DiskMINT: A Self-Consistent Disk Modeling Framework

• DiskMINT is a self-consistent thermochemical framework that models gas and dust properties in protoplanetary disks using ALMA continuum and C18O line observations.
• It integrates vertical hydrostatic equilibrium with detailed dust and gas interactions, yielding realistic temperature, density, and chemical structures.
• DiskMINT-GARDEN expands this approach with a public model grid and machine-learning tool to enable rapid, survey-scale disk mass inference.

DiskMINT, short for Disk Model for INdividual Targets, is an open-source, self-consistent thermochemical disk-modeling framework for inferring the gas and dust properties of protoplanetary disks from continuum and CO isotopologue observations. In its initial form, it was introduced as a Python tool to estimate gas disk masses from optically thin $\mathrm{C^{18}O}$ rotational lines by coupling an SED-derived dust structure to vertical hydrostatic equilibrium, isotope-selective chemistry, and grain-surface conversion of CO into $\mathrm{CO_2}$ ice (Deng et al., 2023). Subsequent developments extended the framework to radially and vertically decoupled gas and dust distributions for individual disks and, later, to DiskMINT-GARDEN, a public grid of 480 self-consistent models plus a fast machine-learning inference tool for large ALMA samples (Deng et al., 18 Sep 2025, Deng et al., 24 Jun 2026).

1. Origins and scientific scope

DiskMINT was developed to address a persistent interpretation problem in protoplanetary-disk observations: faint CO isotopologue emission had often been taken to imply either very low gas masses or severe CO depletion. The framework instead models the vertical density, temperature, and chemistry together, so that the emitting layer is not imposed ad hoc but derived from the same physical assumptions used to interpret the observables (Deng et al., 2023).

The original formulation focused on individual targets and used long-wavelength SED constraints, multiple dust grain sizes, and $\mathrm{C^{18}O}$ line emission to infer disk gas masses. In later work, DiskMINT was extended to infer not only total gas mass but also its radial distribution, while allowing gas and dust surface densities to be spatially decoupled both radially and vertically (Deng et al., 18 Sep 2025). DiskMINT-GARDEN generalized this target-specific framework into a survey-scale inference system by combining the DiskMINT physics with a precomputed model grid and an XGBoost regressor that maps a small set of ALMA observables to $(M_{\rm gas}, \epsilon, R_c)$ (Deng et al., 24 Jun 2026).

In the DiskMINT literature, “self-consistent” has a specific meaning. The pressure-supported vertical density profile, the dust and gas temperature fields, and the chemistry that sets the molecular abundances are computed together and mutually constrained; no piece is held fixed or imposed ad hoc. The resulting thermochemical structure places the CO-emitting layers, their temperatures and optical depths, and the continuum optical depths in a single internally consistent model (Deng et al., 24 Jun 2026).

2. Dynamical and thermal structure

A central component of DiskMINT is vertical hydrostatic equilibrium. In the grid formulation, the standard relation

$$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$

is solved together with the energy balance so that the gas and dust temperature fields and the density structure are updated iteratively (Deng et al., 24 Jun 2026). The local scale height follows

$$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$

In the extended individual-target framework, DiskMINT solves the same vertical hydrostatic problem with the actual $T_g(r,z)$ rather than assuming vertical isothermality, and evaluates the midplane scale height as $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$ (Deng et al., 18 Sep 2025).

Disk surface densities are represented with tapered power-law profiles. DiskMINT-GARDEN adopts

$$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$

with $\gamma=1$, $\mathrm{CO_2}$0 au, and the same profile for gas and dust with a constant $\mathrm{CO_2}$1 across radii (Deng et al., 24 Jun 2026). In the radially decoupled framework, the corresponding similarity-solution form is

$\mathrm{CO_2}$2

with $\mathrm{CO_2}$3 and $\mathrm{CO_2}$4 treated as independent functions constrained by spatially resolved ALMA radial profiles (Deng et al., 18 Sep 2025).

