DALI: Dust And LInes Disk Model

• DALI is a 2D thermo-chemical modeling framework that simulates protoplanetary disks by coupling detailed continuum radiative transfer with dust and gas chemistry.
• It incorporates realistic dust evolution—accounting for grain growth, fragmentation, radial drift, and vertical settling—to self-consistently compute dust opacities and molecular excitation.
• The framework produces synthetic continuum and line observables that help infer disk structures, gas masses, and CO chemistry by simultaneously fitting multi-wavelength diagnostic data.

DALI, conventionally expanded as Dust And LInes, is a 2D thermo-chemical code and modeling framework for protoplanetary disks that combines detailed continuum radiative transfer, thermo-chemical modeling of gas and ice, and line radiative transfer from optical to cm wavelengths in order to produce synthetic continuum and line observables from a single physical model (Facchini et al., 2017, Woitke et al., 2015). In its later implementations, DALI incorporates realistic grain growth, fragmentation, radial drift, and vertical settling, so that dust opacities, grain surface area, chemistry, molecular excitation, and thermal balance are treated self-consistently rather than through fixed dust parametrizations (Facchini et al., 2017). The framework is used to interpret disk structure, gas and dust radial extents, CO isotopologue emission, gas masses, and snowline-related chemistry in Class II and III disks.

1. Physical formulation and model domain

DALI is constructed around the premise that continuum and line observations should be fitted simultaneously, because dust properties, disk geometry, and gas chemistry are mutually coupled. The standard assumptions summarized for Class II and III disks include detailed continuum radiative transfer, thermo-chemical modeling of gas and ice, and line radiative transfer from optical to cm wavelengths, together with simplified but explicit treatments of dust settling and PAHs (Woitke et al., 2015). This architecture is intended to connect observables such as the SED, mm-slope, continuum visibilities, and emission lines including [OI] $63\,\mu\mathrm{m}$, high-$J$ CO lines, (sub-)mm CO isotopologue lines, and CO fundamental ro-vibrational lines within a single disk model (Woitke et al., 2015).

A standard structural prescription used in DALI-based studies is the exponentially tapered power law for the gas surface density,

$$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$

with a corresponding Gaussian vertical distribution,

$$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$

These prescriptions appear repeatedly in DALI applications to dust evolution, CO emission, and gas-mass inference (Facchini et al., 2017, Stapper et al., 2023).

The framework is explicitly designed for physically consistent forward modeling. Rather than inferring gas or dust properties from a single tracer under optically thin assumptions, it uses the coupled disk structure, chemistry, and radiative transfer to compute synthetic observables for direct comparison with interferometric and spectroscopic data (Stapper et al., 2023).

2. Computational architecture

In DALI implementations that include dust evolution, the dust structure is built first and then passed through a sequence of coupled radiative-transfer and chemistry calculations. A representative workflow is summarized below.

Stage	Treatment	Principal role
Dust structure	Semi-analytical prescription from Birnstiel et al. (2015) after $1$ Myr of dust evolution	Sets radial grain size distribution
Continuum transfer	Continuum radiative transfer using local opacities from size distribution	Determines radiation field and dust heating
Chemistry	Time-dependent chemistry with gas-grain interactions, including size-dependent effects	Sets abundances and freeze-out/desorption behavior
Excitation and thermal balance	Non-LTE line excitation and gas temperature calculation, iteratively to convergence	Produces molecular excitation and gas temperatures
Synthetic observables	Continuum and line emission maps, spectra, and visibilities	Enables comparison to ALMA and related data

This sequence follows the description in which DALI uses a semi-analytical prescription from Birnstiel et al. (2015) to set the radial grain size distribution after $1$ Myr of dust evolution; only dust properties are evolved, while the gas-to-dust mass ratio is kept at $100$; after building the dust structure, DALI performs continuum radiative transfer, time-dependent chemistry, and non-LTE line excitation with gas temperature iteration to convergence (Trapman et al., 2019).

A major extension in the newer DALI version is the direct ingestion of outputs from grain growth and radial drift models, replacing simpler two-population dust parametrizations. The grain size distribution becomes radius dependent, and for every grain-size bin the vertical distribution is calculated via steady-state solutions to the vertical advection-diffusion equation, accounting for settling under gravity balanced by turbulent stirring (Facchini et al., 2017). At each location, the actual grain size distribution sets both the dust opacity used in radiative transfer and the total grain surface area used in chemistry and thermal coupling (Facchini et al., 2017).

