On the accuracy of mass and size measurements of young protoplanetary disks

Knowing the masses and sizes of protoplanetary disks is of fundamental importance for the contemporary theories of planet formation. However, their measurements are associated with large uncertainties. In this proof of concept study, we focus on the very early stages of disk evolution, concurrent with the formation of the protostellar seed, because it is then that the initial conditions for subsequent planet formation are likely established. Using three-dimensional hydrodynamic simulations of a protoplanetary disk followed by radiation transfer postprocessing, we constructed synthetic disk images at millimeter wavelengths. We then calculated the synthetic disk radii and masses using an algorithm that is often applied to observations of protoplanetary disks with ALMA, and compared the resulting values with the actual disk mass and size derived directly from hydrodynamic modeling. We paid specific attention to the effects of dust growth on the discrepancy between synthetic and intrinsic disk masses and radii. We find that the dust mass is likely underestimated in Band 6 by factors of 1.4–4.2 when Ossenkopf & Henning opacities and typical dust temperatures are used, but the discrepancy reduces in Band 3, where the dust mass can be even overestimated. Dust growth affects both disk mass and size estimates via the dust-size-dependent opacity, and extremely low values of dust temperature ($\approx$ several Kelvin) are required to recover the intrinsic dust mass when dust has grown to mm-sized grains and its opacity has increased. Dust mass estimates are weakly sensitive to the distance to the source, while disk radii may be seriously affected. We conclude that the accuracy of measuring the dust mass and disk radius during the formation of a protoplanetary disk also depends on the progress in dust growth. The same disk, but observed at different stages of dust growth and with different linear resolutions, can have apparent radii that differ from the intrinsic value by up to a factor of two. Multi-wavelength observations that can help to constrain the maximum dust size would be useful when inferring the disk masses and sizes.

1 Introduction

Knowing the masses of protoplanetary disks is of fundamental importance for the contemporary theories of planet formation. The mass of gas in the disk influences its tendency to undergo gravitational instability and form planets through disk gravitational fragmentation (Mayer et al., 2007; Vorobyov, 2013; Nayakshin, 2017; Mercer and Stamatellos, 2020; Boss and Kanodia, 2023; Xu et al., 2025). The mass of dust in the disk sets the upper limits on the masses of terrestrial planets and cores of giant planets to be formed via planetesimal hierarchical growth or pebble accretion (Pollack et al., 1996; Lambrechts and Johansen, 2012; Jin and Li, 2014). Dust particles at the upper part of the size spectrum are the main carriers of dust mass in the disk (Birnstiel et al., 2016), but they also contribute strongly to the (sub-)millimeter opacity in the outer cold and gravitationally unstable disk regions if their size exceeds a certain threshold value (Pavlyuchenkov et al., 2019).

Knowing the sizes of protoplanetary disks is of no less importance for planet formation theories. Gravitational fragmentation is known to operate in the disk at distances beyond tens of astronomical units, because at smaller distances the high rate of shear and slow cooling prevents disk fragmentation (Gammie, 2001; Rice et al., 2003; Meru and Bate, 2012). The size of the dust disk, which is usually smaller than that of the gas disk (Ansdell et al., 2018; Trapman et al., 2019; Hsieh et al., 2024), carries information about the efficiency of radial dust drift and, indirectly, about the efficiency of dust growth in the system. The size of the dust disk also defines the spatial extent within which planetesimals – the first building blocks of protoplanets – are expected to form. Interestingly, recent observational data on FU Orionis-type objects revealed several features that do not fit into the contemporary models of disk evolution, dust growth, and planet formation, indicating that the disks of these outburst objects tend to be of a smaller size but higher mass than disks around their quiescent counterparts (Kóspál et al., 2021). The time evolution of both masses and sizes reflects the dominant transport and loss mechanisms of mass and angular momentum in a protoplanetary disk (Manara et al., 2023).

While the masses and sizes of the gas disk are usually obtained via the observations of CO isotopologues, sometimes complemented with ${\mathrm{N}\mathrm{2}\mathrm{H}}^{+}$, or hydrogen deuteride (HD) (McClure et al., 2016; Trapman et al., 2025), the dust disk is probed via observations of thermal dust emission at (sub-)mm wavelengths (Tobin et al., 2020; Tychoniec et al., 2020; Anderson et al., 2022). Observational estimates of gas masses suffer from uncertainties in the abundance of gaseous tracers, their freezeout on dust grains, and strong dependence of population levels on the gas temperature (Bergin and Williams, 2017; Molyarova et al., 2017). Alternatively, dynamical measurements that involve fitting of the disk rotation curve to the Keplerian law can provide estimates on the gas mass in the disk (Veronesi et al., 2021), but such exercises require high resolution observations.

Estimates of dust mass in the disk are no less affected by uncertainties in the dust opacity, inclination, and dust temperature (Dunham et al., 2014; Tobin et al., 2020; Tychoniec et al., 2020), particularly, if the optically thin assumption is made. The latter may be justified for evolved Class II systems, but not for younger Class 0/I counterparts. The situation may become even more complicated in protostellar systems that are in their earliest stages of formation. Observationally, these objects may reveal themselves as very low luminosity objects or VELLOs (Dunham et al., 2010; Kim et al., 2019), which may be the first hydrostatic cores (FHSCs) surrounded by nascent disks (Vorobyov et al., 2017a). Indeed, numerical hydrodynamics simulations indicate that a protoplanetary disk may start forming before the FHSC collapses due to molecular hydrogen dissociation to form the protostellar seed (Inutsuka, 2012; Tomida et al., 2015; Vorobyov et al., 2024).

In this work, we aim to determine the accuracy of the dust mass and disk radius measurements in a protoplanetary disk in its very early stages that are concurrent with the formation of the protostellar seed. This is important because this stage is likely to set the initial dust repository for subsequent planet formation. To this end, we produce synthetic observables of the protoplanetary disk made with a dust distribution derived from three-dimensional numerical hydrodynamics simulations of cloud collapse and disk formation.

We also consider models with artificially imposed limits on the maximum dust size to account for electrostatic or bouncing barriers, which are not considered self-consistently in the dust growth model (Vorobyov et al., 2025). This allows us to connect numerical simulations with synthetic observations and test the basic assumptions underlying dust mass and disk radius estimates in real observations. In Sect. 2 the numerical hydrodynamics model is briefly reviewed. The methods to estimate disk masses and sizes from numerical data are described in Sect. 3. The main results are presented in Sect. 4. Model limitations and comparison with previous work are provided in Sect. 5. Main conclusions are summarized in Sect. 6.

2 Numerical model

In this section, we provide a basic description of the numerical hydrodynamics model employed to simulate the formation of a protoplanetary disk. We used the three-dimensional ngFEOSAD code to simulate the gravitational collapse of a pre-stellar cloud and early evolution of a nascent protoplanetary disk. The code solves the equations of gas and dust dynamics including self-gravity and dust growth in the polytropic approximation on nested Cartesian meshes. The detailed description of the code and the pertinent equations can be found in Vorobyov et al. (2024), and also in a concise form in Supplementary Appendix 7. Here, we provide the basic information that is relevant to calculating the synthetic observables of our model protoplanetary disk.

The numerical simulations start from the gravitational collapse of a slowly rotating Bonnor-Ebert sphere with a mass of $0.87M_{\odot }$. The initial temperature of the cloud is 10 K and it is given an initial positive density perturbation of 30% to initiate the collapse. The ratios of rotational-to-gravitational and thermal-to-gravitational energies are 0.5% and 70%, respectively. Initially, the dust-to-gas mass ratio is set equal to 1:100 and the maximum dust size is $a_{\text{max}}=2.0$ $\mu$m. The minimum dust size is kept fixed at $5.0\times 10^{-3}$ $\mu$m and the slope of the dust size distribution is kept constant at $p=-3.5$ throughout the simulation for simplicity.

