arXiv subjects

Takayuki Muto

Publications and source records attributed to Takayuki Muto.

At least 19 recordsLinked to original sources

arXiv2026

Probing disk dynamics and dust evolution through shadows in protoplanetary disks: A case study of the HD 142527 disk

Planet formation begins with dust growth and planetesimal formation within protoplanetary disks surrounding young stars. To understand these processes, it is essential to estimate dust grain sizes from disk observations. In this study, we develop a new method to constrain grain size based on the estimation of cooling timescales. Our approach applies to transitional disks that possess an inclined inner disk casting shadows on the outer disk, whose temperature variations serve as a tracer of dust properties. By constructing a three-dimensional model of the disk surface using near-infrared scattering light images and comparing it with submillimeter dust continuum maps, we estimate the spatial offset between the irradiated and shadowed regions to derive the cooling timescale. We then build an analytic model that calculates the cooling timescale at the dust thermal emission height with an assumed turbulent diffusion intensity to infer the dust surface density and dust grain size. Applying this method to the protoplanetary disk around HD~142527, we find that the disk's northern shadowed region cools on a timescale of a few percent of the orbital period and that the maximum grain size consistent with the observations is approximately 0.1-1 mm. We also find that the conditions required for the vertical shear instability, which needs a short cooling timescale, are satisfied, allowing turbulence with an intensity consistent with near-infrared observations. This study demonstrates that estimating cooling timescales is an effective tool for constraining dust grain size. Our approach can be generally applied to other transition disks with inner-disk-induced shadows.

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arXiv2026

ALMA 2D super-resolution imaging survey of Ophiuchus Class I/flat spectrum/II disks. II. Statistical analysis of stellar and disk properties

We present a statistical study of stellar and dust disk properties for young stellar objects in the Ophiuchus star-forming region. Building on our previous paper (Shoshi et al. 2025b), which applied two-dimensional super-resolution imaging with PRIISM to ALMA archival Band 6 continuum data and spatially resolved 78 disks, we analyze a sample of 67 systems with robust dust-radius measurements. We combine stellar parameters from the literature, including bolometric temperature $T_{\rm bol}$, stellar mass $M_\ast$, and mass accretion rate $\dot{M}_{\rm acc}$, with disk parameters derived from the super-resolution images, including inclination $i_{\rm disk}$, millimeter luminosity $L_{\rm mm}$, and dust radius $R_{95\%}$. We quantify pairwise correlations and compare their behavior across evolutionary stages (Class I/FS and Class II) and between disks with and without detectable substructures. We identify substructure dependencies in $L_{\rm mm}$ and $R_{95\%}$, indicating that substructures tend to be found preferentially in relatively massive and extended disks. Moreover, we find a tight size-luminosity relation between $R_{95\%}$ and $L_{\rm mm}$. In particular, only Class II disks with substructures exhibit a steeper scaling, $R_{95\%}\propto L_{\rm mm}^{0.8}$, while the other subsamples are broadly consistent with $R_{95\%}\propto L_{\rm mm}^{0.4\text{-}0.5}$. This behavior is qualitatively consistent with disk evolution models in which disks with planet-induced pressure bumps follow a steeper size-luminosity relation than smooth disks. Overall, our results suggest that disk substructures play an important role in shaping the evolution of dust and global disk properties, while providing empirical constraints on accretion, dust trapping, and possible gravitational instability in young disks.

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arXiv2026

A Hybrid Origin for the Multiple Ring-Gap Structures in the Large Protoplanetary Disk V1094 Sco: A Low-Mass Planet and Secular Gravitational Instability