Dust settling is included in the more advanced DiskMINT models through a steady-state turbulent diffusion equation,

$\mathrm{CO_2}$5

with

$\mathrm{CO_2}$6

and $\mathrm{CO_2}$7 (Deng et al., 18 Sep 2025). This produces distinct vertical distributions for grains of different sizes, rather than a single well-mixed dust layer.

The thermal treatment differs slightly across published implementations. In the 2023 RU Lup application, the gas temperature was derived from a cross-section-weighted mean of dust temperatures plus viscous heating,

$\mathrm{CO_2}$8

with

$\mathrm{CO_2}$9

(Deng et al., 2023). In the IM Lup extension, the gas temperature in the dense molecular layer where $\mathrm{C^{18}O}$0 originates is computed to be in thermal equilibrium with the dust grains, which is treated as appropriate for a massive disk (Deng et al., 18 Sep 2025).

3. Chemistry and radiative transfer

DiskMINT’s reduced CO chemistry is tailored to the dominant pathways controlling CO in disks. The network includes CO, $\mathrm{C^{18}O}$1, $\mathrm{C^{18}O}$2, atomic C/C$\mathrm{C^{18}O}$3, O/O$\mathrm{C^{18}O}$4, $\mathrm{C^{18}O}$5, $\mathrm{C^{18}O}$6, $\mathrm{C^{18}O}$7, $\mathrm{C^{18}O}$8, and hydrocarbons needed to close dominant pathways (Deng et al., 24 Jun 2026). Three processes are treated as critical.

First, CO freeze-out onto grains is included, with the accretion rate per unit volume written as

$\mathrm{C^{18}O}$9

where

$(M_{\rm gas}, \epsilon, R_c)$0

Thermal desorption, photodesorption, and cosmic-ray–induced desorption are also included (Deng et al., 24 Jun 2026).

Second, DiskMINT incorporates grain-surface conversion of CO ice to $(M_{\rm gas}, \epsilon, R_c)$1 ice on water-rich ice mantles. This process moves the effective CO snow surface upward by tens of K compared to pure CO ice, so the gaseous CO layer becomes thinner and located higher, around $(M_{\rm gas}, \epsilon, R_c)$2 K in the self-consistent vertical temperature structure (Deng et al., 24 Jun 2026). In the individual-target studies, this reaction is treated as the principal grain-surface pathway lowering gaseous CO above the midplane snowline (Deng et al., 18 Sep 2025).

Third, the framework includes isotope-selective photodissociation. Because self-shielding and line overlap are weaker for rare isotopologues, $(M_{\rm gas}, \epsilon, R_c)$3 becomes selectively underabundant in UV-exposed layers. DiskMINT therefore computes vertical abundance structures in which photodestruction competes with freeze-out and grain-surface chemistry, rather than assuming a fixed isotopologue abundance (Deng et al., 24 Jun 2026).

Continuum radiative transfer is solved with RADMC-3D. The transfer equation along a ray is

$(M_{\rm gas}, \epsilon, R_c)$4

and for an isothermal slab the emergent intensity reduces to

$(M_{\rm gas}, \epsilon, R_c)$5

DiskMINT explicitly computes $(M_{\rm gas}, \epsilon, R_c)$6 from the converged vertical structure, rather than relying on the optically thin estimator

$(M_{\rm gas}, \epsilon, R_c)$7

which is valid only when $(M_{\rm gas}, \epsilon, R_c)$8 (Deng et al., 24 Jun 2026).

For line radiative transfer, the individual-target workflows use non-LTE calculations with LIME to generate synthetic cubes and images of CO isotopologues (Deng et al., 2023, Deng et al., 18 Sep 2025). By contrast, DiskMINT-GARDEN uses LTE line luminosities for inference, with LTE verified in the grid because gas densities in the emitting layers exceed $(M_{\rm gas}, \epsilon, R_c)$9–$$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$0 for $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$1 $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$2–1 and $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$3–2 (Deng et al., 24 Jun 2026).