The dust-to-gas mass ratio throughout the computational domain begins to deviate from the initial value as collapse proceeds, the disk forms, and dust starts growing and settling to the disk midplane. Dust enhancement of the innermost cloud regions occur already in the predisk stage due to differential collapse of gas and dust in the cloud (Bate, 2022). Mild dust growth also occurs in the collapse stage (Vorobyov et al., 2025), but its main phase begins when the FHSC forms and the disk begins to build around the FHSC owing to conservation of angular momentum of infalling matter. We terminate the simulations just before the FHSC is about to collapse due to molecular hydrogen dissociation and to form the protostellar seed. We therefore address the very early stages of evolution when the luminosity of the central source is still low but the protoplanetary disk may already have started to form. Considering these early stages of disk evolution also makes it easier to model the synthetic disk images because uncertainties with the luminosity and radius of a young stellar object are lifted (Vorobyov et al., 2017b). We note that while compact disks around FHSCs were theoretically and numerically predicted (Tomida et al., 2015; Wurster et al., 2021; Vorobyov et al., 2025), they are still observationally elusive. The more advanced stage when the central protostar reaches a mass of $0.1-0.2M_{\odot }$ will be addressed in a follow-up study.

Relevant to our model are the choices of the turbulent viscosity ${\alpha }_{\text{visc}}=10^{-3}$ and the dust collisional fragmentation velocity $v_{\text{frag}}=5.0$ m ${\text{s}}^{-1}$. The former influences the dust growth efficiency and maximum dust size via the turbulent relative velocity of dust-to-dust collisions (Ormel and Cuzzi, 2007), while the latter sets the fragmentation barrier and also the maximum dust size in our simplified monodisperse dust growth model (Birnstiel et al., 2012). We note that the drift barrier is self-consistently treated via the numerical solution of the dust dynamics equations with dust-to-gas friction, including the backreaction of dust on gas. The choices of ${\alpha }_{\text{visc}}$ and $v_{\text{frag}}$ were motivated by observations of dust settling in protoplanetary disks (Rosotti, 2023) and laboratory experiments on dust survival during collisions (Gundlach and Blum, 2015; Blum, 2018). Apart from the actual dust size distribution obtained in our numerical simulations, we also consider several models in which the maximum dust size is artificially reset to smaller values to imitate the possible effects of electrostatic and bouncing barriers (Okuzumi, 2009; Wada et al., 2011), not considered self-consistently in ngFEOSAD (Vorobyov et al., 2025).

We used 12 nested grids with the linear size of the outermost grid equal to 0.09 pc. The number of grid cells per Cartesian coordinate direction of each nested grid is $N=64$. The effective numerical resolution in the inner 4.5 au is 0.14 au and it remains at a sub-au level up to 36 au. The disk at the end of simulations extends to about 50–60 au.

3 Estimates of disk masses and sizes

In this section, we describe the method we used to derive the disk masses and sizes directly from our numerical hydrodynamics simulations. We also explain how we obtained synthetic observables of the protoplanetary disk with a radiation transfer tool and used these observables to compare the synthetic disk masses and sizes with those derived directly from hydrodynamic modeling.

3.1 Deriving the disk mass and size from hydrodynamic simulations

The first step in our procedure is to derive the disk mass using the three-dimensional distribution of gas densities and velocities in our computational domain. To do that in a numerical model that self-consistently computes the cloud-to-disk transition, we have to develop a means of distinguishing the disk from the infalling cloud. For this purpose, we adopted the disk tracking conditions outlined in Joos et al. (2012). In particular, we used the following criteria to determine if a particular computational cell in the entire computational domain belongs to the disk, and not to the infalling cloud:

• the gas rotational velocity must be faster than twice the radial infall velocity, $v_{\mathrm{\varphi }}>2v_{\text{r}}$,

• the gas rotational velocity must be faster than twice the vertical infall velocity, $v_{\varphi }>2v_{\text{z}}$,

• gas must not be thermally supported, $\frac{{\rho }_{\text{g}}{v}_{\varphi }^2}{2}>2P$,

• the gas number density must be higher than $10^9$ ${\text{cm}}^{-3}$.

Here, $v_{\text{z}}$, $v_{\text{r}}$, and $v_{\varphi }$ are the components of gas velocity in the cylindrical coordinates, ${\rho }_{\text{g}}$ the volume density of gas, and $P$ the gas pressure. If any of the first three conditions fails, a particular grid cell is not qualified as belonging to the disk. The forth condition must always be fulfilled. We also tried a higher threshold for the gas number density $10^{10}$ ${\text{cm}}^{-3}$ but found little difference.

At the earliest stages of disk evolution considered in this work ($t\le 15$ kyr after disk formation), some radial drift and vertical settling of dust towards the disk midplane may have already occurred (Bate, 2022; Lebreuilly et al., 2020; Vorobyov et al., 2024). Therefore, the gas and dust disks may not be having identical extents. Since we use the volume density and velocity of gas to identify the disk-to-envelope interface, our algorithm identifies the gas disk rather than the dust disk. We neglect this difference in the present work, as we do not expect it to be decisive at this early evolution stage. Indeed, Hsieh et al. (2024) found that the ratio of gas to dust disk radii for Class 0 objects does not deviate much from unity.

3.2 Construction of synthetic disk images in dust continuum emission

The second step is to obtain the simulated images of our model disks in millimeter wavebands. For this purpose, we employed the RADMC-3D radiative transfer tool (Dullemond et al., 2012) and postprocessed the resulting synthetic fluxes with the ALMA Observational Support Tool (ALMA OST) to account for atmospheric effects and finite interferometer resolution. The input parameters into RADMC-3D are the three-dimensional dust volume density, temperature, and dust size distributions obtained from ngFEOSAD and the output is the radiation intensity distribution at a specific wavelength. Since we modeled the earliest stages of evolution, the radiation of the central star was neglected.

In the RADMC-3D radiation transfer simulations we used the Cartesian grid with the mesh refinement option. It allows us to directly map the ngFEOSAD nested meshes onto the RADMC-3D grid layout. We considered eight inner nested meshes of ngFEOSAD, which encompass a cube with an edge size of 1,100 au centered at the FHSC. Importing the dust content of the disk into RADMC-3D calculations requires specifying its density, temperature, and dust opacity. While the first two were taken directly from hydrodynamic modeling, the absorption and scattering dust opacities were obtained using the opTool (Woitke et al., 2016) for dust grains with a fixed minimum dust size of $5\times 10^{-3}$ $\mu$m and a maximum dust size $a_{\text{max}}$ (see also Supplementary Appendix 11), which value was either fixed or directly taken from our hydrodynamic simulations. In the latter case, $a_{\text{max}}$ is a continuous distribution and varies from computational cell to cell, which is too complex to implement in RADMC-3D. To circumvent this problem, we introduce in each cell a number of dust sub-populations, each characterized by the same minimum size $5\times 10^{-3}$ $\mu$m but different maximum sizes: $a_{\text{max}}^{\text{sub}}$ = 2.0 $\mu$m, 10 $\mu$m, 100 $\mu$m, 1 mm, and 1 cm. We then retain in each cell only the sub-population whose maximum size $a_{\text{max}}^{\text{sub}}$ is closest to $a_{\text{max}}$.