High spatial resolution observations reveal that some protoplanetary disks host multiple ring-gap pairs at large stellocentric radii, yet their physical origin remains unsettled. We present a multi-wavelength analysis of the V1094~Sco disk using Atacama Large Millimeter/submillimeter Array Band~6 continuum and $^{12}$CO and $^{13}$CO $J=2-1$ emission, together with a Very Large Telescope/SPHERE near-infrared scattered light image. The continuum image shows four narrow dust ring-gap pairs extending to exceptionally large radii ($r \sim 380$ au), while the CO isotopologues trace a spatially extended gas disk ($r \sim 760$ au) in Keplerian rotation. From the dust ring widths, we place conservative upper limits on the turbulent viscosity parameter, $\alpha \lesssim 10^{-3}$ and potentially $\lesssim 10^{-4}$, implying weak turbulence. The ensemble of gap widths and depths is inconsistent with a simple one-planet-per-gap interpretation. At $r \simeq 100$~au, a double gap and its scattered light counterpart are consistent with multi-gap excitation by a single low-mass companion of $(55 \pm 35)\,M_{\oplus}$. At $r \simeq 170$-$230$~au, the outer ring system shows regular spacing and no clear scattered light counterpart, indicating mechanisms that operate primarily at the disk midplane. These outer rings are quantitatively compatible with secular gravitational instability. V1094~Sco therefore supports a hybrid pathway in which weak turbulence in an extended disk allows secular gravitational instability to assemble long-lived midplane dust concentrations that can cradle planet formation beyond $\sim100$~au, alongside planet-driven substructures at intermediate radii.

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arXiv2026

Bridging the gap: consistent modeling of protoplanetary disk heating and gap formation by planet-induced spiral shocks

A giant planet embedded in a protoplanetary disk excites spiral density waves, which steepen into shocks as they propagate away from the planet. These shocks lead to secular disk heating and gap opening, both of which can have important implications for the evolution of solids near the planet. To date, these two effects have largely been modeled independently. In this study, we present a self-consistent model that unifies these processes by linking shock heating and angular momentum deposition through the entropy jumps across the spiral shocks. We show that this model accurately reproduces the temperature and surface density profiles around the planet's orbit, as obtained from two-dimensional hydrodynamic simulations with standard $α$ viscosity and $β$ thermal relaxation prescriptions. Furthermore, by incorporating an empirically derived scaling law for the radial distribution of the entropy jump, we construct a fully analytic model that self-consistently predicts the temperature and surface density structures of disks hosting a giant planet. This work represents a first step toward understanding how a giant planet forming in the inner disk region influences the distribution and composition of second-generation planets and planetesimals in its vicinity.

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arXiv2025

Axisymmetric Modeling of DSHARP Dusty Disks: Asymmetric Structures and Inner-Disk Dispersal

High-resolution observations of Class II protoplanetary disks frequently reveal annular structures that may indicate the presence of embedded planets. In this study, we model brightness profiles and geometries for 16 dusty disks in the DSHARP observations to identify asymmetric substructures, including possible planet-induced signatures. We find no compelling evidence for circumplanetary emission in these systems. We identify a possible one-armed spiral in Elias 27; while previous studies report an $m=2$ spiral, the morphology of the newly identified spiral agrees with a spiral for a possible protoplanet. Although non-detection of circumplanetary emission is consistent with low expected luminosities, the absence of additional dust spirals except for Elias 27 may constrain properties of potential embedded planets given their theoretical detectability. The analyses further suggest that spiral amplitudes and phases correlated with gap and ring locations in WaOph 6 and IM Lup, appearing as deflections of spirals in images. Five disks exhibit strong residual asymmetries attributable to the vertical extent of their dust layers. We find that the brightness temperature of inner disks ($1$-$5$ au) declines with stellar age on a $\sim 3$ Myr timescale, while the total flux shows no clear decreasing trend. This trend is consistent with the presence of pressure bumps that retain large dust grains at outer radii, while allowing the inner disks to disperse. The universality of disk dispersal timescale at millimeter wavelengths, observed in both the extended DSHARP disks and the compact disks from previous demographic surveys, may constrain timing of planet formation, including habitable planets.

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arXiv2025

ALMA High-resolution Observation for the Transitional Disk around IRAS 04125+2902

Recently, the youngest transiting planet was discovered around the T Tauri star, IRAS 04125+2902, in the Taurus-Auriga star-forming region. This system is crucial for understanding the early stages of planet formation. We used Atacama Large Millimeter/submillimeter Array Band 6 data to investigate the IRAS 04125+2902 system in detail. The dust continuum emission reveals a ring-gap transitional disk structure with an inclination of 35.6$^{\circ}$. In addition, two-dimensional super-resolution imaging based on Sparse Modeling and the one-dimensional modeling of disk brightness distribution suggest the existence of an inner emission, which may be attributed to an inner disk, although free-free emission from the central star is not ruled out. Furthermore, we identified the $^{12}$CO $J$=2-1 emission, and the dynamical mass of the central star is estimated to be 0.7-1.0 $M_{\odot}$. The asymmetry of the dust ring and the velocity distortion around the central star are, if at all, weak, suggesting that the inner disk, if it exists, is not highly inclined with respect to the outer disk. Radiative transfer calculations of dust continuum emission suggest that the inner and the outer disk may be misaligned by $\sim$10$^\circ$, which may be confirmed in future observations with higher resolution and sensitivity. Our results suggest that IRAS 04125+2902 is a dynamically complex system, where the binary orbit, outer disk, inner disk, and planetary orbit are mutually misaligned, providing insight into the early orbital evolution of young systems.

