' therefore overstates the evidence. The conclusion should be rephrased as 'among the model variants we consider' unless a broader parameter exploration or a statistical model-comparison test is added.
[Sec. 5.3, Eq. (3)] The flux-size tracks are computed for face-on discs, with no dust diffusion, and using only absorption (Eq. 3) with no scattering. As the authors note in Sec. 5.3, dust diffusion can increase the 90% flux radius by up to a factor of two (Pinilla et al. 2021), scattering can reduce the flux from optically thick regions (Zhu et al. 2019), and inclination affects the measured size. The argument that the effect of diffusion on the 95% radius would still leave the agreement 'good' is not demonstrated. Given that the central discrimination depends on the radii of the class 0/I discs, a test with at least one of these effects included (or a clear justification for ignoring them) is needed to establish robustness.

minor comments (4)

[Abstract] The phrase 'the evolution of the cumulative evolution of the observable dust masses' contains a duplicated word; it should be something like 'the evolution of the cumulative distribution of observable dust masses'.
[Sec. 3.2, Fig. 4 caption] The caption states that the class 0/I discs report the 95% flux radius, but the text in Sec. 3.2 says the eDisk sizes were estimated from Gaussian fitting; please clarify how the reported 95% radius relates to Gaussian-derived sizes, or refer to the observed quantity as 'reported size' rather than '95% flux radius'.
[Sec. 5.1] There is a typo: 'Viscous heating as also been inferred' should be 'Viscous heating has also been inferred'.
[Sec. 4, Fig. 7] The agreement between the model and observed cumulative dust mass distributions is assessed visually. A two-sample Kolmogorov-Smirnov test or a similar quantitative comparison would strengthen the statement that the model 'agrees well' with the observations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the flux-radius agreement is a forward-model parameter comparison, not a fitted prediction, and the population synthesis provides an external check.

full rationale

The paper's central claim is that a disc model with R1=40 au and alpha_vh=1e-3 best reproduces observed class 0/I fluxes and radii. These values are stated as model inputs in Sec. 2.1 and Table 1, not derived from the data by a fitting procedure, and the conclusion is a qualitative selection among two pre-defined parameter combinations. The disc flux and flux radius are computed from the model intensity profile via Eqs. (3)-(6), so the comparison is forward-modeled rather than constructed to match the data. The subsequent population synthesis in Sec. 4 uses the selected model to compare an independent observable, the cumulative dust-mass distribution in star-forming regions, which is not the same data used to motivate the parameter choice. The main admitted limitation in Sec. 5.3 is that Gaussian-fitted eDisk sizes may overestimate true sizes by up to a factor of 2, which could weaken the model preference; this is a measurement and systematic uncertainty, not a circular reduction. Appendix C corrects a parameter error in the authors' prior A23 nominal run and states that the conclusions of that paper are unchanged; this self-correction is not load-bearing for the present derivation. No equation in the paper reduces an output to an input by construction, and no fitted parameter is renamed as a prediction. Hence no significant circularity.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a chain of standard disc physics assumptions plus two hand-selected parameters (R1, alpha_vh). None of the assumptions are new or independently verified here; the model is a numerical forward model, not a first-principles derivation.

free parameters (7)

R1 (centrifugal radius scaling) = 40 au (mod-a-3-r40) and 10 au (mod-a-2-r10)
Sets the angular momentum budget of the collapsing core. The high value is chosen because it yields the best match to class 0/I flux radii; the low value is from A23. This is a hand-selected parameter for the central claim.
alpha_vh (viscous heating efficiency) = 1e-3 (weak) and 1e-2 (full)
Controls midplane temperature and thus flux at fixed radius. Weak heating is chosen to reduce fluxes to match class 0/I observations; this is a hand-selected parameter.
alpha_nu (angular momentum transport coefficient) = 1e-2
Drives disc spreading and accretion. Fixed from A23; chosen, not fitted here, but affects the flux-radius tracks and gap opening time.
alpha_t (turbulent stirring) = 1e-4
Sets the fragmentation-limited maximum particle size through the Birnstiel model. A standard choice from the literature.
v_f (fragmentation velocity) = 1 m/s
Maximum collision velocity for growth in the fragmentation limit. Standard assumption; determines dust sizes and opacities.
T_disc (assumed disc temperature for mass estimates) = 30 K for class 0/I, 20 K for class II
Used in Eq. (1) to convert flux to optically thin dust mass. Standard observational assumptions, not fitted here.
tau_disc (disc lifetime for fdisc scaling) = 2.5 Myr
Used in Eq. (7) to scale observed cumulative dust mass distributions to cluster age; from Mamajek (2009) and A23.