In each Monte-Carlo RADMC-3D simulation we set the number of photons to $N=10^8$. The synthetic radiation fluxes obtained after radiation transfer simulations are then post-processed with the ALMA OST with parameters corresponding to ALMA Band 6 or Band 3. In particular, the bandwidth is set to 7.5 GHz, the beam size is varied from $0.042^{\prime \prime }\times 0.046^{\prime \prime }$ to $0.134^{\prime \prime }\times 0.146^{\prime \prime }$, and the on-source time is equal to 1.0 h of observation. We consider the atmospheric conditions corresponding to precipitable water vapor of 1.796 mm, which results in a theoretical RMS-noise of $\sigma \simeq 1.11\times 10^{-5}$ Jy. We note that the adopted angular resolution is close to the highest limit¹, and may not be available at every ALMA cycle.

3.3 Estimates of disk masses and sizes from radiation fluxes

The final step is to derive the disk masses and sizes from the radiation fluxes obtained with RADMC-3D and postprocessed with ALMA OST as described above. For this purpose, we follow an approach usually employed when inferring the masses and sizes of dust disks from observations in (sub)-mm wavebands (e.g., Tobin et al., 2020; Kóspál et al., 2021). First, we determine the radial extent within which 90% or 97% of the total flux at a given wavelength is confined, $R_{90\text{\%}}^{\text{obs}}$ or $R_{97\text{\%}}^{\text{obs}}$. The former value is often used in observational astronomy because the low signal-to-noise ratio near the disk outer boundary may interfere with the measurements, while the latter value may be considered as an idealized upper limit that can be achieved with high signal-to-noise observations. The value of $R_{90\text{\%}}^{\text{obs}}$ is further regarded as the nominal radius of our synthetic disk, while $R_{97\text{\%}}^{\text{obs}}$ is used for comparison.

Once the radius of our synthetic disk is calculated, the dust mass of the disk is estimated using the following equation

$M_{90\text{\%}}^{\text{obs}}=\frac{d^2{F}_{\lambda }^{90\text{\%}}}{B_{\lambda }\left(T_{\text{d}}\right){\kappa }_{\lambda }},(1)$

where $d$ is the assumed distance to the object, $F_{\lambda }^{90\text{\%}}$ is the total flux at wavelength $\lambda$ contained within the disk extent defined by $R_{90\text{\%}}^{\text{obs}}$, $B_{\lambda }\left(T_{\text{d}}\right)$ is the Planck function at an average dust temperature $T_{\text{d}}$, and ${\kappa }_{\lambda }$ is the dust opacity per gram of gas. We also considered dust masses $M_{97\text{\%}}^{\text{obs}}$ contained within 97% of the total flux as an idealized limit. For the wavelength of observations we choose 1.3 mm, which corresponds to the center of the B6 band on ALMA. The dust opacity ${\kappa }_{\lambda =1.3\mathrm{m}\mathrm{m}}$ is set equal to 0.89 ${\text{cm}}^2$ ${\text{g}}^{-1}$ for $\lambda =1.3$ mm (Ossenkopf and Henning, 1994), which is a frequent observer’s choice (Tobin et al., 2020; Kóspál et al., 2021), and is scaled accordingly for $\lambda =3.0$ mm. Details on the determination of the dust temperature $T_{\text{dust}}$ are provided in Sect. 4.3. We note here that Equation 1 is derived in the limit of negligible dust scattering. Nevertheless, we will use in this work the synthetic fluxes and dust opacities that consider the effects of dust scattering. This is done because fluxes from real disks are by default affected by dust scattering. However, in Supplementary Appendix 10 we conducted several experiments and turned off dust scattering to assess its effects on the synthetic fluxes. Equation 1 is valid for the optically thin limit, but we will apply it to disk configurations that are partly or fully optically thick for the lack of a better alternative. We note that this equation is often used in large surveys of star-forming regions and can lead to serious mass miscalculations, as we will demonstrate later in this work.

4 Results

In this section, the comparison between the disk masses and sizes derived directly from hydrodynamic modeling with those derived from synthetic radiation fluxes is carried out.

4.1 Model disk characteristics and synthetic images

Figure 1 presents the spatial distribution of the main disk characteristics in the disk midplane and at the vertical slice taken through the $x-z$ plane at $y=0$ as obtained from numerical hydrodynamics modeling with ngFEOSAD at 15 kyr after the instance of disk formation. A clear two-armed spiral pattern is evident in both gas and dust volume density distributions in the disk midplane. The Toomre $Q$-parameter is less than unity in the spiral arms, indicating that the disk is strongly gravitationally unstable, but gravitational fragmentation does not yet occur (see Supplementary Appendix 8 for more details). The spatial distribution of grown dust in the disk midplane mostly follows that of gas. Dust evaporation in the hot center of the FHSC is accounted for (see appendix in Das et al. (2025)). The dust size in this early stage already exceeds 1.0 mm throughout most of the disk midplane, reaching a few centimeters in the innermost regions. We note, however, that fast dust growth may be somewhat tampered by electrostatic and bouncing dust growth barriers that were not considered in our numerical model (Vorobyov et al., 2025). The spiral arms are warmer than the interspiral regions and the highest temperature is achieved in the innermost regions occupied by the FHSC.

Figure 1

Figure 1. Model disk in the midplane and across a vertical slice. Shown are from top to bottom and from left to right: gas volume density, grown dust volume density, maximum dust size, and temperature.

The vertical slices reveal a radially flared gas density profile with local bulges, which correspond to the position of spiral arms. Dust settling to the disk midplane is manifested by a narrower dust density distribution than that of gas. The maximum dust size $a_{\text{max}}$ quickly drops with increasing vertical distance from $\sim 1-10$ mm in the disk midplane down to 2.0 $\mu$m at the disk-cloud interface, a value that is characteristic of the infalling cloud in our model. The temperature drops with vertical distance. This may not be realistic for evolved disks that are passively heated by the central star but is justified in our case, because the radiation input from the FHSC is expected to be insignificant and the disk is mainly heated in this stage by turbulent viscosity and PdV work, both operating more efficiently near the disk midplane, rather than in the disk atmosphere.

Figure 2 presents synthetic radiation intensity images at 1.3 mm obtained by postprocessing the model disk shown in Figure 1 with RADMC-3D. The simulation box is $1100\times 1100\times 1100$ ${\text{au}}^3$, comprising the inner eight nested meshes of the entire numerical hydrodynamics domain, but we show only the inner $90\times 90$ ${\text{au}}^2$ region with a face-on orientation of the disk. To assess the effect of dust growth on the synthetic disk images and to account for the possible effects of electrostatic and bouncing barriers, we considered several cases of the dust size distribution across the disk. Apart from the dust size distribution directly obtained from hydrodynamic modeling with ngFEOSAD (see Figure 1) and characterized by spatially varying values of the maximum dust size $a_{\text{max}}$ (hereafter, referred to as “actual” distribution), we also considered several fixed values for $a_{\text{max}}$. In particular, we have artificially reset the actual dust sizes throughout the disk to a constant value, keeping the gas and dust densities unchanged. We note that we do not recompute the entire model with fixed dust sizes, but rather reset the maximum dust size alone.

Figure 2

Figure 2. Synthetic intensity distributions assuming different maximum size of the dust grains in the disk. From left to right: $a_{\text{max}}=10\mu$m, $100\mu$m, 1 mm, and spatially varying $a_{\text{max}}$ distribution, directly taken from simulations. The envelope has $a_{\text{max}}=2\mu$m in all models considered.

