exoALMA. XII. Weighing and Sizing exoALMA Disks with Rotation Curve Modelling

This paper addresses a central problem in protoplanetary disk physics: how to measure fundamental disk properties—stellar mass, gas (disk) mass, and characteristic disk size—in a way that is dynamically grounded rather than inferred indirectly from dust emission or simplified Keplerian assumptions. The authors focus on the exoALMA large program, which provides high-quality ALMA observations of CO isotopologues (specifically 12CO and 13CO). Because the disk’s rotation curve encodes the gravitational potential, the rotation curve can be used to “weigh” disks: deviations from pure Keplerian rotation arise from (i) the radial pressure gradient and (ii) the disk’s self-gravity. Correctly modeling these non-Keplerian effects is therefore crucial for accurate dynamical mass and size estimates.

The significance of this work is twofold. First, it advances a methodology for dynamical disk mass measurements that is less dependent on uncertain CO chemistry or dust opacity assumptions than flux-based approaches. Second, it enables downstream tests of disk evolution theories, including whether disks are gravitationally stable and what effective angular-momentum-transport efficiency (parameterized by an effective ${\alpha }_S$) is required to explain observed accretion rates. In the broader context of planet formation, these measurements constrain how much gas is available for planet building and how quickly disks evolve.

Methodologically, the authors model the rotation curves of ten selected exoALMA sources using simultaneously fitted 12CO and 13CO rotation curves extracted with discminer. They adopt a thermally stratified physical model for the disk, building on prior analytic work and generalizing it to include vertical temperature structure. The temperature structure is taken from earlier exoALMA modeling (via disksurf) using the Dartois prescription, parameterized by six thermal parameters (e.g., $T_{\mathrm{mid},100}$, $T_{\mathrm{atm},100}$, and power-law indices, plus a vertical scale-height parameter). The disk surface density is assumed to follow a self-similar form $\Sigma (R)$ with parameters including disk mass $M_d$, scale radius $R_c$, and a fixed steepness parameter $\gamma =1$. The rotation curve model includes contributions from stellar gravity (Keplerian term), pressure-gradient effects, and disk self-gravity (computed via an integral with elliptic functions). The fitting is performed with an MCMC framework (emcee) implemented in the DySc code, sampling the posterior distributions of $M_{\star }$, $M_d$, and $R_c$. For each disk, the authors run multiple MCMC fits and propagate systematic uncertainties from the thermal structure by repeatedly drawing thermal parameters from their posterior distribution.

The sample selection is important: the method requires well-defined CO emitting surfaces and an approximately axisymmetric disk so that azimuthal averaging and centrifugal balance assumptions are valid. The authors exclude sources with strong non-axisymmetric features (MWC 758, CQ Tau) and those with low inclination where the emitting layer extraction is unreliable (HD 135344B, HD 143006, J1604). The final sample is: AA Tau, DM Tau, HD 34282, J1615, J1842, J1852, LkCa 15, PDS66, SY Cha, and V4046 Sgr. They further treat AA Tau as an outlier due to contamination from diffuse backside emission at large radii; AA Tau is excluded from statistical conclusions.

Key findings are reported in several layers.

1) Dynamical disk masses and gravitational stability. The authors obtain dynamical disk masses for the ten sources (with best-fit values and uncertainties listed in their Table 1). For example, DM Tau has $M_d=0.057_{-0.020}^{+0.019}{M}_{\odot }$ and $R_c=240_{-27}^{+42}$ au; LkCa 15 has $M_d=0.108_{-0.016}^{+0.016}{M}_{\odot }$ and $R_c=150_{-16}^{+12}$ au; V4046 Sgr has $M_d=0.058_{-0.006}^{+0.006}{M}_{\odot }$ and $R_c=99_{-5}^{+5}$ au. They note a practical detectability threshold: the minimum measurable disk-to-star mass ratio is about 5%. In their sample, only three sources (J1852, PDS66, V4046 Sgr) have best-fit disk masses below this threshold, and they mark such cases accordingly. To assess gravitational instability, they compute the Toomre $Q$ parameter using the modeled surface density and midplane temperature (extrapolated from the stratified thermal structure). Excluding AA Tau due to large uncertainties, they find that all disks have $Q>1$, implying gravitational stability and consistent with the lack of prominent spiral structures.