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arXiv2025

VLT/MUSE Detection of the AB Aurigae b Protoplanet with $H _{\rm α}$ Spectroscopy

We analyze high-contrast, medium-spectral-resolution $H_{\rm α}$ observations of the star AB Aurigae using the Very Large Telescope's Multi Unit Spectroscopic Explorer (MUSE). In multiple epochs, MUSE detects the AB Aur b protoplanet discovered from Subaru/SCExAO data in emission at wavelengths slightly blue-shifted from the $H_{\rm α}$ line center (i.e. at 6558.88--6560.13 Å; $\sim$ -100 km s$^{-1}$) and in absorption at redshifted wavelengths (6562.8--6565.1 Å; $\sim$ 75 km s$^{-1}$). AB Aur b's $H_{\rm α}$ spectrum is inconsistent with that of the host star or the average residual disk spectrum and is dissimilar to that of PDS 70 b and c. Instead, the spectrum's shape resembles that of an inverse P Cygni profile seen in some accreting T Tauri stars and interpreted as evidence of infalling cold gas from accretion, although we cannot formally rule out all other nonaccretion origins for AB Aur b's MUSE detection. AB Aurigae hosts only the second protoplanetary system detected in $H_{\rm α}$ thus far and the first with a source showing a spectrum resembling an inverse P Cygni profile. Future modeling and new optical data will be needed to assess how much of AB Aur b's emission source(s) originates from protoplanet accretion reprocessed by the disk, a localized scattered-light feature with a unique $H_{\rm α}$ profile, or another mechanism.

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arXiv2025

JVLA Measurement of Grain Size in the Compact Dust Ring around Class I Protostar WL 17

The maximum grain size in protoplanetary disks is a critical parameter for planet formation, as the efficiency of mechanisms like streaming instability and pebble accretion depend on grain size. Even young class 0/I objects, such as HL Tau, show substructures in their disks, indicating the potential for early planet formation. In this study, we investigated the grain size in the dust surrounding the class I object WL 17 using the Karl G. Jansky Very Large Array. Observations were conducted across seven frequency bands (Q, Ka, K, Ku, X, C, and S bands) ranging from 2 to 48 GHz, corresponding to wavelengths of 15 cm to 6.3 mm, with a spatial resolution exceeding 0\farcs5. While the ring structure at 0\farcs1 of WL 17 remains unresolved in our data, its emission is clearly detected at all observed frequencies, except at 2 GHz. To estimate the maximum grain size ($a_{\rm max}$) within the ring, we compared the observed spectral energy distribution (SED) with theoretical SEDs calculated for various $a_{\rm max}$ values using radiative transfer models. Assuming the dust opacity follows the DSHARP model, our analysis suggests that certain structures internal to the ring achieved a maximum grain size of approximately 4.2 mm. Additionally, we discuss the gravitational stability of the ring and the potential planetary core mass that could form through pebble accretion within the structure.

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arXiv2025

ALMA 2D Super-resolution Imaging Survey of Ophiuchus Class I/Flat Spectrum/II Disks -- I: Discovery of New Disk Substructures

This study focuses on Class I, Flat Spectrum (FS), and Class II disks in the Ophiuchus molecular cloud, a nearby active star-forming region with numerous young stellar objects (YSOs), to unveil signs of substructure formation in these disks. We employ two-dimensional super-resolution imaging based on Sparse Modeling (SpM) for ALMA archival Band 6 continuum data, achieving images with spatial resolutions comparable to a few au (0".02-0".2) for 78 dust disks, all of which are spatially resolved. In our sample, we confirm that approximately 30-40% of the disks exhibit substructures, and we identify new substructures in 15 disks (4 Class I, 7 Class FS, and 4 Class II objects). Compared to the eDisk sample in terms of bolometric temperature, Tbol, our targets are in a relatively later accretion phase. By combining our targets with the eDisk sample, we confirm that substructure detection in available data is restricted to objects where Tbol exceeds 200-300 K and the dust disk radius, Rdust, is larger than ~30 au. Moreover, we find that the distribution of inclination angles for Class II disks has a deficit of high values and is not consistent with being random. Analyzing molecular line emission data around these objects will be crucial to constrain disk evolutionary stages further and understand when and how substructures form.