Figure 2 shows the results of our experiments with a fixed $a_{\text{max}}=2.0$ $\mu$m in the cloud, but different $a_{\text{max}}$ in the disk (note that the minimum size stays constant at $a_{\text{min}}=5\times 10^{-3}$ $\mu$m in all cases). As the maximum size of dust in the disk increases, the appearance of the spiral arms in synthetic radiation intensity images becomes more diffuse. The transition is clearly evident between $a_{\text{max}}=100$ $\mu$m and 1.0 mm. More specifically, the transition occurs between $a_{\text{max}}=200\mu$m and 300 $\mu$m, but the intensity maps at these wavelengths are similar to the images at $a_{\text{max}}=100$ $\mu$m and 1.0 mm, respectively. This effect is related to a sharp increase in the dust opacity above a certain maximum dust size, defined as $a_{\text{max}}^{\text{crit}}=\lambda /(2\pi )$, where $\lambda$ is the wavelength of observation (see Supplementary Appendix 11). We note that dust of different size may also have different dynamics and efficiency of dust accumulation in spiral arms, an effect that is not considered in our controlled experiments. However, as was demonstrated in Vorobyov et al. (2024), dust trapping in spiral arms is greatly reduced due to low Stokes numbers in warm and dense environments of protoplanetary disks in the earliest evolutionary stages, an effect which is indirectly confirmed by the lack of spiral patterns in deeply embedded disks (Ohashi et al., 2023).

The effect of sharp increase in opacity is corroborated in Figure 3 showing the optical depth of our model disk at $\lambda =1.3$ mm including contributions from absorption and isotropic scattering. For the case of $a_{\text{max}}$ = 10 $\mu$m and 100 $\mu$m, the outer spiral arms and the interspiral regions are optically thin. Only the FHS, the inner disk regions and the inner parts of spiral arms are optically thick. We note, however, that these optically thick regions may contain a significant fraction of the total radiation flux (see Figure 5). The optically thin medium effectively means that the radiation intensity is proportional to the product of the optical depth and the Planck function. For a larger $a_{\text{max}}=1$ mm and for the actual maximum dust size distribution, the optical depth is above 1.0 almost throughout the entire disk extent, apart from isolated interarmed regions, and the radiation intensity in these optically thick cases is mostly determined by the Planck function. We note also that the spiral arms appear slightly sharper when the actual dust sized distribution is applied, rather than that with a fixed upper limit of $a_{\text{max}}=1.0$ mm.

Figure 3

Figure 3. Optical depth in models with different maximum dust sizes in the disk. From left to right are the cases with $a_{\text{max}}=10$ $\mu$m, $a_{\text{max}}=100$ $\mu$m, $a_{\text{max}}=1.0$ mm, and actual model distribution. The maximum dust size in the envelope is 2.0 $\mu$m in each case.

4.2 Postprocessed synthetic disk images in Band 6

As a next step, we take the synthetic disk images shown in Figure 2 and postprocess them with the ALMA observational support tool (OST) (Heywood et al., 2011) to add atmospheric noise and finite resolution effects. Figure 4 depicts the resulting disk images in Band 6 of ALMA with a beam size of $0.042^{\prime \prime }\times 0.046^{\prime \prime }$ and an exposure time of 1.0 h. The chosen configuration provides an angular resolution that is a factor $\approx 2.3$ lower than the maximum achievable resolution of ALMA in Band 6². Three different distances to the object are chosen: $R=140$ pc, 350 pc, and 700 pc, which correspond to the linear sizes of the beam: $\approx$ 6.0 au, 15 au, and 30 au. Varying the distance to the source but keeping the angular resolution fixed at $0.042^{\prime \prime }\times 0.046^{\prime \prime }$ would imitate the observations of distinct star-forming clusters with the same configuration of ALMA antennas (e.g., Cycle 11, C-8). Each row of panels correspond to dust size distributions in the disk that are characterized by different maximum dust sizes $a_{\text{max}}$ but similar $a_{\text{min}}$, both chosen in accordance with Figure 2. In the infalling cloud, the dust size spectrum is similar in all cases, with minimum and maximum dust sizes of $5\times 10^{-3}$ $\mu$m and 2.0 $\mu$m, respectively. We also note that in some regions beyond the disk, where the detection limit was lower than noise, negative fluxes have occurred. These regions were plotted with the color corresponding the lowest detected limit, $\log I_{\nu }=-14.5$ in CGS units.

Figure 4

Figure 4. Synthetic intensities of the model disk at 1.3 mm obtained with RADMC-3D and postprocessed using the ALMA OST at Band 6. The three columns show, from left to right, the three assumed distances to the object: 140 pc, 350 pc, 700 pc. The rows from top to bottom are models with: $a_{\text{max}}^{\text{disk}}=10\mu$m, $100\mu$m, 1 mm, and actual maximal dust size in the disk. The black and red circles cover the regions of the disk where 90% and 97% of the total radiation flux is contained, defining the disk sizes according to the adopted criteria. In all cases, $a_{\text{max}}$ in the cloud is 2.0 $\mu$m. The white circles in the top-right corner of each panel show the linear size of the beam.

Several interesting features can be noted from our mock observations. Firstly, the spiral arms smear out with increasing distance and are visually indistinguishable at $R=700$ pc. The spiral pattern also becomes less sharp with increasing $a_{\text{max}}$, as was already noted for the unprocessed radiation intensities in Figure 2. This latter finding implies that the spiral pattern caused by gravitational instability may be easier to detect in younger disks with $a_{\text{max}}\le 100\mu$m where the main phase of dust growth is yet to occur (see also Hall et al., 2019). We also note that the fiducial criterion for the disk radius $(R_{90\text{\%}}^{\text{obs}})$ cuts out a large portion of the spiral arms in the disk if the main dust growth phase has not yet occurred (first and second rows in Figure 4).

Secondly, the synthetic disk radii ($R_{90\text{\%}}^{\text{obs}}$ and $R_{97\text{\%}}^{\text{obs}}$) grow with increasing distance to the source. This may be the effect of beam smearing, as the linear size of the beam notably increases with distance, see the white circles in Figure 4. The effect is strongest for $a_{\text{max}}\le$ 100 $\mu$m with an increase in $R_{90\text{\%}}^{\text{obs}}$ and $R_{97\text{\%}}^{\text{obs}}$ of up to $50\%$, but even for the mm-cm sized grains the increase can be up to 20%. We note that subtracting about 1/2 of the linear size of the beam from the synthetic disk radii can help to compensate for the effects of beam smearing (with increasing distance) on the apparent radius of the dust disk, in particular for $R_{90\text{\%}}^{\text{obs}}$.

Thirdly, disks in a more advanced stage of dust growth look somewhat bigger. This can be related to the optical depth effects. For the case of $a_{\text{max}}\le 100$ $\mu$m, the central optically thick region of the disk ($\le 20$ au) dominates the radiation intensity and the optically thin spiral arms are characterized by an order of magnitude weaker values (see Figure 4). For large grains and optically thick disks, the contrast between the inner and outer disk regions is reduced.

Indeed, Figure 5 demonstrates that the radial intensity profiles of the synthetic disk in the $a_{\text{max}}\le 100$ $\mu$m limit are steep and fall off with distance rapidly. A substantial fraction of the integrated radiation flux is, therefore, localized within the inner compact region of the disk. For models with $a_{\text{max}}\ge 1.0$ mm, however, the radiation intensity has a weaker contrast between the inner disk regions and the spiral arms, both being optically thick. As a result, the radial intensity profiles become shallower, making the disk look bigger when the same criterion (90% or 97%) for the integrated radiation flux is applied. At the same time, beam smearing also makes the intensity profiles look shallower as the distance to the source increases, regardless of $a_{\text{max}}$. We also note that for models with $a_{\text{max}}\ge 1.0$ mm the radiation intensity in the central regions of the disk decreases as compared to models with $a_{\text{max}}\le 100$ $\mu$m. This is the effect of dust scattering (see Supplementary Appendix 10, 11), which also contributes to the overall decrease in the integrated radiation flux as discussed later in Sect. 4.3 in the context of dust masses.

Figure 5

Figure 5. Radial profiles of the radiation intensity in CGS units after postropcessing with ALMA OST. The profiles are obtained by azimuthally averaging the corresponding spatial distributions shown in Figure 4. The vertical dashed and dotted lines indicate the radial positions within which 90% or 97% of the total flux is localizes, respectively. Panels from left to right correspond to different adopted distances to the source.