2) Gas-to-dust ratios are high. Combining dynamical gas masses with dust masses from continuum emission (assuming optically thin dust emission), they infer gas-to-dust ratios well above the canonical value of 100. The average gas-to-dust ratio is approximately $\sim 400$ (excluding AA Tau). This is not statistically consistent with 100. The authors argue that the discrepancy is likely driven by dust-mass underestimation: if dust emission is optically thick in the inner disk, the optically thin assumption used in the dust modeling would bias dust masses low, inflating the inferred gas-to-dust ratio. They also remark on a trend that gas-to-dust ratios appear higher in low dust-mass disks, which may relate to compactness and optical depth effects.

3) Scale radii from rotation modeling and comparison to flux-based radii. A major advantage of dynamical modeling is that it yields a physically motivated scale radius $R_c$ tied to the surface density profile and pressure gradient. The authors compare $R_c$ to flux-based radii: radii enclosing 68% of emission for dust and for 12CO and 13CO. They find that flux-based gas radii are larger than $R_c$: the average ratio is about 2.5 for 12CO and 1.75 for 13CO. For dust, the average ratio of dust radius to $R_c$ is about 0.75, meaning dust emission is more compact than the gas scale radius but broadly comparable. They interpret the mismatch between expected dust-to-gas scale-radius ratios from self-similar viscous evolution (which would predict $\gtrsim 5$) and their observed values as evidence that substructures can slow radial drift and alter dust size evolution. They also find that theoretical predictions for CO radii (based on an assumed CO abundance) systematically overestimate observed CO radii, suggesting CO depletion. When they recompute CO radii using depletion factors inferred from independent forward modeling of N2H+ and rare CO isotopologues in other exoALMA work, agreement improves for most sources.

4) Effective angular momentum transport ${\alpha }_S$. Using the dynamical estimates of $M_d$, $M_{\star }$, and $R_c$, along with accretion rates ${\dot{M}}_{\star }$ from the literature, they compute an effective angular momentum transport parameter ${\alpha }_S$ via a self-similar viscous disk relation. They find ${\alpha }_S$ values spanning a broad range from $10^{-5}$ to $10^{-2}$, with uncertainties dominated primarily by accretion-rate uncertainties (adopted fractional uncertainty of about 0.35 dex). They emphasize that ${\alpha }_S$ is an effective transport efficiency rather than a direct measurement of turbulence.

They also compare their ${\alpha }_S$ estimates with independent constraints from line broadening and radiative transfer modeling for several disks (e.g., DM Tau, V4046 Sgr, MWC480, IM Lup, HD 163296). In some cases, literature values differ by up to orders of magnitude, and they discuss potential reasons such as differences in assumed stellar mass and the possibility of vertical $\alpha$ gradients (e.g., higher effective transport in disk surface layers).

Limitations include: (i) systematic uncertainties from thermal structure modeling, which they address via posterior sampling but cannot eliminate; (ii) the assumption of a self-similar surface density profile with $\gamma$ fixed to 1, which they acknowledge could bias $R_c$ (the most sensitive parameter); (iii) the assumption of axisymmetry and centrifugal balance, motivating the sample cuts; (iv) MCMC statistical uncertainties do not automatically include all systematics (e.g., beam-smearing correlations in the rotation curve data are assumed uncorrelated); and (v) dust-mass inference relies on optically thin assumptions, which likely drives the high gas-to-dust ratios.

Practically, the results matter for anyone using ALMA CO kinematics to infer disk masses and sizes, for interpreting gas availability for planet formation, and for calibrating disk evolution models. Observationally, the paper provides a roadmap for dynamical disk weighing using CO rotation curves with thermal stratification and self-gravity. Theoretically, it suggests that many disks are gravitationally stable at the measured epochs, that dust evolution is strongly influenced by substructures, that CO depletion is common enough to affect size inferences, and that effective angular momentum transport efficiencies vary widely across systems.