4. Observables and inference workflows

DiskMINT supports two inference regimes. The first is a full forward-modeling mode for individual targets. In the 2023 workflow, the long-wavelength SED is fitted first; DiskMINT then constructs a self-consistent dust-plus-gas structure, computes $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$4 abundances with the reduced chemistry, performs non-LTE line radiative transfer with LIME, and compares the resulting total luminosity, velocity profile, and radial intensity profile to the data (Deng et al., 2023). In the structured 2025 formulation, $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$5 and $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$6 are updated iteratively via

$$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$7

and

$$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$8

while solving vertical hydrostatic equilibrium and dust settling at each step (Deng et al., 18 Sep 2025).

The second regime is DiskMINT-GARDEN, which converts the same physics into a survey-scale estimator. The public grid contains 480 self-consistent models spanning stellar mass $$\frac{dP}{dz} = -\rho \Omega_K^2 z,\qquad \Omega_K=\sqrt{\frac{GM_\star}{r^3}},$$9–$$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$0, gas disk mass $$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$1–$$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$2, dust-to-gas ratio $$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$3–$$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$4, and characteristic radius $$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$5–$$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$6 au (Deng et al., 24 Jun 2026). For each model it provides the Band 6 continuum luminosity $$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$7 near 234 GHz, the $$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$8 $$H=\frac{c_s}{\Omega_K},\qquad c_s=\sqrt{\frac{k_B T}{\mu m_H}}.$$9–1 and $T_g(r,z)$0–2 line luminosities $T_g(r,z)$1, and the dust size metric $T_g(r,z)$2, the radius enclosing 90% of the continuum flux (Deng et al., 24 Jun 2026).

The machine-learning stage uses a gradient-boosted decision tree regressor in log space to map

$T_g(r,z)$3

at fixed $T_g(r,z)$4 to

$T_g(r,z)$5

Training uses a 90/10 split with early stopping and regularization; the validation performance is reported as $T_g(r,z)$6 with residual scatter $T_g(r,z)$7 dex across parameters (Deng et al., 24 Jun 2026). Statistical uncertainties are propagated by Monte Carlo sampling in observable space and are typically $T_g(r,z)$8, increasing to $T_g(r,z)$9 at $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$0, while the dominant uncertainty is systematic and is conservatively taken as a factor of $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$1 on inferred $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$2 (Deng et al., 24 Jun 2026).

A recurrent inference issue is optical-depth degeneracy. $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$3 increases with $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$4 but saturates when $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$5 rises, especially for small $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$6; $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$7 is therefore used to distinguish compact optically thick disks from extended optically thin ones, while $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$8 constrains $H(r)=c_s(r,\mathrm{mid})/\Omega_K(r)$9 and, with $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$0, informs shielding and temperature in the emitting layers (Deng et al., 24 Jun 2026).

5. Applications to RU Lup, IM Lup, and ALMA survey samples

DiskMINT was first demonstrated on RU Lup, a high-accreting star whose disk had previously been inferred to have $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$1 and gas-to-dust ratio $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$2. The best-fit DiskMINT model with vertical hydrostatic equilibrium yielded $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$3, while a parametric Gaussian vertical distribution that better matched the IR SED yielded $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$4 (Deng et al., 2023). Both models reproduced the total $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$5 luminosities for $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$6–1 and $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$7–2, the $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$8–1 velocity profile, and the deprojected radial intensity profile. The comparison indicated that the inferred gas mass was about an order of magnitude higher than earlier DALI-grid estimates for the same source (Deng et al., 2023).

In IM Lup, DiskMINT was used in its radially and vertically decoupled form. By fitting the multi-wavelength SED, the millimeter continuum, and the $$\Sigma(r)=\Sigma_1 (r/1\,{\rm au})^{-\gamma}\exp[-(r/R_c)^{2-\gamma}]$$9 radial emission profiles, the study derived a gas disk mass of $\gamma=1$0–$\gamma=1$1, with a well-mixed best fit of $\gamma=1$2 and a structured best fit of $\gamma=1$3 (Deng et al., 18 Sep 2025). The gas disk was found to extend beyond the dust disk, with $\gamma=1$4 au versus $\gamma=1$5–$\gamma=1$6 au, and the outer disk was inferred to be drift-dominated, with $\gamma=1$7–$\gamma=1$8 between $\gamma=1$9 and $\mathrm{CO_2}$00 au (Deng et al., 18 Sep 2025). The same work concluded that this dust-to-gas ratio is likely insufficient for strong streaming-instability-driven clumping in the outer disk.