The entire disk becomes optically thick in the $a_{\text{max}}\ge 1.0$ mm models (see Figure 3) because the maximum dust size surpasses the critical value $a_{\text{max}}^{\text{crit}}=\lambda /(2\pi )$. This effect is known as the opacity cliff, see also Supplementary Appendix 11. As dust grows and the optical depth transits from the optically thin $({\tau }_{\nu }<1.0)$ to the optically thick $({\tau }_{\nu }>1.0)$ regime, the radiation intensity becomes proportional to the Planck function, rather than to the product of the Planck function and optical depth. Since both quantities decline with radius, the product of the two (optically thin limit) would produce a steeper distribution of the radiation intensity than just the Planck function (optically thick limit). As a result, the extent within which 90% (or 97%) of the flux is contained (the disk radius by our definition) becomes bigger in the optically thick case. The effect can be substantial, with an increase of more than $50\%$ in $R_{90\text{\%}}^{\text{obs}}$ when the maximum dust sizes grows from $a_{\text{max}}$ = 10 $\mu$m to mm-cm sized grains in the disk. The trend is also present but less expressed when $R_{97\text{\%}}^{\text{obs}}$ is used.

4.3 Synthetic disk masses

The first step in calculating the synthetic disk masses is to determine the flux $F_{\lambda }$ in Equation 1. Table 1 presents the integrated synthetic fluxes contained within $R_{90\text{\%}}^{\text{obs}}$ and $R_{97\text{\%}}^{\text{obs}}$ for all three adopted distances. As expected, $F_{90\%}^{\text{obs}}$ and $F_{97\%}^{\text{obs}}$ decrease with distance, but their behavior with increasing $a_{\text{max}}$ is more complex. The highest values are found for $a_{\text{max}}\le 100$ $\mu$m and the lowest are around $a_{\text{max}}=1.0$ mm. The decrease in the flux at the advanced stages of dust growth is likely caused by flux dilution due to strong dust scattering at mm-sized grains, see Supplementary Appendix 10,11.

Table 1

Table 1. Integral radiation fluxes in Band 6.

To determine dust masses from the synthetic disk images shown in Figure 4, we should also know the value of the average dust temperature $T_{\text{d}}$ that enters Equation 1. This quantity is hard to constrain directly from observations and therefore the average dust temperature is often determined for young protoplanetary disks (Class 0 and I) following Tobin et al. (2020) as:

$T_{\text{d}}=43K{\left(\frac{L_{*}}{1.0{\mathrm{L}}_{\odot }}\right)}^{0.25},(2)$

where $L_{*}$ is the luminosity of the central source. This relation is based on radiation transfer simulations of a protoplanetary disk embedded into an envelope and heated only by the central star. This case is different from the earliest evolutionary stage considered in this study, in which the disk is actively heated by internal hydrodynamic processes and stellar irradiation is neglected. Therefore, Equation 2 is not directly applicable in our case.

We could have used the actual hydrodynamic data to derive $T_{\text{d}}$ as an alternative. However, we adopted a barotropic equation of state, which is a simplification compared to the solution of the full energy balance equation including radiation transfer. We also note that the ngFEOSAD code makes no distinction between the gas and dust temperatures. More sophisticated simulations demonstrate that deviations may occur at the disk-envelope interface where a higher temperature of gas than that of dust may be expected due to heating by the shock wave caused by the infalling envelope (Pavlyuchenkov et al., 2015; Bate and Keto, 2015; Vorobyov et al., 2020).

Considering these uncertainties we decided not to focus on a particular value of the mean dust temperature but instead used a range of $T_{\text{d}}=10-50$ K when calculating the dust mass in Equation 1. We note that Dunham et al. (2014) found that an average temperature declines from 30 K for Class 0–15 K for Class I objects, and an upper limit of 50 K for a very young disk considered in this work is consistent with a general temperature decline with age (Commerçon et al., 2012).

Using Equation 1, we can now calculate the dust mass contained within the synthetic disk shown in Figure 4, with its radius defined by either $R_{90\text{\%}}^{\text{obs}}$ or $R_{97\text{\%}}^{\text{obs}}$. We note that the dust opacity is that of Ossenkopf and Henning (1994) for all calculations in this section.

The resulting values are plotted in Figure 6. The synthetic masses of dust vary with $a_{\text{max}}$ in the same manner as the integrated radiation fluxes, namely, they decrease at the advanced stages of dust growth when $a_{\text{max}}$ transits from 10 $\mu$m to 1.0 mm, likely due to flux dilution by scattering on grown dust grains. An opposite trend is observed when $a_{\text{max}}$ changes from 1.0 mm to the actual dust size distribution, which is on average higher than 1.0 mm, see Figure 1. Interestingly, the peak in the dust scattering opacity is also located around 1.0 mm, see Supplementary Appendix Figure 16, thus reinforcing our conclusion about the effects of scattering on the dust mass estimates. At the same time, $M_{90\text{\%}}^{\text{obs}}$ and $M_{97\text{\%}}^{\text{obs}}$ only insignificantly vary with distance to the source, and this small variation is probably due to beam smearing.

We now compare the synthetic dust masses shown in Figure 6 with the actual mass of dust in our model disk. The latter is calculated by summing up all dust within the disk extent defined by the red contour line in Supplementary Appendix Figure 14 (see Supplementary Appendix 9). The resulting value is $M_{\text{dust}}^{\text{mod}}\approx 673M_{\oplus }$ (evaporated dust in the FHSC is not counted), almost independent of the threshold gas number density ($10^9$ or $10^{10}$ ${\text{cm}}^{-3}$). Figure 6 demonstrates a strong dependence of the synthetic dust mass on the dust temperature. In all models considered, the dust mass is heavily underestimated for $T_{\text{d}}\ge 20$ K by factors of several. The mismatch is somewhat sensitive to the adopted radius of the disk: $R_{90\text{\%}}^{\text{obs}}$ or $R_{97\text{\%}}^{\text{obs}}$, or to the dust growth phase, but the trend to underestimate the intrinsic dust mass persists. For the reader’s convenience, Table 2 provides the synthetic dust masses at several $T_{\text{d}}$. The underestimate of dust mass in the disk found for very young disks in our study is in agreement with similar conclusions by Dunham et al. (2014) for Class 0 and I disks, see also Sect. 5. A good match between the synthetic and intrinsic dust masses can only be achieved for $T_{\text{d}}<20$ K, and in particular for $T_{\text{d}}=$ 15.9 K, 16.0 K, 13.1 K, and 14.4 K in models with $a_{\text{max}}=10$ $\mu$m, $a_{\text{max}}=100$ $\mu$m, $a_{\text{max}}$ = 1.0 mm, and in the model with the actual dust size distribution, respectively. These values were derived using $R_{90\%}^{\text{obs}}$ as the disk radius. We emphasize that the best-fit temperatures are lower than what is usually assumed when recovering dust masses from observations, $T_{\text{d}}=20-40$ K (Ansdell et al., 2017; Tobin et al., 2020; Kóspál et al., 2021), but are comparable to the average temperature in the envelope and the optically thin disk regions. As we will see below, the proper choice of average dust temperature may also depend on $a_{\text{max}}$ and the ALMA band.

Figure 6

Figure 6. Masses of dust in the synthetic disk. The left column shows the masses as a function of $a_{\text{max}}$ but for the fixed distance to the source $d=140$ pc, while the right column corresponds to distinct $d$ but fixed $a_{\text{max}}$ = 1.0 mm. Solid and dotted lines represent the mass estimates using $R_{90\%}^{\text{obs}}$ and $R_{97\%}^{\text{obs}}$, respectively. The horizontal dashed line defines the intrinsic dust mass in the model disk.

