arXiv subjects

Shangjia Zhang

Publications and source records attributed to Shangjia Zhang.

At least 19 recordsLinked to original sources

arXiv2026

A Framework to Model Stellar Irradiated Disks with Frequency-dependent Absorption and Scattering Opacities in Athena++

The frequency dependence of opacity is crucial for determining the thermal structure of protoplanetary disks, which in turn influences disk dynamics and planet formation. Yet many disk models adopt simplified thermodynamics, and common radiation-hydrodynamic approaches often use gray opacities, ignore scattering, and yield inaccurate results in regions with intermediate optical depth. We present a comprehensive framework that models stellar irradiation with frequency-dependent absorption and scattering across all optical depths using the Athena++ finite-volume code, extended with multigroup radiation transport and newly implemented radial rays to more accurately represent the stellar flux. To calibrate this framework, we focus exclusively on hydrostatic disk models, allowing us to isolate radiative effects and evaluate the method without additional dynamical complexity. Because dust opacity increases strongly with frequency, ultraviolet stellar irradiation heats the tenuous disk atmosphere while the optically thick midplane remains cooler. This vertical temperature gradient is captured more accurately when more frequency bands are used or when scattering is included. Our hydrostatic models achieve equilibrium temperatures that differ from Monte Carlo radiative-transfer benchmarks on average by 2--5% with 64 frequency bands and 7--11% with 3 bands. Reducing the number of bands lowers computational cost by at least an order of magnitude while increasing the maximum possible temperature deviation only from 8% to 19%. This calibration demonstrates the accuracy and efficiency of the framework and provides a solid foundation for future self-consistent studies of irradiated protoplanetary disks, including fully dynamical simulations and applications involving chemical processes and time-dependent stellar luminosity.

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arXiv2025

Shadow-Induced Warps in Protoplanetary disks

Shadows are commonly observed in protoplanetary disks in near-infrared and (sub)millimeter images, often cast by misaligned inner disks or other obscuring material. While recent studies show that shadows can alter disk dynamics, only the case symmetric across the midplane (e.g., from a polar-aligned inner disk) has been studied. Here we study shadows cast by an inner disk with a $30^\circ$ mutual inclination using 3D radiation-hydrodynamical simulations. Given the same shadow shape and amplitude, the $30^\circ$ inclined shadow leads to a much stronger accretion compared with the polar case, reaching $α\sim$ 1, because the disk is squeezed twice in one azimuth, leading to shocks and strong radial flows near the midplane. The outer disk develops a warp: the inner disk region tilts toward alignment with the shadow, while the outer, exponentially tapered disk tilts and twists in a different direction, inclined $\sim$ 32$^\circ$ relative to the inner region. Locally isothermal simulations with a prescribed temperature structure reproduce the effect, confirming that it is thermally driven. Fourier-Hermite analysis shows that it is the m=1, n=1 temperature perturbation that drives the warp by launching bending waves, with the tilting response of the disk approximately proportional to the modal amplitude. This mode always exists unless the shadow is coplanar or polar. Given a fixed temperature contrast, the m=1,n=1 mode peaks at $\sim$15$^\circ$ mutual inclination, but still contributes substantially across 3$^\circ$ to 30$^\circ$. Shadows cause disk warps--they are not only a consequence of them. We discuss testable predictions for current and future ALMA and NIR observations.

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arXiv2025

Puffed-up Inner Rings and Razor-thin Outer Rings in Structured Protoplanetary Disks

The vertical distribution of pebbles in protoplanetary disks is a fundamental property influencing planet formation, from dust aggregation to the assembly of planetary cores. In the outer region of protoplanetary disks, the intensity of the optically thin but geometrically thick dust ring decreases along the minor axis due to reduced line-of-sight optical depth. Multi-ring disks thus provide an excellent opportunity to study the radial variation of the vertical properties of dust. We investigate the vertical dust distribution in 6 protoplanetary disks with resolved double rings, using high-resolution ALMA Band 6 continuum observations. By modeling the azimuthal intensity variations in these rings, we constrain the dust scale heights for each ring. Our results reveal a dichotomy: inner rings exhibit puffed-up dust layers with heights comparable to the gas scale height, while outer rings are significantly more settled, with dust scale heights less than 20\% of the gas scale height. This suggests a radial dependence in dust settling efficiency within the disks, potentially driven by localized planetary interactions or the global radial dependence of the Vertical Shear Instability (VSI). We discuss the implications of these findings for dust trapping, planet formation, and protoplanetary disk evolution. Our work highlights the importance of vertical dust distribution in understanding the early stages of planet formation and suggests that outer ($>80$~au), settled rings are preferred sites for planet formation over inner ($<80$~au), turbulent rings.