3. Dust evolution, settling, and gas-chemistry feedback

The defining physical advance of DALI relative to fixed-dust models is the explicit feedback of realistic dust particle distributions onto gas chemistry and molecular emissivity. In these models, the radial dust grain size distribution is determined by grain growth, fragmentation, and radial drift, with the inner disk typically fragmentation-limited and the outer disk drift-limited (Facchini et al., 2017, Trapman et al., 2019). Vertical settling is solved for each grain-size bin, producing strong stratification in which small grains remain higher in the disk while large grains concentrate toward the midplane (Facchini et al., 2017).

This redistribution of dust has several coupled consequences. First, the local dust opacity changes the penetration of stellar and interstellar radiation. Second, the total grain surface area changes gas-grain collision rates, adsorption, desorption, $\mathrm{H}_2$ formation, and other surface processes. Third, the gas and dust temperature structure can decouple substantially. In particular, grain growth and settling concur in thermally decoupling the gas and dust components, due to the low collision rate with large grains, so that the gas can be much colder than the dust at intermediate heights, especially for low turbulence values (Facchini et al., 2017).

The same coupling modifies CO chemistry and excitation. DALI-based models find that more small grains in the outer disk can enhance FUV shielding and allow CO to survive farther out, while lower grain surface area in settled regions suppresses gas-dust thermal coupling and can reduce CO excitation (Trapman et al., 2019, Facchini et al., 2017). The code also predicts that, due to disk mid-plane shadowing, a second CO thermal desorption front can occur in the warmer outer mid-plane disk (Facchini et al., 2017).

These results depend sensitively on turbulence, commonly parametrized by $\alpha$. Representative DALI grids span $\alpha = 10^{-2}, 10^{-3}, 10^{-4}$, disk masses from $J$0 to $J$1, and characteristic radii $J$2 AU in the gas/dust-size study, while the Herbig-disk gas-mass study uses a grid spanning five orders of magnitude in gas mass, $J$3 to $J$4 (Trapman et al., 2019, Stapper et al., 2023).

4. Synthetic observables and measurement diagnostics

DALI is used to compute both continuum and line observables. On the continuum side, its outputs include the SED, mm-slope, and continuum visibilities. On the line side, it is used for [OI] $J$5, high-$J$6 CO, (sub-)mm CO isotopologue, and CO ro-vibrational lines, with non-LTE level populations computed from the local densities, radiation field, chemistry, and temperature (Woitke et al., 2015). The line-transfer step therefore depends directly on the dust solution that determined the UV field, shielding, and thermal balance.

For disk-size work, DALI studies define outer radii observationally through cumulative flux fractions. The radius $J$7 enclosing a fraction $J$8 of the total flux is defined by

$J$9

In practice, $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$0 is measured from $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$1 mm continuum and $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$2 from $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$3CO $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$4 moment-zero maps (Trapman et al., 2019). For the gas, an analytic relation derived in the same study connects the measured gas radius to the CO photodissociation edge,

$$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$5

where $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$6 is the radius at which the CO column drops to the critical value and CO becomes photodissociated (Trapman et al., 2019).

The same work gives explicit observational conditions for robust size measurements. To minimize the uncertainties due to observational factors requires $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$7 the characteristic radius and a peak SNR $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$8 on the $$\Sigma_{\rm gas}(R) = \Sigma_c \left(\frac{R}{R_c}\right)^{-\gamma} \exp\left[-\left(\frac{R}{R_c}\right)^{2-\gamma}\right],$$9CO emission moment-zero map. For the dust outer radius to enclose most of the disk mass, it should be defined using a high fraction, $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$0–$$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$1, of the total flux, whereas for the gas any radius enclosing $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$2 of the $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$3CO flux will contain most of the disk mass (Trapman et al., 2019).

A recurring result of the DALI framework is that continuum-only modeling is insufficient. Strong dust settling and missing disk flaring have similar effects on continuum observations but opposite effects on far-IR gas emission lines, and PAH molecules can shield the gas from stellar UV radiation because of their strong absorption and negligible scattering opacities (Woitke et al., 2015). This is why DALI applications emphasize simultaneous fitting of continuum and line diagnostics.