Table 2

Table 2. Synthetic dust masses in Band 6 at particular $T_{\text{d}}$.

4.4 The effect of dust-size-dependent opacities

The value of the mean dust temperature depends on the intricacies of the averaging procedure (Tobin et al., 2020) and on the disk evolutionary stage (Dunham et al., 2014; Ansdell et al., 2017), and thus may vary in wide limits. However, this may not be the only cause of mismatch between the intrinsic and synthetic disk masses. Dust opacity is known to depend on the dust size spectrum and, in particular, on the maximum dust size. Since dust growth has proven to be efficient already in the very early stages of evolution, uncertainties with the actual sizes of dust grains and, thus, with the dust opacity may also affect the dust mass estimates. Instead of using the OH5 opacity ${\kappa }_{\lambda =1.3\mathrm{m}\mathrm{m}}=0.89$ ${\text{cm}}^2$ ${\text{g}}^{-1}$ from Ossenkopf and Henning (1994), as was done in Sect. 4.3 following the common practice in observational astronomy (Tobin et al., 2020; Kóspál et al., 2021), we now adopt the dust-size dependent absorption and scattering opacities from Woitke et al. (2016) (see Supplementary Appendix Figure 16 in Supplementary Appendix 11). We note that Ossenkopf and Henning (1994) opacities are proper to use for cold dark clouds, rather than for very high-density environments of disks.

Figure 7 shows the synthetic dust masses in the disk $M_{90\text{\%}}^{\text{obs}}$, derived using Equation (1) for a wide range of dust temperatures and maximum dust sizes. To guide the eye, the black line shows the intrinsic dust mass in the disk, $M_{\text{disk}}^{\text{mod}}=673M_{\oplus }$. The slope of the dust size distribution is always kept equal $p=-3.5$ and the distance to the source is 140 pc.

Figure 7

Figure 7. Synthetic dust mass in the disk, $M_{90\text{\%}}^{\text{obs}}$, as a function of dust temperature and maximum dust size. Radiation fluxes for the actual dust size distribution in Band 6 were used to derive the masses. The solid line indicates the true dust mass in the disk, $M_{\text{disk}}^{\text{mod}}=673M_{\oplus }$. The dotted lines delineate the synthetic dust masses that differ by factors 2.0, 0.5, 0.1 from the true disk mass.

Clearly, there is no universal dust temperature that would provide the best match between the synthetic and intrinsic dust masses in the disk if dust growth is considered and the corresponding dust-size-dependent opacities are used. The dust temperature for which both masses match $(M_{90\text{\%}}^{\text{obs}}=M_{\text{disk}}^{\text{mod}}=673M_{\oplus })$ for a particular dust size distribution varies from $\approx 23$ K in disks with low to moderate dust growths ($a_{\text{max}}\le 100$ $\mu$m) to as low as 3.0 K for disks with $a_{\text{max}}\approx 1.0$ mm, and again increases to approx 8–20 K for disks with advanced dust growth ($a_{\text{max}}\ge 10$ cm). Extremely low dust temperatures required to match the intrinsic dust mass around $a_{\text{max}}\approx 1.0$ mm are caused by the increase in dust opacity at the opacity cliff (see Supplementary Appendix 16). We also note that an assumption of $T_{\text{d}}>23$ K would always underestimate the intrinsic dust mass, regardless of the progress in dust growth, while lower temperatures may also overestimate dust mass in the limit of $a_{\text{max}}\le 100$ $\mu$m.

4.5 Effects of ALMA configuration

In the previous sections, model disks located at different distances of 140 pc, 350 pc, and 700 pc but observed with a similar beam size of $0.042^{\prime \prime }\times 0.046^{\prime \prime }$ were considered.

This setup imitated distinct star-forming clusters located at different distances from the Sun but observed with the same ALMA configuration. In this section, we explore another setup and consider a model disk observed at a fixed distance of 140 pc but with different angular resolution. In particular, two beam sizes are considered: $0.042^{\prime \prime }\times 0.046^{\prime \prime }$ and $0.134^{\prime \prime }\times 0.146^{\prime \prime }$. The linear sizes of the beam in this case are 6 au and 16 au, respectively. This case imitates observations of the same star-forming cluster but with different ALMA configurations (e.g., Cycle 11, Configurations C-8 and C-6, respectively).

Figure 8 compares the resulting synthetic disk images. The effect of changing the beam size on the synthetic disk masses and sizes is similar to that seen in Figure 4 where the beam size was fixed but distance to the source was changing. In particular, the synthetic disk radius grows with increasing beam size by about 10%, likely due to the effect of beam smearing, while the dust mass is only slightly affected and increases by less than 1%. It seems that the beam smearing insignificantly alters the total flux, which determines the dust mass estimates (see Equation 1). Although the disk radius increases, this effect has little consequence for the mass estimates because the radiation intensity rapidly declines with radial distance near the disk outer edge (see, Figure 5). We also note that the spiral pattern is significantly smoothed out and barely visible for a beam size of $0.134^{\prime \prime }\times 0.146^{\prime \prime }$, even for the closest possible distance of 140 pc.

Figure 8

Figure 8. Synthetic intensity distribution of the model disk at 1.3 mm postprocessed using the ALMA OST for Band 6. The two columns show the effects of different beam sizes: $0.042^{\prime \prime }\times 0.046^{\prime \prime }$ (left) and $0.134^{\prime \prime }\times 0.146^{\prime \prime }$ (right). The distance to the object is 140 pc in both cases. The models shown from top to bottom: $a_{\text{max}}^{\text{disk}}=10\mu$m, $100\mu$m, 1 mm, and variable maximal size in the disk. The black and red circles cover the regions of the disk where 90% and 97% of the total flux is contained. The linear size of the beam is indicated with white circles.

4.6 Other ALMA bands

In this section, we show synthetic observations of the disk at the ALMA Band 3 ($\lambda =3.0$ mm) for comparison. The beam size in this case is set equal to $0.105^{\prime \prime }\times 0.115^{\prime \prime }$ and the corresponding configuration provides an angular resolution that is a factor $\approx 2.5$ lower than the maximum achievable resolution in Band 3. The linear sizes of the beam at the adopted distances of 140 pc, 350 pc, and 700 pc are $\approx$15 au, 38 au, and 75 au. Dust opacities in Ossenkopf and Henning (1994) were tabulated only out to $\lambda =1.3$ mm. For consistency, we extrapolate the opacity to longer wavelengths using the following relation

${\kappa }_{\lambda =3.0\mathrm{m}\mathrm{m}}={\kappa }_{\lambda =1.3\mathrm{m}\mathrm{m}}{\left(\frac{\lambda =1.3\mathrm{m}\mathrm{m}}{\lambda =3.0\mathrm{m}\mathrm{m}}\right)}^{\beta },(3)$

where the dust opacity index is set equal to $\beta =1.8$ as in Ossenkopf and Henning (1994).

Figure 9 presents the resulting synthetic images. The radiation intensity in Band 3 notably decreases compared to Band 6, as expected for a longer wavelength.

Figure 9

Figure 9. Similar to Figure 4 but for Band 3 with a beam size of $0.105^{\prime \prime }\times 0.115^{\prime \prime }$.

At the same time, the integrated flux increases with growing $a_{\text{max}}$ as Table 3 demonstrates, unlike Band 6 where we saw the opposite trend caused by flux dilution due to a sharp increase in dust scattering around $a_{\text{max}}=1.0$ mm. Apparently, the increase of radiation flux due to transition from the optically thin regime to the optically thick one (see Figure 11 below) outweighs the effect of dust scattering in Band 3.

Table 3

Table 3. Integral radiation fluxes in Band 3.