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arXiv2025

Discovery of H$α$ Emission from a Protoplanet Candidate Around the Young Star 2MASS J16120668-3010270 with MagAO-X

2MASS J16120668-3010270 (hereafter 2MJ1612) is a young M0 star that hosts a protoplanetary disk in the Upper Scorpious star-forming region. Recent ALMA observations of 2MJ1612 show a mildly inclined disk ($i$=37$^\circ$) with a large dust-depleted gap (R$_\text{cav}\approx$0.4" or 53 au). We present high-contrast H$α$ observations from MagAO-X on the 6.5m Magellan Telescope and new high resolution sub-mm dust continuum observations with ALMA of 2MJ1612. On both 2025 April 13 and 16, we recovered a point source with H$α$ excess with SNR $\gtrsim$5 within the disk gap in our MagAO-X Angular and Spectral Differential (ASDI) images at a separation of 141.96$\pm$2.10 mas (23.45$\pm$0.29 au deprojected) from the star and position angle (PA)= 159.00$\pm$0.55$^\circ$. Furthermore, this H$α$ source is within close proximity to a K band point source in SPHERE/IRDIS observation taken on 2023 July 21 \citep{sphere2025sub}. The astrometric offset between the K band and H$α$ source can be explained by orbital motion of a bound companion. Thus our observations can be best explained by the discovery of an accreting protoplanet, 2MJ1612 b, with an estimated mass of 4$M_\text{Jup}$ and H$α$ line flux ranging from (29.7 $\pm$7.5)$\times$10$^{-16}$ ergs/s/cm$^2$ to (8.2$\pm$3.4)$\times$10$^{-16}$ ergs/s/cm$^2$. 2MJ1612 b is likely the third example of an accreting H$α$ protoplanet responsible for carving the gap in its host disk, joining PDS 70b and c. Further study is necessary to confirm and characterize this protoplanet candidate and to identify any additional protoplanets that may also play a role in shaping the gap.

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arXiv2025

JWST/NIRCam Coronagraphic Search for Hidden Planets in the HD~163296 Protoplanetary Disk

HD~163296 is a Herbig Ae/Be star with multiple signposts of on-going planet formation on its disk, such as prominent rings and gaps, as well as kinematic features as identified by previous ALMA observations. We carried out JWST/NIRCam coronagraphic imaging using the F410M and F200W NIRCam filters, with the goal of detecting the emission from the putative young planets in this system. Our F410M observations did not detect the putative planets at the predicted locations of the ALMA velocity kinks, but detected a point-like source candidate at a separation of $\approx0\farcs75$ and a position angle of $\approx231\fdg4$ that is unlikely a background star because of the measured flux in the F410M filter and the detection limit in the F200W filter. These data achieved unprecedented contrast levels at $\sim4~\micron$ at stellocentric separations $ρ\gtrsim0\farcs8$. This allowed us to derive stringent constraints at the outer velocity kink ($Δ{\rm F410M}=15.2~{\rm mag}$) on the mass of the putative planet with or without a circumplanetary disk, and considering different possible initial entropies for the planet.

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arXiv2025

Asymmetric Temperature Variations In Protoplanetary disks: I. Linear Theory, Corotating Spirals, and Ring Formation

Protoplanetary disks can exhibit asymmetric temperature variations due to phenomena such as shadows cast by the inner disk or localized heating by young planets. We investigate the disk features induced by these asymmetric temperature variations. We find that spirals are initially excited, then break into two and reconnect to form rings. By carrying out linear analyses, we first study the spiral launching mechanism, and find that the effects of azimuthal temperature variations share similarities with effects of external potentials. Specifically, rotating temperature variations launch steady spiral structures at Lindblad resonances, which corotate with the temperature patterns. When the cooling time exceeds the orbital period, these spiral structures are significantly weakened, and a checkerboard pattern may appear. A temperature variation of about 10\% can induce spirals with order unity density perturbations, comparable to those generated by a thermal mass planet. We then study ring formation and find it is related to the coupling between azimuthal temperature variations and spirals outside the resonances. Such coupling leads to a radially varying angular momentum flux, which produces anomalous wave-driven accretion and forms dense rings separated by the wavelength of the waves. Finally, we speculate that spirals induced by temperature variations may contribute to disk accretion through non-linear wave steepening and dissipation. Overall, considering that irradiation determines the temperature structure of protoplanetary disks, the change of irradiation both spatially or/and temporarily may produce observable effects in protoplanetary disks, especially spirals and rings in outer disks beyond tens of AU.