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arXiv2025

Observationally derived magnetic field strength and 3D components in the HD 142527 disk

In protoplanetary disks around young stars, magnetic fields play an important role for disk evolution and planet formation. Polarized thermal emission from magnetically aligned grains is one of the reliable methods to trace magnetic fields. However, it has been difficult to observe magnetic fields from dust polarization in protoplanetary disks because other polarization mechanisms involving grown dust grains become efficient. Here, we report multi-wavelength (0.87 mm, 1.3 mm, 2.1 mm, and 2.7 mm) observations of polarized thermal emission in the protoplanetary disk around HD 142527, showing the lopsided dust distribution. We revealed that the smaller dust still exhibits magnetic alignment in the southern part of the disk. Furthermore, angular offsets between the observed magnetic field and the disk azimuthal direction were discovered, which can be used as a method to measure the relative strengths of each component (radial ($B_r$), azimuthal ($B_ϕ$), and vertical ($B_z$)) of 3D magnetic field. Applying this method, we derived the magnetic field around a 200-au radius from the protostar as $|B_r |:|B_ϕ|:|B_z | \sim 0.26:1:0.23$ and a strength of $\sim 0.3$ milli-Gauss. Our observations provide some key parameters of magnetic activities including the plasma beta, which have only been assumed in theoretical studies. In addition, the radial and vertical angular momentum transfer are found to be comparable, which poses a challenge to theoretical studies of protoplanetary disks.

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arXiv2025

Revealing asymmetry on midplane of proto-planetary disc through modelling of axisymmetric emission: methodology

This study proposes an analytical framework for deriving the surface brightness profile and geometry of a geometrically-thin axisymmetric disc from interferometric observation of continuum emission. Such precise modelling facilitates the exploration of faint non-axisymmetric structures, such as spirals and circumplanetary discs. As a demonstration, we simulate interferometric observations of geometrically-thin axisymmetric discs. The proposed method can reasonably recover the injected axisymmetric structures, whereas Gaussian fitting of the same data yielded larger errors in disc orientation estimation. To further test the applicability of the method, it was applied to the mock data for $m=1,2$ spirals and a point source, which are embedded in a bright axisymmetric structure. The injected non-axisymmetric structures were reasonably recovered except for the innermost parts, and the disc geometric parameter estimations were better than Gasussian fitting. The method was then applied to the real data of Elias 20 and AS 209, and it adequately subtracted the axisymmetric component, notably in Elias 20, where substantial residuals remained without our method. We also applied our method to continuum data of PDS 70 to demonstrate the effectiveness of the method. We successfully recovered emission from PDS 70 c consistently with previous studies, and also tentatively discovered new substructures. The current formulation can be applied to any data for disc continuum emission, and aids in the search of spirals and circumplanetary discs, whose detection is still limited.

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arXiv2024

Predictions of Dust Continuum Emission from a Potential Circumplanetary Disk: A Case Study of the Planet Candidate AB Aurigae b

Gas accreting planets embedded in protoplanetary disks are expected to show dust thermal emission from their circumplanetary disks (CPDs). However, a recently reported gas accreting planet candidate, AB Aurigae b, has not been detected in (sub)millimeter continuum observations. We calculate the evolution of dust in the potential CPD of AB Aurigae b and predict its thermal emission at 1.3 mm wavelength as a case study, where the obtained features may also be applied to other gas accreting planets. We find that the expected flux density from the CPD is lower than the 3-sigma level of the previous continuum observation by ALMA with broad ranges of parameters, consistent with the non-detection. However, the expected planet mass and gas accretion rate are higher if the reduction of the observed near-infrared continuum and H-alpha line emission due to the extinction by small grains is considered, resulting in higher flux density of the dust emission from the CPD at (sub)millimeter wavelength. We find that the corrected predictions of the dust emission are stronger than the 3-sigma level of the previous observation with the typical dust-to-gas mass ratio of the inflow to the CPD. This result suggests that the dust supply to the vicinity of AB Aurigae b is small if the planet candidate is not the scattered light of the star but is a planet and has a CPD. Future continuum observations at shorter wavelength are preferable to obtain more robust clues to the question whether the candidate is a planet or not.

