DSHARP Model: Dust Opacity in Disks

• DSHARP Model is a standardized framework that connects ALMA 1.25 mm continuum intensities to dust surface density and properties in protoplanetary disks.
• It utilizes a fiducial dust prescription with compact grains, effective medium theory, and Mie scattering to derive opacities and interpret complex disk substructures.
• The model reveals that optical depth and scattering effects can significantly bias dust mass estimates, impacting interpretations of dust trapping and planet–disk interactions.

Searching arXiv for DSHARP modeling papers and related applications. The DSHARP model is the standardized dust-opacity and radiative-transfer framework developed within the Disk Substructures at High Angular Resolution Project (DSHARP) to link ALMA $1.25$ mm continuum intensities to the properties and quantities of emitting dust in protoplanetary disks. It emerged from a survey of 20 nearby Class II disks observed at $\sim 0.035''$ ($\sim 5$ au FWHM) resolution at 240 GHz, a program that established that rings, gaps, spirals, and azimuthal asymmetries are ubiquitous in large, bright disks and that many of those structures reside in regions of intermediate to high optical depth even at millimeter wavelengths (Andrews et al., 2018, Birnstiel et al., 2018). In its narrow sense, the DSHARP model denotes the fiducial compact-grain opacity prescription plus a thin-slab transfer treatment with scattering; in practice, it also anchors a wider interpretive program in which continuum morphologies are modeled in terms of dust trapping, planet–disk interactions, vertical structure, and dust-mass inference (Dullemond et al., 2018, Zhang et al., 2018, Liu et al., 2022).

1. Survey context and motivation

DSHARP, the Disk Substructures at High Angular Resolution Project, was designed as a systematic, homogeneous census of disk substructure rather than a collection of individual high-profile detections. Its primary observational goal was to characterize the prevalence, morphologies, spatial scales, spacings, symmetry, and amplitudes of small-scale structures in 20 nearby protoplanetary disks using very high resolution $240$ GHz continuum imaging. The survey found that substructures are ubiquitous in this sample, most frequently as concentric bright rings and dark gaps, with spiral patterns and arc-like azimuthal asymmetries in a minority of systems (Andrews et al., 2018, Huang et al., 2018).

The need for a dedicated DSHARP model followed directly from those observations. Interpreting measured continuum intensity $I_\nu$ in terms of dust surface density $\Sigma_d$ is non-trivial because dust opacities are uncertain, because many DSHARP disks are not optically thin even at millimeter wavelengths, and because scattering can be important when grains are large and have high albedo. The DSHARP dust-model paper was therefore framed as a continuity tool for the survey and its follow-up work: a practical, reproducible framework for converting ALMA continuum measurements into approximate dust masses and surface densities while explicitly accounting for optical depth and albedo effects (Birnstiel et al., 2018).

2. Fiducial dust prescription and opacity construction

The fiducial DSHARP dust model assumes compact grains with no porosity, homogeneous composition, spherical grains, a vacuum embedding medium combined through effective medium theory, a geometrically thin emitting dust layer approximated as a one-dimensional slab at the disk mid-plane, and vertically integrated grain-size distributions. When discussing steady-state size distributions and mean opacities, it adopts a dust-to-gas mass ratio of $0.01$ (Birnstiel et al., 2018).

The fiducial composition uses water ice, astronomical silicates, troilite, and refractory organics, with the following mass fractions and material densities (Birnstiel et al., 2018):

Material	Fiducial fraction	Bulk density $(\mathrm{g\,cm^{-3}})$
Water ice	20.00%	0.92
Astronomical silicates	32.91%	3.30
Troilite	7.43%	4.83
Refractory organics	39.66%	1.50

These choices give a mixture bulk density of

$\rho_{\rm mix}=1.675~{\rm g~cm^{-3}}.$

The optical constants are taken from Warren (2008) for ice, Draine (2003) for astronomical silicates, and Henning (1996) for troilite and refractory organics. For the compact fiducial mixture, the appropriate effective-medium prescription is the Bruggeman rule,

$$\sum_{i=1}^N f_i \frac{\epsilon_i - \bar\epsilon}{\epsilon_i + 2\bar\epsilon} = 0,$$

after which absorption and scattering opacities are computed with Mie theory. The DSHARP implementation averages over 40 linearly spaced sub-bins within each grain-size bin, with each grain-size bin $\sim 0.035''$0 dex wide, to suppress artificial resonances (Birnstiel et al., 2018).