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arXiv2024

High Resolution ALMA Observations of Richly Structured Protoplanetary Disks in $σ$ Orionis

ALMA has detected substructures in numerous protoplanetary disks at radii from a few to over a hundred au. These substructures are commonly thought to be associated with planet formation, either by serving as sites fostering planetesimal formation or arising as a consequence of planet-disk interactions. Our current understanding of substructures, though, is primarily based on observations of nearby star-forming regions with mild UV environments, whereas stars are typically born in much harsher UV environments, which may inhibit planet formation in the outer disk through external photoevaporation. We present high resolution ($\sim8$ au) ALMA 1.3 mm continuum images of eight disks in $σ$ Orionis, a cluster irradiated by an O9.5 star. Gaps and rings are resolved in the images of five disks. The most striking of these is SO 1274, which features five gaps that appear to be arranged nearly in a resonant chain. In addition, we infer the presence of gap or shoulder-like structures in the other three disks through visibility modeling. These observations indicate that substructures robustly form and survive at semi-major axes of several tens of au or less in disks exposed to intermediate levels of external UV radiation as well as in compact disks. However, our observations also suggest that disks in $σ$ Orionis are mostly small and thus millimeter continuum gaps beyond a disk radius of 50 au are rare in this region, possibly due to either external photoevaporation or age effects.

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arXiv2024

3D Radiation-Hydrodynamical Simulations of Shadows on Transition Disks

Shadows are often observed in transition disks, which can result from obscuring by materials closer to the star, such as a misaligned inner disk. While shadows leave apparent darkened emission as observational signatures, they have significant dynamical impact on the disk. We carry out 3D radiation hydrodynamical simulations to study shadows in transition disks and find that the temperature drop due to the shadow acts as an asymmetric driving force, leading to spirals in the cavity. These spirals have zero pattern speed following the fixed shadow. The pitch angle is given by tan$^{-1}$($c_s$/$v_ϕ$) (6$^{\circ}$ if $h/r$=0.1). These spirals transport mass through the cavity efficiently, with $α\sim 10^{-2}$ in our simulation. Besides spirals, the cavity edge can also form vortices and flocculent streamers. When present, these features could disturb the shadow-induced spirals. By carrying out Monte Carlo Radiative Transfer simulations, we show that these features resemble those observed in near-infrared scattered light images. In the vertical direction, the vertical gravity is no longer balanced by the pressure gradient alone. Instead, an azimuthal convective acceleration term balances the gravity-pressure difference, leading to azimuthally periodic upward and downward gas motion reaching 10% of the sound speed, which can be probed by ALMA line observations.

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arXiv2024

A Survey of Protoplanetary Disks Using the Keck/NIRC2 Vortex Coronagraph

Recent Atacama Large Millimeter/submillimeter Array (ALMA) observations of protoplanetary disks in the millimeter continuum have shown a variety of radial gaps, cavities, and spiral features. These substructures may be signposts for ongoing planet formation, and therefore these systems are promising targets for direct imaging planet searches in the near-infrared. To this end, we present results from a deep imaging survey in the $L'$-band (3.8 $μ$m) with the Keck/NIRC2 vortex coronagraph to search for young planets in 43 disks with resolved features in the millimeter continuum or evidence for gaps/central cavities from their spectral energy distributions. Although we do not detect any new point sources, using the vortex coronagraph allows for high sensitivity to faint sources at small angular separations (down to ${\sim}$0$^{\prime\prime}$.1), allowing us to place strong upper limits on the masses of potential gas giant planets. We compare our mass sensitivities to the masses of planets derived using ALMA observations, and while we are sensitive to $\sim$1 M$_{Jup}$ planets in the gaps in some of our systems, we are generally not sensitive to planets of the masses expected from the ALMA observations. In addition to placing upper limits on the masses of gas giant planets that could be interacting with the dust in the disks to form the observed millimeter substructures, we are also able to map the micron-sized dust as seen in scattered light for 8 of these systems. Our large sample of systems also allows us to investigate limits on planetary accretion rates and disk viscosities.