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arXiv2024

Modeling protoplanetary disk heating by planet-induced spiral shocks

We investigate the heating of protoplanetary disks caused by shocks associated with spiral density waves induced by an embedded planet. Using two-dimensional hydrodynamical simulations, we explore the dependence of shock heating rates on various disk and planetary parameters. Our results show that the shock heating rates are primarily influenced by the planet's mass and the disk's viscosity, while being insensitive to the thermal relaxation rate and the radial profiles of the disk's surface density and sound speed. We provide universal empirical formulas for the shock heating rates produced by the primary and secondary spiral arms as a function of orbital radius, viscosity parameter $α$, and planet-to-star mass ratio $q$. The obtained formulas are accurate within a factor of a few for a moderately viscous and adiabatic disk with a planet massive enough that its spiral arms are strongly nonlinear. Using these universal relations, we show that shock heating can overwhelm viscous heating when the disk viscosity is low ($α\lesssim 10^{-3}$) and the planet is massive ($q \gtrsim 10^{-3}$). Our empirical relations for the shock heating rates are simple and can be easily implemented into radially one-dimensional models of gas and dust evolution in protoplanetary disks.

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arXiv2024

First JVLA Radio Observation on PDS70

PDS~70 is a protoplanetary system that hosts two actively accreting gas giants, namely PDS~70b and PDS~70c. The system has a $\sim$60--100 au dusty ring that has been resolved by ALMA, along with circumplanetary disks around the two gas giants. Here we report the first JVLA Q (40--48 GHz), Ka (29--37 GHz), K (18--26 GHz), and X (8--12 GHz) bands continuum observations, and the complementary ALMA Bands 3 ($\sim$98 GHz) and 4 ($\sim$145 GHz) observations towards PDS~70. The dusty ring appears azimuthally asymmetric in our ALMA images. We obtained firm detections at Ka and K bands without spatially resolving the source; we obtained a marginal detection at Q band, and no detection at X band. The spectral indices ($α$) are 5$\pm$1 at 33--44 GHz and 0.6$\pm$0.2 at 22--33 GHz. At 10--22 GHz, the conservative lower limit of $α$ is 1.7. The 33--44 GHz flux density is likely dominated by the optically thin thermal emission of grown dust with $\gtrsim$1 mm maximum grain sizes, which may be associated with the azimuthally asymmetric substructure induced by planet-disk interaction. Since PDS~70 was not detected at X band, we found it hard to explain the low spectral index at 22--33 GHz only with free-free emission. Hence, we attribute the dominant emission at 22--33 GHz to the emission of spinning nanometer-sized dust particles, while free-free emission may partly contribute to emission at this frequency range. In some protoplanetary disks, the emission of spinning nanometer-sized dust particles may resemble the 20--50 GHz excess in the spectra of millimeter-sized dust. The finding of strong continuum emission of spinning nanometer-sized particles can complicate the procedure of constraining the properties of grown dust. Future high-resolution, multi-frequency JVLA/ngVLA and SKA observations may shed light on this issue.

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arXiv2024

ALMA 2D Super-resolution Imaging of Taurus-Auriga Protoplanetary Disks: Probing Statistical Properties of Disk Substructures

In the past decade, ALMA observations of protoplanetary disks revealed various substructures including gaps and rings. Their origin may be probed through statistical studies on the physical properties of the substructures. We present the analyses of archival ALMA Band 6 continuum data of 43 disks (39 Class II and 4 Herbig Ae) in the Taurus-Auriga region. We employ a novel 2D super-resolution imaging technique based on sparse modeling to obtain images with high fidelity and spatial resolution. As a result, we have obtained images with spatial resolutions comparable to a few au ($0''.02 - 0''.1$), which is two to three times better than conventional CLEAN methods. All dust disks are spatially resolved, with the radii ranging from 8 to 238 au with a median radius of 45 au. Half of the disks harbor clear gap structures, whose radial locations show a bimodal distribution with peaks at $\lesssim20$ au and $\gtrsim30$ au. We also see structures indicating weak gaps at all the radii in the disk. We find that the widths of these gaps increase with their depths, which is consistent with the model of planet-disk interactions. The inferred planet mass-orbital radius distribution indicates that the planet distribution is analogous to our Solar System. However, planets with Neptune mass or lower may exist in all the radii.