We find that the synthetic disk radius increases with distance somewhat stronger than in the case of Band 6. This can be expected, as the linear sizes of the beam in Band 3 are greater than for the case of Band 6 (the latter is generally characterized by higher resolution) and the effect of beam smearing in Band 3 is stronger. The increase in $R_{90\%}^{\text{obs}}$ and $R_{97\%}^{\text{obs}}$ can be as large as 70% for $a_{\text{max}}\le 100\mu$m and about 90% for bigger grains. We also see an increase in the synthetic disk radius as dust grows. The effect is more pronounced for smaller linear sizes of the beam (better resolution) where the effect of beam smearing is not dominant. The resulting synthetic disk radii are provided in Table 4. We note here that the disk is marginally resolved at 350 pc and is not resolved at 700 pc, thus the corresponding disk size estimates must be taken with care.

Table 4

Table 4. Synthetic disk radii.

The synthetic dust masses are shown in Figure 10 for a range of $T_{\text{d}}$ and Ossenkopf and Henning (1994) opacity. A good agreement with the intrinsic dust mass is achieved for $T_{\text{d}}=30.9$ K, 31.6 K, 36.8 K and 41.4 K in models with $a_{\text{max}}=10$ $\mu$m, 100 $\mu$m, 1.0 mm and in the model with the actual dust size distribution, respectively. All estimates adopt $R_{90\%}^{\text{obs}}$ as the disk radius. These best-fit $T_{\text{d}}$ are within the range of average dust temperatures adopted in observational studies (Ansdell et al., 2017; Tobin et al., 2020; Kóspál et al., 2021). This means that the dust mass estimates in Band 3 are expected to be more accurate than in Band 6, although notable deviations, both underestimates and overestimates, are possible as $T_{\text{d}}$ varies in the 20–50 K limits. In general, the best-fit $T_{\text{d}}$ increases at the advanced stages of dust growth. As was the case for Band 6, the synthetic disk masses weakly depend on the distance to the source. The resulting synthetic disk masses for several $T_{\text{d}}$ are provided in Table 5.

Figure 10

Figure 10. Similar to Figure 6 but for Band 3.

Table 5

Table 5. Synthetic dust masses in Band 3 for particular $T_{\text{d}}$.

The reason for a better recovery of the intrinsic dust mass is in the systematically lower optical depths of the disk in Band 3. Figure 11 shows the optical depth in Band 3 for models with different maximum dust sizes $a_{\text{max}}$, considering both dust absorption and scattering. For models at the initial stages of dust growth ($a_{\text{max}}\le 100$ $\mu$m), the disk is mostly optically thin, apart from the very inner region with a size of several astronomical units. For advanced stages of dust growth ($a_{\text{max}}\ge 1.0$ mm), the bulk of the disk is optically thick, but on average ${\tau }_{\mathrm{\nu }}$ is a factor of several lower than for the Band 6 case (see Figure 3).

Figure 11

Figure 11. Similar to Figure 3 but for Band 3.

Figure 12 shows the effects of dust growth and varying dust temperature on dust mass estimates. Now, we use the dust-size-dependent opacities (rather than those of Ossenkopf and Henning, 1994) to calculate the dust mass using Equation (1). As for the case of Band 6, there is no universal dust temperature that can provide a good match between synthetic and intrinsic dust masses in the disk when dust growth is considered. For the initial stages of dust growth, when $a_{\text{max}}\le \mathrm{a}\mathrm{f}\mathrm{e}\mathrm{w}\times 100$ $\mu$m, any choice of $T_{\text{d}}$ below 50 K overestimates the true dust mass. At larger $a_{\text{max}}$, the dust mass is in general underestimated, unless for very small $T_{\text{dust}}<10$ K.

Figure 12

Figure 12. Similar to Figure 12 but for B and 3.

We also note that the spiral structure is only detectable at a distance of 140 pc and is completely smoothed out at larger distances, unlike Band 6 for which the spiral structure was still visible at a distance of 350 pc (see Figure 4). The $R_{90\%}^{\text{obs}}$ circle cuts out a large fraction of the spiral arms for models with $a_{\text{max}}\le 100$ $\mu$m, as was also the case for Band 6.

5 Discussion

This work is focused on the earliest stages of disk evolution, when the FHSC or the protostar has not yet provided sufficient luminosity output to start dominating the energy balance in the disk and the disk temperature is instead set by internal hydrodynamic processes. This makes the radiative transfer simulations somewhat easier as there is no uncertainty with the intrinsic parameters of the growing star (such as the effective temperature and stellar radius) and the stellar luminosity output such as accretion luminosity. At the same time, this stage is short and is observationally difficult to capture. There are very few FHSC candidates (Chen et al., 2010; Pezzuto et al., 2012; Duan et al., 2023), and none have been reliably verified to date. Nevertheless, we think that it is important to explore these earliest stages, even though this study may currently represent only a theoretical prospective. In a follow-up study, we will explore subsequent Class 0 and I stages of disk evolution, where the protostar significantly contributes to the disk energy balance, with a similar numerical setup of ngFEOSAD. The current proof-of-concept theoretical work provides an important bridge to more observationally motivated studies of the Class 0 and I stages, including the disk inclination effects which were omitted in the present work.

We found that similar disks but located in distinct star-forming regions at different distances from the Sun can have different apparent disk radii and the variation around the intrinsic value may be substantial, up to a factor of two, depending on the dust growth stage in the disk. When using the Ossenkopf and Henning (1994) opacities and dust temperatures typical for Class 0 and I disks, Equation (1) can yield significant dust mass underestimates by factors of several. The problem is more severe in Band 6 than in Band 3 of ALMA. In the latter case, the dust mass can actually be even overestimated. We also demonstrated that by adjusting the rather uncertain dust temperature in Equation (1), it becomes possible to recover the intrinsic mass of dust in the disk. The best-fit values of $T_{\text{d}}$ strongly depend on the ALMA band, but are weakly sensitive to the distance to the source.

The situation becomes more complicated when the dust-size-dependent opacities are considered. The best-fit $T_{\text{d}}$ begin to strongly depend on $a_{\text{max}}$, that is, on the dust growth stage in the disk. A good match between the synthetic and intrinsic dust masses for $a_{\text{max}}\ge 1.0$ mm may require a dust temperature that is much lower than usually adopted in observational studies. We note that the best-fit $T_{\text{d}}$ also depends on $a_{\text{max}}$ via the corresponding variations in the integrated radiation flux (see Tables 1, 3), but the dependence via the varying dust opacity is stronger. We also note that the intricate relation between $T_{\text{d}}$ and $a_{\text{max}}$ implies some dependence of the former on the latter. This is possible considering that $T_{\text{d}}$ depends on dust opacity and dust opacity in turn depends on $a_{\text{max}}$. In addition, the gas-to-dust energy transfer rate depends on the dust properties, such as the total surface area which changes as dust grows, but these effects are beyond the present study.

Our model and synthetic disk radii (see Table 6) are broadly consistent with those inferred for Class 0 disks by the CAMPOS and VANDAM surveys of the Aquila, Corona Australis, Ophiuchus North, Ophiuchus, Orion, and Serpens molecular clouds (Tobin et al., 2020; Hsieh et al., 2024). Our model, however, considers the disk at the FHSC to protostar transition and may not be representative in terms of disk mass and radius due to short duration of this phase. In any case, the comparison of the synthetic disk radii and masses with the observationally derived samples was not the purpose of this study. Here, we explored the accuracy of inferring the true disk radii and masses from synthetic millimeter fluxes, excluding the associated uncertainties with the luminosity of the growing star but taking dust growth into consideration.

Table 6

Table 6. Synthetic disk radii in Band 6.