Later radiative-transfer applications used the DSHARP opacity tables as a physically motivated alternative to the fixed $\sim 0.035''$1 convention. In that usage, the dust mixture is described by the same four components in volume fractions 36%, 17%, 3%, and 44%, with bulk density $\sim 0.035''$2. Opacities are computed with Mie theory for 32 logarithmically spaced grain-size bins from $\sim 0.035''$3 up to $\sim 0.035''$4, with $\sim 0.035''$5 mm in fiducial models. At $\sim 0.035''$6 mm, the DSHARP opacity curve is not constant: it varies with grain size and shows resonant structure around $\sim 0.035''$7–0.7 mm (Liu et al., 2022).

3. Radiative-transfer formalism and departure from the optically thin approximation

For a grain-size distribution $\sim 0.035''$8, the DSHARP model defines the size-averaged opacity as

$\sim 0.035''$9

with size distributions often written as $\sim 5$0. In the optically thin limit, the continuum intensity obeys

$\sim 5$1

but DSHARP explicitly emphasized that many survey targets do not satisfy that limit (Birnstiel et al., 2018).

Scattering enters through an effective scattering opacity

$\sim 5$2

where $\sim 5$3 is the forward-scattering parameter, and an effective absorption probability

$\sim 5$4

The mean intensity in the slab then satisfies

$\sim 5$5

with source function

$\sim 5$6

and line-of-sight transfer equation

$\sim 5$7

The modified Eddington–Barbier approximation is

$\sim 5$8

which reduces for $\sim 5$9 to

$240$0

For large optical depth, the emergent intensity saturates below the Planck function if the albedo is non-zero, so scattering can make the slab appear colder than it truly is (Birnstiel et al., 2018).

The contrast with the standard literature mass conversion is central. The traditional analytic estimate is

$240$1

often evaluated at $240$2 GHz with $240$3 and $240$4 K, or with the commonly used luminosity scaling

$240$5

The DSHARP-based reinterpretation stresses that this expression assumes optically thin emission and therefore effectively converts only the flux that escapes above the $240$6 surface; if the disk is not actually optically thin, the result is merely a lower limit (Liu et al., 2022).

4. Use in interpreting rings, gaps, and planet-driven structure

A major DSHARP application was the dust-trapping interpretation of narrow rings. Isolated rings were fit with Gaussian profiles,

$240$7

with deconvolved intrinsic width

$240$8

The physical comparison scale was the gas pressure scale height,

$240$9

using an irradiated flared-disk temperature law

$I_\nu$0

Within that framework, rings in AS 209, Elias 24, GW Lup, and the outer ring of HD 163296 were found to be narrower than the pressure scale height, which was treated as strong evidence for dust trapping. Across the analyzed subset, all rings were consistent with dust trapping, peak absorption optical depths were between $I_\nu$1 and $I_\nu$2, the dust masses stored in individual rings were of the order of tens of Earth masses, and the data excluded the combination of very low $I_\nu$3 and very large grains $I_\nu$4 for all rings studied (Dullemond et al., 2018).

A complementary DSHARP modeling branch interpreted annular substructures as the signatures of embedded planets. Those calculations used 2-D hydrodynamical simulations with Dusty FARGO-ADSG, assuming a locally isothermal disk, no self-gravity, no radiative cooling, no dust feedback on gas in the baseline grid, and a planet on a fixed circular orbit. Synthetic $I_\nu$5 mm continuum maps were then generated using DSHARP opacities from Birnstiel et al. (2018) and two extreme dust size distributions, DSD1 and DSD2. The resulting framework calibrated observable gap width, gap depth, ellipticity, asymmetry, and secondary-gap position against planet mass, disk aspect ratio, and turbulence. It found excellent agreement between observations and simulations for AS 209 and concluded that, under the assumption that the detected gaps are induced by young planets, DSHARP disk substructures probe a wide-orbit planet population from Neptune- to Jupiter-mass planets beyond $I_\nu$6 au (Zhang et al., 2018).

5. Consequences for dust-mass inference

The most direct reassessment of the DSHARP model’s astrophysical consequences concerned dust-mass underestimation. A self-consistent radiative-transfer study examined how disk structure, dust properties, scattering, settling, and substructures bias the optically thin analytic conversion. In its fiducial disk, the true dust mass was $I_\nu$7, the simulated $I_\nu$8 mm flux was $I_\nu$9 mJy, the mass-averaged dust temperature was $\Sigma_d$0 K, and the mass-averaged opacity was $\Sigma_d$1. Even if those modeled temperature and opacity values are inserted back into the analytic formula, the result is

$\Sigma_d$2

so the mass-underestimation factor,

$\Sigma_d$3

is $\Sigma_d$4. The point is that optical-depth effects alone hide mass below the $\Sigma_d$5 surface even when $\Sigma_d$6 and $\Sigma_d$7 are known exactly (Liu et al., 2022).