DiskMINT-GARDEN was applied to archival ALMA observations of 34 disks with $\mathrm{CO_2}$01 detections of Band 6 continuum and $\mathrm{CO_2}$02: 14 from AGE-PRO, 14 large disks from exoALMA plus ancillary measurements, and 6 from DSHARP, MAPS, and other programs (Deng et al., 24 Jun 2026). The inferred gas masses agreed with independent dynamical disk-mass measurements and HD-based masses to within a factor $\mathrm{CO_2}$03 for reliable cases, consistent with the adopted systematic floor (Deng et al., 24 Jun 2026). When compared with DALI-based chemical modeling, DiskMINT recovered similar disk masses without invoking large uniform CO or elemental C depletion. Instead, grain-surface CO$\mathrm{CO_2}$04 conversion and the upward-shifted effective CO snow surface were found to reduce CO columns in the emitting layers naturally (Deng et al., 24 Jun 2026). The survey application therefore concluded that extant data suggest little chemical processing due to disk evolutionary processes.

6. Assumptions, limitations, and significance

DiskMINT assumes axisymmetric, flared disks under vertical hydrostatic equilibrium. The survey grid does not include explicit non-axisymmetric substructures or spirals, and warns that models approaching gravitational instability, as flagged by the Toomre parameter $\mathrm{CO_2}$05 at $\mathrm{CO_2}$06, require caution (Deng et al., 24 Jun 2026). In compact, high-surface-density disks, both continuum and $\mathrm{CO_2}$07 can become optically thick, and line luminosities then become less sensitive to mass and more sensitive to temperature, increasing the systematic floor (Deng et al., 24 Jun 2026).

The treatment of dust microphysics varies by implementation. The RU Lup study used dsharp_opac with an ISM-like mixture of 64% astronomical silicates and 36% graphite (Deng et al., 2023), whereas the later work adopted the DIANA standard mixture and size distribution, with opacities computed via Optool (Deng et al., 18 Sep 2025). This suggests that DiskMINT’s mass estimates are not tied to a single dust-opacity package, although $\mathrm{CO_2}$08 and $\mathrm{CO_2}$09 remain sensitive to dust assumptions.

The grid-based inference assumes co-spatial gas and dust surface densities with a global $\mathrm{CO_2}$10, and does not include explicit turbulent mixing or radial drift, even though targeted DiskMINT runs can treat radially varying $\mathrm{CO_2}$11 (Deng et al., 24 Jun 2026). The dense-layer gas temperature is tied closely to dust in the individual-target applications; the IM Lup study explicitly notes that omission of more complete gas heating and cooling introduces at most a factor $\mathrm{CO_2}$12 mass uncertainty for lower-density disks (Deng et al., 18 Sep 2025). For this reason, lower-mass or more UV-irradiated systems may require fuller thermochemical treatments or additional tracers.

Within those limits, DiskMINT’s principal significance is methodological. It recasts $\mathrm{CO_2}$13 from a potentially misleading proxy into a robust tracer of total gas mass, and in extended form of radial gas distribution, provided that the line is interpreted in a model where vertical structure, dust temperatures, shielding, freeze-out, and grain-surface chemistry are solved together (Deng et al., 2023, Deng et al., 18 Sep 2025). DiskMINT-GARDEN further shows that the same physics can be compressed into a reproducible survey pipeline based only on widely available ALMA observables, without requiring $\mathrm{CO_2}$14 (Deng et al., 24 Jun 2026).

Planned extensions follow directly from the current limitations. The 2026 framework identifies additional tracers such as HD and $\mathrm{CO_2}$15, simultaneous inference of vertical CO emitting surfaces, radially varying $\mathrm{CO_2}$16 and substructure, and coupling to spatially resolved line moments as natural directions for reducing degeneracies and improving robustness, especially in compact or optically thick disks (Deng et al., 24 Jun 2026).