The mismatch between intrinsic and synthetic disk masses was also reported by Dunham et al. (2014) in the context of Class 0 and I disks, though without considering the effects of dust growth. It was also found that the disk masses may be underestimated by up to factors of two to three at mm wavelengths and up to an order of magnitude at sub-mm wavelengths, in particular, due to uncertainties in the optical depth and dust temperatures, as was also found in our study. In contrast, Harsono et al. (2015) found that disk masses inferred from synthetic observations in millimeter wavelength agree rather well with the actual disk masses of young embedded disks obtained from magnetohydrodynamics simulations. However, the adopted beam size of 5–${15}^{\prime \prime }$ may have encompassed the large-scale emission of the surrounding envelope, rather than the disk itself. They also retrieved the disk mass from the total envelope + disk flux (Equations 1–3 in Jørgensen et al., 2009). Mass underestimates of gravitationally unstable disks by factors 2.5–30 using synthetic (sub)-mm observations at different ALMA bands was also reported in Evans et al. (2017). In a more recent study by Liu et al. (2022), focused rather on Class II analytic disk models but with sophisticated dust composition and considering dust growth, possible severe underestimation of dust mass was also reported. The requirement of multi-wavelength observations for more accurate measurements of usually underestimated dust masses was also reported in Viscardi et al. (2025).

We also note that multi-wavelengths observations of polarized dust emission can help to distinguish between small grains with $a_{\text{max}}\le 100$ $\mu$m and larger grains with $a_{\text{max}}\sim 1.0$ mm (e.g., Guillet et al., 2020), thanks to the change in the direction of polarization from perpendicular to parallel with respect to magnetic field lines. Interestingly, this is about a dust size range where a qualitative transition in the synthetic disk images occurs in our model (see Figures 3, 4). To conclude, most theoretical studies report a substantial underestimate of the dust mass when using the conventional techniques (see Equation 1), and this issue may be inherent to all stages of disk evolution, including the very earliest considered in this work.

We finally note that the magnetic field and turbulence can have significant impacts on disk masses and sizes, especially in disk’s early stages of evolution (Seifried et al., 2012; Santos-Lima et al., 2013; Lam et al., 2019; Maury et al., 2022). For instance, magnetic braking and magnetic disk winds can reduce the disk mass and size, and magnetohydrodynamic instabilities can shape the infalling envelope into streamers instead of axially symmetric infall (Machida and Basu, 2025). While the peculiar structure of the infalling envelope is unlikely to affect the synthetic images at mm-wavelengths because of its rarefied nature, a reduced disk mass could also result in lower optical depths, and, as a consequence, in more accurate mass estimates via Equation (1). This effect can be mimicked by our collapse simulations with lower initial cloud core masses, which we leave for future studies.

6 Conclusion

In this work, we investigated the accuracy with which the mass and radius of a young protoplanetary disk can be inferred using its dust thermal emission at millimeter wavelengths. We started with a three-dimensional hydrodynamic simulation of protoplanetary disk formation with the ngFEOSAD code. We then exported the resulting density and temperature distributions of dust in the disk and envelope into the RADMC-3D code, using either the realistic dust size distribution derived directly from hydrodynamic simulations, or making simplifying assumptions about the maximum dust size to explore the possible effects of dust growth. Next, we postprocessed the radiation fluxes with the ALMA observational support tool to generate realistic synthetic images of our model disk at different ALMA bands and with different resolution. These images were finally used to calculate the synthetic disk radii and the dust masses contained with these radii using the conventional method (see Equation 1) adopted in observational astronomy.

Four models with the maximum dust size corresponding to moderate and advanced stages of dust growth were considered: $a_{\text{max}}=10$ $\mu$m, $a_{\text{max}}=100$ $\mu$m, $a_{\text{max}}=1.0$ mm, and the model with the spatially varying distribution of $a_{\text{max}}$ obtained from hydrodynamic simulations. We focused on a very young protoplanetary disk that formed in our simulations around a first hydrostatic core, just before it transits to a protostar. This allowed us to exclude uncertainties with the protostellar radiation output and focus on other parameters that may influence the disk mass and radius measurements. The main findings can be summarized as follows.

• When choosing the Ossenkopf and Henning (1994) opacity, dust mass can be underestimated by factors of $1.4-4.2$ in Band 6 for the mean dust temperatures $T_{\text{d}}=20-40$ K, a typical range adopted in observational studies. A good match between the synthetic and intrinsic dust masses in Band 6 can be recovered for dust temperatures $T_{\text{dust}}=13-16$ K. Observations in Band 3 can both underestimate or overestimate the dust mass by up to a factor of two depending on the choice of $T_{\text{d}}$. The best-bit $T_{\text{d}}$ lie in the 31–41 K limits. Synthetic disk masses in ALMA Bands 3 and 6 seem to be weakly sensitive to the distance to the source.

• When more realistic dust-size-dependent opacities are considered (see Supplementary Appendix Figure 16), the discrepancy between the synthetic and intrinsic dust masses begins to strongly depend on the maximum dust size, in addition to strong dependence on $T_{\text{d}}$. To achieve a fair agreement, information on the maximum dust size in the disk is desirable, and extremely low values of dust temperature may be required to reconcile the synthetic and intrinsic masses for $a_{\text{max}}\ge 1.0$ mm.

• Synthetic disks look bigger at the advanced stages of dust growth $a_{\text{max}}\ge 1.0$ mm. This may be due to the fact that most of the disk becomes optically thick as dust grows and the dust opacity reaches a peak value near $\lambda =a_{\text{max}}/(2\pi )$.

• Synthetic disk sizes grow with increasing distance to the source and deteriorating linear resolution, likely due to the effect of beam smearing.

• Dust scattering becomes significant at the advanced stages of dust growth ($a_{\text{max}}\simeq 1.0$ mm), affecting the radial intensity profiles and dust mass estimates, in particular in Band 6.

• Spiral pattern generated by gravitational instability is easier detected at the early stages of dust growth ($a_{\text{max}}\le 100\mu$m) than at the advanced stages ($a_{\text{max}}\ge 1.0$ mm) and in Band 6 rather than in Band 3.

Both adopted definitions for the disk radius, 90% or 97% of the total flux, can either underestimate or overestimate the intrinsic disk radius, depending on the linear resolution of observations and maximum dust size. This means that a young protoplanetary disk viewed at different resolution and at different stages of dust growth would have different apparent disk radii, and the variation around the intrinsic value can be substantial, up to a factor of two. The effect of beam smearing with increasing distance can be offset by subtracting about 1/2 of the linear size of the beam from the synthetic disk radii. This work provides an important bridge to more observationally accessible Class 0 and I stages of disk evolution. In a follow-up study, we plan to use the developed algorithm to study the accuracy with which Class 0 and I disk masses and sizes can be recovered using thermal dust emission.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

EV: Conceptualization, Funding acquisition, Methodology, Software, Supervision, Validation, Writing – original draft, Writing – review and editing. AS: Data curation, Methodology, Software, Validation, Visualization, Writing – review and editing. VE: Data curation, Methodology, Visualization, Writing – review and editing. MD: Formal Analysis, Validation, Writing – review and editing. MG: Formal Analysis, Validation, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work was supported by the Austrian Science Fund (FWF), project I4311-N27, DOI:10.55776/I4311. AS and VE also acknowledge support by the Ministry of Science and Higher Education of the Russian Federation (State assignment in the field of scientific activity 2023, GZ0110/23-10-IF). Open access funding provided by University of Vienna.

Acknowledgements

We are thankful to the reviewers for constructive comments that helped to improve the manuscript. Simulations were performed on the Austrian Scientific Cluster (https://asc.ac.at/).

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2026.1728292/full#supplementary-material

Footnotes

¹https://almascience.nrao.edu/about-alma/alma-basics

²https://almascience.eso.org/about-alma

References

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