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arXiv2024

Thermal Structure Determines Kinematics: Vertical Shear Instability in Stellar Irradiated Protoplanetary Disks

Turbulence is crucial for protoplanetary disk dynamics, and Vertical Shear Instability (VSI) is a promising mechanism in outer disk regions to generate turbulence. We use Athena++ radiation module to study VSI in full and transition disks, accounting for radiation transport and stellar irradiation. We find that the thermal structure and cooling timescale significantly influence VSI behavior. The inner rim location and radial optical depth affect disk kinematics. Compared with previous vertically-isothermal simulations, our full disk and transition disks with small cavities have a superheated atmosphere and cool midplane with long cooling timescales, which suppresses the corrugation mode and the associated meridional circulation. This temperature structure also produces a strong vertical shear at $\mathrm{τ_*}$ = 1, producing an outgoing flow layer at $τ_* < 1$ on top of an ingoing flow layer at $τ_* \sim 1$. The midplane becomes less turbulent, while the surface becomes more turbulent with effective $α$ reaching $\sim10^{-2}$ at $τ_* \lesssim$1. This large surface stress drives significant surface accretion, producing substructures. Using temperature and cooling time measured/estimated from radiation-hydro simulations, we demonstrate that less computationally-intensive simulations incorporating simple orbital cooling can almost reproduce radiation-hydro results. By generating synthetic images, we find that substructures are more pronounced in disks with larger cavities. The higher velocity dispersion at the gap edge could also slow particle settling. Both properties are consistent with recent Near-IR and ALMA observations. Our simulations predict that regions with significant temperature changes are accompanied by significant velocity changes, which can be tested by ALMA kinematics/chemistry observations.

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arXiv2023

Porous Dust Particles in Protoplanetary Disks: Application to the HL Tau Disk

Dust particle sizes constrained from dust continuum and polarization observations by radio interferometry are inconsistent by at least an order of magnitude. Motivated by porous dust observed in small Solar System bodies (e.g., from the Rosetta mission), we explore how the dust particle's porosity affects the estimated particle sizes from these two methods. Porous particles have lower refractive indices, which affect both opacity and polarization fraction. With weaker Mie interference patterns, the porous particles have lower opacity at mm wavelengths than the compact particles if the particle size exceeds several hundred microns. Consequently, the inferred dust mass using porous particles can be up to a factor of six higher. The most significant difference between compact and porous particles is their scattering properties. The porous particles have a wider range of particle sizes with high linear polarization from dust self-scattering, allowing mm-cm-sized particles to explain polarization observations. With a Bayesian approach, we use porous particles to fit HL Tau disk's multi-wavelength continuum and mm-polarization observations from ALMA and VLA. The moderately porous particles with sizes from 1 mm-1 m can explain both continuum and polarization observations, especially in the region between 20-60 au. If the particles in HL Tau are porous, the porosity should be from 70% to 97% from current polarization observations. We also predict that future observations of the self-scattering linear polarization at longer wavelengths (e.g., ALMA B1 and ngVLA) have the potential to further constrain the particle's porosity and size.

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arXiv2023

Substructures in Compact Disks of the Taurus Star-forming Region

Observations of substructure in protoplanetary disks have largely been limited to the brightest and largest disks, excluding the abundant population of compact disks which are likely sites of planet formation. Here, we reanalyze ~0.1'', 1.33 mm ALMA continuum observations of 12 compact protoplanetary disks in the Taurus star-forming region. By fitting visibilities directly, we identify substructures in 6 of the 12 compact disks. We then compare the substructures identified in the full Taurus sample of 24 disks in single star systems and the ALMA DSHARP survey, differentiating between compact (R_eff,90% < 50 au) and extended (R_eff,90% > 50 au) disk sources. We find that substructures are detected at nearly all radii in both small and large disks. Tentatively, we find fewer wide gaps in intermediate-sized disks with R_eff,90% between 30 and 90 au. We perform a series of planet-disk interaction simulations to constrain the sensitivity of our visibility-fitting approach. Under an assumption of planet-disk interaction, we use the gap widths and common disk parameters to calculate potential planet masses within the Taurus sample. We find that the young planet occurrence rate peaks near Neptune masses, similar to the DSHARP sample. For 0.01 $M_J/M_\odot$ $\lesssim$ $M_p/M_*$ $\lesssim$ 0.1 $M_J/M_\odot$, the rate is 17.4$\pm$8.3%; for 0.1 $M_J/M_\odot$ $\lesssim$ $M_p/M_*$ $\lesssim$ 1 $M_J/M_\odot$, it is 27.8$\pm$8.3%. Both of them are consistent with microlensing surveys. For gas giants more massive than 5 $M_J$, the occurrence rate is 4.2$\pm$4.2%, consistent with direct imaging surveys.

