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arxiv: 2604.11642 · v1 · submitted 2026-04-13 · 🌌 astro-ph.EP · astro-ph.SR

Investigation of gravitational stability of protoplanetary disks based on statistical analysis of their masses

Sophia A. Drobchik , Sergey A. Khaibrakhmanov This is my paper

Pith reviewed 2026-05-10 15:24 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.SR
keywords protoplanetary disksgravitational stabilityToomre parameterdisk mass distributionstar-forming regionsobservational biasespower-law profile
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The pith

Statistical analysis of 1155 protoplanetary disks finds only 1.2 percent formally gravitationally unstable.

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

The paper gathers disk masses from ten different surveys covering star-forming regions and fits a power-law form to the overall mass distribution. For each disk it computes the Toomre stability parameter Q by assuming the surface density falls off as a power law with radius. Only a small percentage of the disks fall below Q = 1 or sit in the narrow marginal band, which appears at odds with theoretical models that anticipate more frequent gravitational collapse. The authors argue that current mass values are systematically too low because of optical depth, molecular depletion, and uncertain gas-to-dust ratios, so the true share of unstable disks is probably much larger.

Core claim

A combined sample of 1155 protoplanetary disks yields a mass distribution dN/dM proportional to M to the power of minus 1.36. When the Toomre parameter is evaluated for every disk under the assumption of a power-law radial density profile, 1.2 percent satisfy Q less than 1 and another 1.7 percent satisfy 1 less than or equal to Q less than or equal to 2. The scarcity of unstable systems is ascribed to observational underestimation of disk masses arising from optical-depth effects, CO depletion, and uncertainty in the gas-to-dust ratio, implying that the actual fraction of gravitationally unstable disks is substantially higher.

What carries the argument

The Toomre parameter Q evaluated for each disk from its total mass and an assumed power-law radial density profile.

Load-bearing premise

The assumed power-law density profiles and the observed disk masses together give an accurate picture of gravitational stability without large hidden biases from temperature structure or non-axisymmetric motions.

What would settle it

New observations that return disk masses several times higher than current estimates and that produce a much larger fraction of objects with Toomre Q below 2 would support the claim that instability is more common once biases are removed.

Figures

Figures reproduced from arXiv: 2604.11642 by Sergey A. Khaibrakhmanov, Sophia A. Drobchik.

Figure 1
Figure 1. Figure 1: Density distribution of disk masses by stability type. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overall mass distribution of the sample disks. The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: illustrates the dependence of the Toomre pa￾rameter Q on the disk-to-star mass ratio. As the mass ratio Mdisk/M⋆ increases from 10−3 to 1, Q decreases from ap￾proximately 102 to 10−1 . When the mass ratio exceeds 0.132, the Q parameter falls below the critical value Q = 1, indi￾cating the transition from gravitationally stable to unsta￾ble systems. This result aligns with theoretical predictions, which req… view at source ↗
Figure 3
Figure 3. Figure 3: Power-law fit to the mass distribution of disks with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

We compiled a sample of $1155$ protoplanetary disks, combining data from ten surveys of star-forming regions. Based on the sample, we constructed a power-law approximation of the disk mass distribution: $dN/dM \propto M^{-\beta}$, $\beta = 1.36 \pm 0.14$. We used the sample for a statistical analysis of the gravitational stability of protoplanetary disks. To analyze the stability of the disks, we calculated the Toomre parameter ($Q$) for each of them. In the calculations, it was assumed that the radial density distribution in the disks is described by a power-law profile. The calculations of the Toomre parameter show that only $1.2$ % of the disks in the sample are formally unstable ($Q < 1$), while $1.7$ % are in a state of marginal stability ($1 \leq Q \leq 2$). The low observed abundance of unstable disks contradicts theoretical expectations and may be explained by a systematic underestimation of disk masses due to limitations of observational methods. Considering the effects of optical depth, CO depletion, as well as uncertainty in the gas-to-dust ratio, we conclude that the actual fraction of the gravitationally unstable systems may be significantly higher.

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 / 1 minor

Summary. The paper compiles a sample of 1155 protoplanetary disks from ten surveys, constructs a power-law mass distribution dN/dM ∝ M^{-β} with β = 1.36 ± 0.14, and computes the Toomre Q parameter for each disk assuming a power-law radial density profile. It reports that only 1.2% of disks are formally unstable (Q < 1) and 1.7% are marginally stable (1 ≤ Q ≤ 2), concluding that this low abundance contradicts theoretical expectations and is likely due to systematic underestimation of disk masses from observational limitations including optical depth, CO depletion, and gas-to-dust ratio uncertainties.