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arXiv2024

Forming localized dust concentrations in a dust ring: DM Tau case study

The previous, high angular resolution 225 GHz ($\sim$1.3 mm) continuum observations on the transitional disk DM Tau have resolved an outer ring at 20-120 au radii that is weakly azimuthally asymmetric. We aimed to examine dust growth and filtration in the outer ring. We performed the $\sim$0$''$.06 ($\sim$8.7 au) resolution Karl G. Jansky Very Large Array (JVLA) 40-48 GHz ($\sim$7 mm; Q band) continuum observations and the complementary observations at lower frequencies. In addition, we analyzed the archival JVLA observations that were taken since 2010. Intriguingly, the Q band image resolved the azimuthally highly asymmetric, knotty dust emission sources close to the inner edge of the outer ring. Fitting the 8-700 GHz spectral energy distribution (SED) with two dust components indicates that the maximum grain size in these knotty dust emission sources is likely $\gtrsim$300 $μ$m while it is $\lesssim$50 $μ$m in the rest of the ring. These results may be explained by trapping of inward migrating grown dust close to the ring inner edge. The exact mechanism for developing the azimuthal asymmetry has not yet been identified, which may be due to planet-disk interaction that might also be responsible for the creation of the dust cavity and pressure bump, or the fluid instabilities and vortex formation due to shear motions. Finally, we remark that the asymmetries in DM Tau are hard to diagnose from the $\gtrsim$225 GHz observations owing to a high optical depth at the ring. In other words, the apparent symmetric or asymmetric morphology of the transitional disks may be related to the optical depths of those disks at the observing frequency.

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arXiv2024

Shadowing in the protoplanetary disk of ZZ Tau IRS with HST

An inner component misaligned from an outer component in a protoplanetary disk can result in the former casting shadows on the latter. We present a new instance of shadowing on the outer disk around a very low mass star, ZZ~Tau~IRS. Through the analysis of near-infrared (NIR) archival data at $λ=1.6$~$μ$m acquired with the Wide Field Camera 3 on the Hubble Space Telescope, we identified brightness asymmetries in the top and bottom halves of the highly inclined outer disk, separated by a dark lane. The brighter sides in the top and bottom halves are on the opposite sides, which we attributed to shadows cast by a misaligned inner disk. Radiative transfer modeling of the system with a misaligned angle of 15~deg between the inner and outer disks well reproduced the observations. Additionally, we found an elevated brightness temperature of $^{12}$CO~(3-2) at $r\sim30$~au on the brighter side in NIR wavelengths in the top half disk, which can be explained by the shadowing effect too. While the origin of the misaligned inner disk remains unclear, future monitoring observations to search for temporal variations in brightness asymmetries will likely provide useful clues.

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arXiv2023

Panchromatic (Sub)millimeter Polarization Observations of HL Tau Unveil Aligned Scattering Grains

Polarization is a unique tool to study the properties of dust grains of protoplanetary disks and detail the initial conditions of planet formation. Polarization around HL Tau was previously imaged using the Atacama Large Millimeter/submillimeter Array (ALMA) at Bands 3 (3.1 mm), 6 (1.3 mm), and 7 (0.87 mm), showing that the polarization orientation changes across wavelength $λ$. The polarization morphology at Band 7 is predominantly parallel to the disk minor axis but appears azimuthally oriented at Band 3, with the morphology at Band 6 in between the two. We present new ~0.2" (29 au) polarization observations at Q-Band (7.0 mm) using the Karl G. Jansky Very Large Array (VLA) and at Bands 4 (2.1 mm), 5 (1.5 mm), and 7 using ALMA, consolidating HL Tau's position as the protoplanetary disk with the most complete wavelength coverage in dust polarization. The polarization patterns at Bands 4 and 5 continue to follow the morphological transition with wavelength previously identified in Bands 3, 6, and 7. Based on the azimuthal variation, we decompose the polarization into contributions from scattering ($s$) and thermal emission ($t$). We find that $s$ decreases slowly with increasing $λ$, and $t$ increases more rapidly with $λ$ which are expected from optical depth effects of toroidally aligned, scattering prolate grains. The relatively weak $λ$ dependence of $s$ is consistent with large, porous grains. The sparse polarization detections from the Q-band image are also consistent with toroidally aligned prolate grains.

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