assumptions (7)

domain assumption Gas disc evolution follows the classical viscous alpha-disc equations (Shakura & Sunyaev 1973; Pringle 1981) with a single alpha parameter for angular momentum transport.
Used throughout Sec. 2.1; standard but unproved within this paper.
domain assumption Disc formation from collapse of a Bonnor-Ebert sphere with the Takahashi et al. (2013) rotationally supported collapse model.
Sets the initial gas surface density and centrifugal radius; Sec. 2.1.
domain assumption Dust growth is fragmentation-limited with v_f=1 m/s and turbulent stirring alpha_t=1e-4; only the largest particles are traced.
Sec. 2.1; determines max particle size and opacity evolution.
domain assumption The dust opacity is given by the Birnstiel et al. (2018) opacity tables for a particle size distribution n(a) proportional to a^-3.5.
Sec. 2.2, Fig. 1; needed to convert surface density to optical depth and flux.
domain assumption Disc emission is computed as I_nu = B_nu(T)(1 - exp(-tau_nu)) with face-on viewing (i=0) and no scattering.
Sec. 2.2, Eqs. (3)-(5); ignores inclination, scattering, and vertical temperature gradients.
domain assumption The observed survey flux radii (Gaussian-fitted sizes or 68%/95% radii) are directly comparable to the model flux radii from Eq. (6).
Sec. 3.2 and 5.3; the paper notes Gaussian fitting can overestimate embedded disc sizes by up to a factor of 2.
domain assumption Photoevaporation follows the Picogna et al. (2021) X-ray prescription when calculating population synthesis.
Sec. 4; adopted from prior work.

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Cite this review

Pith. "Pith review of The evolution of the flux-size relationship in protoplanetary discs by viscous evolution and radial pebble drift." pith.science (2026). https://pith.science/paper/JDBQEKOT

@misc{pith2026250104411,
  author       = {Pith},
  title        = {Pith review of: The evolution of the flux-size relationship in protoplanetary discs by viscous evolution and radial pebble drift},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDBQEKOT}},
  note         = {Machine review of arXiv:2501.04411}
}

read the original abstract

In this paper we study the evolution of radiative fluxes, flux radii and observable dust masses in protoplanetary discs, in order to understand how these depend on the angular momentum budget and on the assumed heat sources. We use a model that includes the formation and viscous evolution of protoplanetary gas discs, together with the growth and radial drift of the dust component. We find that we are best able to match the observed fluxes and radii of class 0/I discs when we assume (i) an initial total angular momentum budget corresponding to a centrifugal radius of 40 au around solar-like stars, and (ii) inefficient viscous heating. Fluxes and radii of class II discs appear consistent with disc models with angular momentum budgets equivalent to centrifugal radii of both 40 au or 10 au for solar like stars, and with models where viscous heating occurs at either full efficiency or at reduced efficiency. During the first 0.5 Myr of their evolution discs are generally optically thick at a wavelength of 1.3 mm. However, after this discs are optically thin at mm-wavelengths, supporting standard means of dust mass estimates. Using a disc population synthesis model, we then show that the evolution of the cumulative evolution of the observable dust masses agrees well with that observed in young star forming clusters of different ages.

Figures

Figures reproduced from arXiv: 2501.04411 by the authors.

**Figure 2.** Evolution of the temperature (left panel), the gas surface density and dust surface density (solid and dotted lines, middle [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The true dust mass of the model (blue), the optically thin [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: The total emitted flux at λ “ 1.3 mm versus the flux radius, comparing to observed class II disc in the left panel and class 0/I discs in the right panel (gray symbols). The left and right panel show the 68% and 95% flux radius because the observational samples we comp…

**Figure 3.** Figure 3: During the first 0.25 Myr the disc is optically thick at [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

**Figure 5.** Figure 5: Emitted flux λ “ 1.3 mm as a function of the 68% (left) and 95% (right) flux radius, similar to [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: Disc fluxes at λ “ 1.3 mm as a function of flux radius, similar to [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: Cumulative distributions of dust masses in our model (left panel) and dust masses estimated from ALMA observations (right [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: Spectral index of the whole disc as a function of time. [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: Spectral index across the disc as a function of radial po [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

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write newline

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