5. Scientific results obtained with DALI

One major DALI application concerns the longstanding difference between gas and dust radial extents in protoplanetary disks. DALI models with realistic dust evolution show that the difference of dust and gas radial sizes is largely due to differences in the optical depth of CO lines and millimeter continuum, without the need to invoke radial drift, while the effect of radial drift is primarily visible in the sharp outer edge of the continuum intensity profile (Facchini et al., 2017). In the related size-ratio study, $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$4 is directly related to the radius where the CO column density drops below $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$5 and CO becomes photodissociated, and $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$6 scales with the total CO content of the disk (Trapman et al., 2019). Very large ratios, $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$7, are identified as a clear sign for dust evolution and radial drift, but these cases are rare in current observations; for smaller ratios, interpreting $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$8 requires modeling the disk structure including the total CO content (Trapman et al., 2019).

A second major application is the derivation of gas masses from CO isotopologues. In a study of $$\rho_{\rm gas}(R, z) = \frac{\Sigma_{\rm gas}(R)}{\sqrt{2\pi} R h}\exp\left[-\frac{1}{2}(z/Rh)^2\right].$$9 Herbig disks within $1$0 pc observed in $1$1CO, $1$2CO, and C$1$3O, DALI is used to infer masses by matching observed integrated line fluxes and emitting region sizes to a thermo-chemical model grid (Stapper et al., 2023). The key result is that the majority of Herbig disks for which $1$4CO and C$1$5O are detected are optically thick in both, so computing the gas mass using a simple optically thin relation between line flux and column density underestimates the gas mass by at least an order of magnitude compared to the masses obtained with DALI (Stapper et al., 2023). The DALI-inferred gas masses are consistent with a gas-to-dust ratio of at least $1$6, and these gas-to-dust ratios are two orders of magnitude higher than those found for T Tauri disks using similar techniques, illustrating the determining role of temperature in CO chemistry and emission (Stapper et al., 2023).

DALI has also been used as a testbed for individual disks. For HD 163296, the models are compared directly to ALMA observations, and in order to reproduce the observed CO snowline a binding energy for CO typical of ice mixtures needs to be used rather than the lower pure CO value (Facchini et al., 2017). This type of result is characteristic of DALI’s role: it does not treat line emission as a passive tracer, but as the outcome of coupled dust evolution, thermal structure, and chemistry.

6. Distinctions, limitations, and related usages of the acronym

In the protoplanetary-disk literature, DALI refers to the thermo-chemical Dust And LInes code, but the acronym has also been used for a distinct Dust and Ice Lines approach to inferring disk surface density from the observed locations of dust and volatile lines (Powell et al., 2017). That method is based on the assumption that, at a given location, the timescales for particle drift, growth, and disk lifetime are equal, $1$7, and it uses dust lines and ice lines as empirical diagnostics of $1$8 (Powell et al., 2017). The two usages are related by scientific domain but differ methodologically: the Dust and Ice Lines approach is an analytic or semi-analytic inversion scheme, whereas DALI is a forward thermo-chemical radiative-transfer framework.

This distinction matters because the two approaches have different dependencies and different strengths. The dust-line method is described as not depending on assumed dust opacity or dust-to-gas ratio and as providing an empirical benchmark for total mass at specific radii, whereas DALI or similar radiative transfer models require assumed dust opacities, temperature, and global dust-to-gas/CO-to-$1$9 ratios (Powell et al., 2019). A plausible implication is that the two approaches are complementary rather than interchangeable: DALI is stronger where line optical depth, excitation, freeze-out, and chemical conversion must be modeled self-consistently, while dust-line methods offer an external check on inferred surface densities.

Within DALI itself, several modeling degeneracies are explicit. Evolved dust properties often needed to fit the SED have important consequences for disk chemistry and heating/cooling balance, leading to stronger emission lines in general; strong dust settling and missing disk flaring have similar effects on continuum observations, but opposite effects on far-IR gas emission lines (Woitke et al., 2015). For that reason, DALI-based analyses recommend using line observations of robust chemical tracers of the gas, such as O, CO, and $1$0, as additional constraints to determine disk shape, mass, opacities, and dust/gas ratio by simultaneously fitting continuum and line observations (Woitke et al., 2015).