Significance. The compilation of a large, multi-survey sample and the explicit power-law fit with uncertainty constitute a strength for statistical analysis of disk masses. If the Toomre Q results are shown to be robust, the finding of a low unstable fraction would be significant for protoplanetary disk studies, as it would point to potential biases in mass measurements that affect models of gravitational instability and early planet formation.

major comments (3)
  1. [Abstract and Toomre parameter calculations] Abstract and Toomre parameter calculations: The radial density distribution is assumed to follow a power-law profile to derive local surface density Σ from total mass for each of the 1155 disks, but the manuscript provides no details on the adopted power-law index, outer radius normalization, or sound speed (temperature) values. Because Σ scales as 1/R_outer² and Q ∝ c_s/Σ, variations in these parameters can shift disks across the Q=1 and Q=2 thresholds, directly affecting the reported 1.2% and 1.7% fractions.
  2. [Statistical analysis of gravitational stability] Statistical analysis of gravitational stability: No description is given of error propagation, Monte Carlo sampling, or sensitivity tests for the Toomre Q values that incorporate the mass uncertainties from the ten surveys or the ±0.14 uncertainty on the fitted β. This omission leaves the central instability fractions without quantified reliability.
  3. [Discussion section] Discussion section: The inference that the actual unstable fraction may be significantly higher rests on qualitative discussion of optical depth, CO depletion, and gas-to-dust ratio effects, but the manuscript contains no quantitative model or estimated correction factor for mass underestimation to support the claim.
minor comments (1)
  1. A histogram or cumulative distribution plot of the computed Q values across the full sample would help readers assess the overall distribution and the robustness of the threshold fractions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed report. Their comments highlight important aspects of clarity, robustness, and support for our conclusions. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: Abstract and Toomre parameter calculations: The radial density distribution is assumed to follow a power-law profile to derive local surface density Σ from total mass for each of the 1155 disks, but the manuscript provides no details on the adopted power-law index, outer radius normalization, or sound speed (temperature) values. Because Σ scales as 1/R_outer² and Q ∝ c_s/Σ, variations in these parameters can shift disks across the Q=1 and Q=2 thresholds, directly affecting the reported 1.2% and 1.7% fractions.

    Authors: We agree that these computational assumptions must be stated explicitly for reproducibility. In the revised manuscript we will add a dedicated paragraph in the Methods section specifying the adopted power-law index for the surface density profile, the outer radius used to normalize Σ from total mass, and the temperature (hence sound speed) values employed. We will also include a brief sensitivity test demonstrating how the reported instability fractions respond to plausible variations in these parameters. revision: yes

  2. Referee: Statistical analysis of gravitational stability: No description is given of error propagation, Monte Carlo sampling, or sensitivity tests for the Toomre Q values that incorporate the mass uncertainties from the ten surveys or the ±0.14 uncertainty on the fitted β. This omission leaves the central instability fractions without quantified reliability.

    Authors: The referee correctly identifies the absence of formal uncertainty quantification on the instability fractions. We will revise the statistical analysis section to incorporate a Monte Carlo procedure: 10,000 realizations will be drawn by sampling each disk mass within its reported survey uncertainty and β within its ±0.14 error. The resulting distribution of unstable and marginally stable fractions will be reported, providing 1σ uncertainties on the 1.2 % and 1.7 % values. revision: yes

  3. Referee: Discussion section: The inference that the actual unstable fraction may be significantly higher rests on qualitative discussion of optical depth, CO depletion, and gas-to-dust ratio effects, but the manuscript contains no quantitative model or estimated correction factor for mass underestimation to support the claim.

    Authors: We acknowledge that the present discussion of mass underestimation remains qualitative. A full quantitative correction model for every disk would require extensive additional radiative-transfer and chemical modeling that lies outside the scope of this statistical compilation. In the revision we will nevertheless strengthen the Discussion by citing literature-based order-of-magnitude correction factors for optical depth, CO depletion, and gas-to-dust ratio variations, and we will show how even modest average corrections would raise the unstable fraction substantially. This provides more concrete support for our conclusion while remaining within the paper’s statistical framework. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper compiles 1155 disk masses from surveys, fits the distribution parameter β = 1.36 ± 0.14 directly to that sample via dN/dM ∝ M^{-β}, and separately applies the standard Toomre Q formula to each disk under an assumed power-law radial density profile. The reported 1.2 % unstable and 1.7 % marginal fractions are obtained by direct evaluation of Q < 1 and 1 ≤ Q ≤ 2 on the input masses and profile; they are not predictions derived from β or forced by construction. The inference of systematic mass underestimation is an interpretive comparison to theoretical expectations, not a mathematical identity or fitted output. No self-citations, uniqueness theorems, or ansatzes from prior author work appear in the provided text to justify core steps. The chain is therefore self-contained against external data and standard formulas.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The analysis rests on a fitted power-law index for the mass distribution and a domain-standard assumption of power-law radial density; no new entities are introduced.

free parameters (1)
  • beta = 1.36
    Power-law index fitted to the disk mass distribution from the compiled sample.
axioms (1)
  • domain assumption Radial density distribution in protoplanetary disks follows a power-law profile
    Invoked explicitly for Toomre Q calculations on each disk.

pith-pipeline@v0.9.0 · 5538 in / 1348 out tokens · 54871 ms · 2026-05-10T15:24:47.757145+00:00 · methodology

discussion (0)

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