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arXiv2022

ALMA Detection of Dust Trapping around Lagrangian Points in the LkCa 15 Disk

We present deep high-resolution ($\sim$50 mas, 8 au) ALMA 0.88 and 1.3 mm continuum observations of the LkCa 15 disk. The emission morphology shows an inner cavity and three dust rings at both wavelengths, but with slightly narrower rings at the longer wavelength. Along a faint ring at 42 au, we identify two excess emission features at $\sim$10$σ$ significance at both wavelengths: one as an unresolved clump and the other as an extended arc, separated by roughly 120 degrees in azimuth. The clump is unlikely to be a circumplanetary disk (CPD) as the emission peak shifts between the two wavelengths even after accounting for orbital motion. Instead, the morphology of the 42 au ring strongly resembles the characteristic horseshoe orbit produced in planet--disk interaction models, where the clump and the arc trace dust accumulation around Lagrangian points $L_{4}$ and $L_{5}$, respectively. The shape of the 42 au ring, dust trapping in the outer adjacent ring, and the coincidence of the horseshoe ring location with a gap in near-IR scattered light, are all consistent with the scenario of planet sculpting, with the planet likely having a mass between those of Neptune and Saturn. We do not detect point-like emission associated with a CPD around the putative planet location ($0.''27$ in projected separation from the central star at a position angle of $\sim$60\degr), with upper limits of 70 and 33 $μ$Jy at 0.88 and 1.3 mm, respectively, corresponding to dust mass upper limits of 0.02--0.03 $M_{\oplus}$.

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arXiv2022

Investigating the Future Potential of an Upgraded ALMA to Image Planet Forming Disks at Sub-au Scales

In recent years, ALMA has been able to observe large-scale substructures within protoplanetary disks. Comparison with the predictions from models of planet-disk interaction has indicated that most of these disk substructures can be explained by the presence of planets with the mass of Neptune or larger at orbital radii of $\approx 5 - 100$ au. Better resolution is needed to observe structures closer to the star, where terrestrial planets are expected to form, as well as structures opened by planets with masses lower than Neptune. We investigate the capabilities of a possible extension to ALMA that would double the longest baseline lengths in the array to detect and resolve disk substructures opened by Earth-mass and Super Earth planets at orbital radii of $1-5$ au. By simulating observations of a family of disk models using this extended configuration in ALMA Bands 6 and 7, we show that an upgraded ALMA would detect gaps in disks formed by super-Earths as close as 1 au, as well as Earth-mass planets down to $2-3$ au from the young host stars in nearby star forming regions.

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arXiv2021

PGNets: Planet mass prediction using convolutional neural networks for radio continuum observations of protoplanetary disks

We developed Convolutional Neural Networks (CNNs) to rapidly and directly infer the planet mass from radio dust continuum images. Substructures induced by young planets in protoplanetary disks can be used to infer the potential young planets' properties. Hydrodynamical simulations have been used to study the relationships between the planet's properties and these disk features. However, these attempts either fine-tuned numerical simulations to fit one protoplanetary disk at a time, which was time-consuming, or azimuthally averaged simulation results to derive some linear relationships between the gap width/depth and the planet mass, which lost information on asymmetric features in disks. To cope with these disadvantages, we developed Planet Gap neural Networks (PGNets) to infer the planet mass from 2D images. We first fit the gridded data in Zhang et al. (2018) as a classification problem. Then, we quadrupled the data set by running additional simulations with near-randomly sampled parameters, and derived the planet mass and disk viscosity together as a regression problem. The classification approach can reach an accuracy of 92\%, whereas the regression approach can reach 1$σ$ as 0.16 dex for planet mass and 0.23 dex for disk viscosity. We can reproduce the degeneracy scaling $α$ $\propto$ $M_p^3$ found in the linear fitting method, which means that the CNN method can even be used to find degeneracy relationship. The gradient-weighted class activation mapping effectively confirms that PGNets use proper disk features to constrain the planet mass. We provide programs for PGNets and the traditional fitting method from Zhang et al. (2018), and discuss each method's advantages and disadvantages.