In the DoAr 33 application, one of the observed DSHARP disks, the authors constructed a grid of 6,552 radiative-transfer models varying $\Sigma_d$8, $\Sigma_d$9, $0.01$0, and $0.01$1, with $0.01$2 fixed from the observed surface-brightness slope, and analyzed the grid with flat priors following the SED-fitting procedure of Pinte et al. (2008). The dust masses inferred for DoAr 33 were:

Method	$0.01$3	Ratio to analytic
Analytic conversion	$0.01$4	1.0
Radiative transfer, DSHARP opacities	$0.01$5	7.3
Radiative transfer, DIANA opacities	$0.01$6	2.3

For the DSHARP-opacity solution, about 70% of the total mass lies below the $0.01$7 surface, the model mass-averaged temperature is $0.01$8 K rather than the analytic $0.01$9 K, and the best-fit grain size reaches $(\mathrm{g\,cm^{-3}})$0 mm, lowering the average opacity to $(\mathrm{g\,cm^{-3}})$1. That opacity change alone contributes a factor of $(\mathrm{g\,cm^{-3}})$2 to the mass increase relative to the standard analytic conversion. Across the wider parameter study, $(\mathrm{g\,cm^{-3}})$3 ranges from a few to hundreds, with true dust mass, disk outer radius, and inclination as the strongest drivers; $(\mathrm{g\,cm^{-3}})$4 mm masses are less underestimated than $(\mathrm{g\,cm^{-3}})$5 mm masses because the optical depth is lower at longer wavelength (Liu et al., 2022).

6. Limitations and subsequent extensions

The original DSHARP dust model was presented as a standard reference, not as a uniquely correct description of disk solids. Its stated limitations include uncertain grain shape and porosity, the spherical-particle assumption inherent to Mie theory, non-unique effective-medium approximations, an approximate scattering treatment through the $(\mathrm{g\,cm^{-3}})$6 correction rather than full 3-D radiative transfer, and a strongest applicability in fragmentation-limited steady-state regimes rather than drift-limited regions. The model paper also emphasized that short-wavelength opacities are especially sensitive to the small-grain population, and that CO line-extinction estimates can be weakened if back-side line emission is not far behind a dust ring (Birnstiel et al., 2018).

Subsequent DSHARP analyses extended the basic modeling program rather than replacing it. A super-resolution reanalysis with the 1D visibility-fitting code Frankenstein ($(\mathrm{g\,cm^{-3}})$7) showed that DSHARP data contain more recoverable radial structure than standard CLEAN images reveal: the $(\mathrm{g\,cm^{-3}})$8 profiles match the data to a mean factor of $(\mathrm{g\,cm^{-3}})$9 longer baseline than the Fourier transform of the CLEAN images and $\rho_{\rm mix}=1.675~{\rm g~cm^{-3}}.$0 longer baseline than the transform of the CLEAN component models, with high-contrast gaps on average 14% wider and 44% deeper, and high-contrast rings on average 26% narrower (Jennings et al., 2021). A radiative-transfer study of projection-driven gap filling used DSHARP images to constrain dust scale heights and inferred low dust layers, generally $\rho_{\rm mix}=1.675~{\rm g~cm^{-3}}.$1 AU at 100 AU, implying weak turbulence with $\rho_{\rm mix}=1.675~{\rm g~cm^{-3}}.$2 under the assumption $\rho_{\rm mix}=1.675~{\rm g~cm^{-3}}.$3 (Pizzati et al., 2023). More recently, axisymmetric visibility-space modeling of 16 DSHARP disks found no compelling evidence for circumplanetary emission, identified a possible one-armed spiral in Elias 27, attributed strong residual asymmetries in five disks to the vertical extent of their dust layers, and reported that the brightness temperature of inner disks declines with stellar age on a $\rho_{\rm mix}=1.675~{\rm g~cm^{-3}}.$4 Myr timescale while total flux shows no clear decreasing trend (Aizawa et al., 18 Dec 2025).

Within protoplanetary-disk studies, the enduring significance of the DSHARP model is therefore methodological as much as compositional. It established a survey-wide standard for opacity calculation and slab radiative transfer with scattering; it supplied the dust prescription used in DSHARP analyses of trapping, gap opening, and mass inference; and it shifted continuum interpretation away from the assumption that millimeter emission can generally be treated as optically thin, pure-absorption radiation (Birnstiel et al., 2018, Dullemond et al., 2018, Zhang et al., 2018, Liu et al., 2022).