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arXiv2021

Self-consistent ring model in protoplanetary disks: temperature dips and substructure formation

Rings and gaps are ubiquitous in protoplanetary disks. Larger dust grains will concentrate in gaseous rings more compactly due to stronger aerodynamic drag. However, the effects of dust concentration on the ring's thermal structure have not been explored. Using MCRT simulations, we self-consistently construct ring models by iterating the ring's thermal structure, hydrostatic equilibrium, and dust concentration. We set up rings with two dust populations having different settling and radial concentration due to their different sizes. We find two mechanisms that can lead to temperature dips around the ring. When the disk is optically thick, the temperature drops outside the ring, which is the shadowing effect found in previous works adopting a single-dust population in the disk. When the disk is optically thin, a second mechanism due to excess cooling of big grains is found. Big grains cool more efficiently, which leads to a moderate temperature dip within the ring where big dust resides. This dip is close to the center of the ring. Such temperature dip within the ring can lead to particle pile-up outside the ring and feedback to the dust distribution and thermal structure. We couple the MCRT calculations with a 1D dust evolution model and show that the ring evolves to a different shape and may even separate to several rings. Overall, dust concentration within rings has moderate effects on the disk's thermal structure, and self-consistent model is crucial not only for protoplanetary disk observations but also for planetesimal and planet formation studies.

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arXiv2021

Limits on Millimeter Continuum Emission from Circumplanetary Material in the DSHARP Disks

We present a detailed analysis for a subset of the high resolution (~35 mas, or 5 au) ALMA observations from the Disk Substructures at High Angular Resolution Project (DSHARP) to search for faint 1.3 mm continuum emission associated with dusty circumplanetary material located within the narrow annuli of depleted emission (gaps) in circumstellar disks. This search used the Jennings et al. (2020) $\tt{frank}$ modeling methodology to mitigate contamination from the local disk emission, and then deployed a suite of injection-recovery experiments to statistically characterize point-like circumplanetary disks in residual images. While there are a few putative candidates in this sample, they have only marginal local signal-to-noise ratios and would require deeper measurements to confirm. Associating a 50% recovery fraction with an upper limit, we find these data are sensitive to circumplanetary disks with flux densities $\gtrsim 50-70$ $μ$Jy in most cases. There are a few examples where those limits are inflated ($\gtrsim 110$ $μ$Jy) due to lingering non-axisymmetric structures in their host circumstellar disks, most notably for a newly identified faint spiral in the HD 143006 disk. For standard assumptions, this analysis suggests that these data should be sensitive to circumplanetary disks with dust masses $\gtrsim 0.001-0.2$ M$_\oplus$. While those bounds are comparable to some theoretical expectations for young giant planets, we discuss how plausible system properties (e.g., relatively low host planet masses or the efficient radial drift of solids) could require much deeper observations to achieve robust detections.

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arXiv2021

Impact of Differential Dust Settling on the SED and Polarization: Application to the Inner Region of the HL Tau Disk

The polarimetric observations on the protoplanetary disk around HL Tau have shown the scattering-induced polarization at ALMA Band 7, which indicates that the maximum dust size is $\sim 100~{\rm μm}$, while the Spectral Energy Distribution (SED) has suggested that the maximum dust size is $\sim$ mm. To solve the contradiction, we investigate the impact of differential settling of dust grains on the SED and polarization. If the disk is optically thick, longer observing wavelength traces more interior layer which would be dominated by larger grains. We find that, the SED of the center part of the HL Tau disk can be explained with mm-sized grains for a broad range of turbulence strength, while $160~{\rm μm}$-sized grains can explain barely only if the turbulence strength parameter $α_{\rm t}$ is lower than $10^{-5}$. We also find that the observed polarization fraction can be potentially explained with the maximum dust size of $1~{\rm mm}$ if $α_{\rm t}\lesssim10^{-5}$, although models with $160~{\rm μm}$-sized grains are also acceptable. However, if the maximum dust size is $\sim3~{\rm mm}$, the simulated polarization fraction is too low to explain the observations even if the turbulence strength is extremely small, indicating the maximum dust size of $\lesssim1$ mm. The degeneracy between 100 ${\rm μm}$-sized and mm-sized grains can be solved by improving the ALMA calibration accuracy or polarimetric observations at (sub-)cm wavelengths.

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