11institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany
11email<EMAIL_ADDRESS>22institutetext: National Astronomical Observatory of
Japan, Osawa 2-21-1, Mitaka, Tokyo 181-8588, Japan 33institutetext: Department
of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo,
152-8551, Japan

# Probing the Temperature Structure of the Inner Region of a Protoplanetary
Disk

Takahiro Ueda Satoshi Okuzumi Akimasa Kataoka 11223322 Mario Flock 11

(Received XXX; accepted YYY)

###### Abstract

Context. Disk temperature structure is crucial for the formation of planets.
Midplane heating induced by disk accretion plays a key role in determining the
disk temperature particularly at the inner disk midplane where planets form.
However, the efficiency of accretion heating has been not well constrained by
observations.

Aims. Our aim is to observationally constrain the physical properties of the
inner region of the CW Tau disk where the midplane heating potentially takes
place.

Methods. We construct two-dimensional physical models of the CW Tau disk,
taking into account the midplane heating. The models are compared with the
ALMA dust continuum observations at Bands 4, 6, 7 and 8, with an angular
resolution of 0$\aas@@fstack{\prime\prime}$1\. The observed brightness
temperatures are almost wavelength-indenpendent at $\lesssim$10 au.

Results. We find that if the maximum dust size $a_{\rm max}$ is $\lesssim
100\leavevmode\nobreak\ {\rm\mu m}$, the brightness temperatures predicted by
the model exceed the observed values, regardless of the efficiency of
accretion heating. The low observed brightness temperatures can be explained
if millimeter scattering reduces the intensity. If the disk is passive,
$a_{\rm max}$ needs to be either $\sim 150\leavevmode\nobreak\ {\rm\mu m}$ or
$\gtrsim$ few ${\rm cm}$. The accretion heating significantly increases the
brightness temperature particularly when $a_{\rm max}\lesssim
300\leavevmode\nobreak\ {\rm\mu m}$, and hence $a_{\rm max}$ needs to be
either $\sim 300\leavevmode\nobreak\ {\rm\mu m}$ or $\gtrsim$ few ${\rm cm}$.
The midplane temperature is expected to be $\sim$1.5–3 times higher than the
observed brightness temperatures, depending on the models. The dust settling
effectively increases the temperature of the dust responsible for the
millimeter emission in the active disk, which makes the model with
$300\leavevmode\nobreak\ {\rm\mu m}$-sized dust overpredicts the brightness
temperatures when strong turbulence is absent. Porous dust (porosity of 0.9)
makes the accretion heating more efficient so that some sort of reduction in
accretion heating is required.

Conclusions. The brightness temperature is not a simple function of the dust
temperature because of the effect of scattering and midplane heating even if
the disk is optically thick. The current data of the CW Tau disk are not
enough to discriminate between the passive and active disk models. Future
longer wavelength and higher angular resolution observations will help us
constrain the heating mechanisms of the inner protoplanetary disks.

###### Key Words.:

protoplanetary disks — accretion, accretion disks — planets and satellites:
formation

##  1 Introduction

The temperature structure of inner protoplanetary disks controls the evolution
of dust grains (e.g., Birnstiel et al. 2010; Okuzumi et al. 2016; Dra̧żkowska
& Alibert 2017; Ueda et al. 2021b), subsequent formation of planets and their
migration (e.g., Kley & Nelson 2012; Bitsch et al. 2014; Bitsch 2019; Savvidou
& Bitsch 2021) and eventually the chemical composition of planets (e.g., Öberg
et al. 2011; Madhusudhan et al. 2014; Ohno & Ueda 2021; Schneider & Bitsch
2021; Notsu et al. 2022). Particularly at the midplane of the inner regions of
disks ($\lesssim 10$ au), the heating resulting from disk gas accretion is
thought to play a key role in determining the disk temperature. In the
classical model of accretion heating, the gravitational energy of the disk gas
is released near the midplane where the energy is hard to escape from the
disk, leading to efficient heating of the disk midplane (e.g., Hubeny 1990;
Nakamoto & Nakagawa 1994). However, recent non-ideal magneto-hydrodynamical
models have shown that accretion heating may be less efficient compared to the
classical model when the accretion is primarily driven by the
magnetorotational instability (MRI) in the upper layer of the disk (Hirose &
Turner 2011) or by magneto-hydrodynamical disk winds (Mori et al. 2019). This
is because the gravitational energy released at the upper layer can escape
from the disk more easily. Still, accretion heating in magneto-
hydrodynamically accreting disks can affect the disk temperature structure
depending on the disk ionization state and opacity (Béthune & Latter 2020;
Kondo et al. 2022).

Even though how the disk midplane is heated is crucial for the formation of
planets, it is poorly constrained by observations. The Atacama Large
Millimeter/submillimeter Array (ALMA) provides an opportunity to probe the
temperature structure of the inner regions of disks, where accretion heating
potentially takes place. The ALMA high-resolution multi-wavelength analysis
allows us to evaluate the dust properties (e.g., dust size, temperature and
surface density) as a function of radial distance based on the spectral
behavior of the (sub-)millimeter dust thermal emission (Carrasco-González et
al. 2019; Macías et al. 2021; Sierra et al. 2021; Ueda et al. 2022; Guidi et
al. 2022). However, previous studies have generally assumed that the observed
dust temperatures at different ALMA wavelengths are identical (i.e.,
vertically isothermal). If the disk is optically thick at the observing
wavelengths, the different observing wavelengths trace different heights
within the disk, which may alter the interpretation of the disk properties
(Sierra & Lizano 2020; Ueda et al. 2021a).

In other words, the vertical disk structure can be probed by the multi-
wavelength sub-millimeter to centimeter observations by leveraging the
difference in optical depth at each wavelength. For instance, recent ALMA
observations of CO isotopologue line emissions have provided insights into the
vertical structure of the outer regions of disks by utilizing the expected
emission heights of different CO isotopologues (Law et al. 2021, 2023). For
the inner regions of disks, dust continuum observations would be more suitable
than the gas molecule observations as it is easier to achieve better angular
resolution and sensitivity with a reasonable observing time. Okuzumi et al.
(2023) demonstrated that the vertical temperature structure of the inner few
au of disks can be inferred from the multi-wavelength dust continuum
observations using ALMA and the next generation Very Large Array (ngVLA).

In this paper, we investigate the temperature structure of the disk around CW
Tau using ALMA multi-wavelength dust continuum observations. The CW Tau disk
has a high accretion rate ($3$–$10\times 10^{-8}M_{\odot}\leavevmode\nobreak\
{\rm yr^{-1}}$; McClure 2019; Robinson et al. 2022; Gangi et al. 2022) and has
been observed at ALMA Bands 4, 6, 7 and 8 (Ueda et al. 2022). This combination
of a high accretion rate and rich observational data makes the CW Tau disk an
excellent target for studying the vertical temperature structure of the inner
region potentially heated by disk accretion. This paper is constructed as
follows. In Section 2, we introduce our observational data and theoretical
models. The models are compared with the observations in Section 3. Discussion
and summary are in Section 4 and 5, respectively.

##  2 Methods

###  2.1 ALMA observations

We make use of the ALMA data of the CW Tau disk taken and calibrated in Ueda
et al. (2022). The observations were carried out at ALMA Bands 4
($\lambda=2.17\leavevmode\nobreak\ {\rm mm}$), 6 ($1.34\leavevmode\nobreak\
{\rm mm}$), 7 ($0.89\leavevmode\nobreak\ {\rm mm}$) and 8
($0.75\leavevmode\nobreak\ {\rm mm}$). The CW Tau disk is a Class II disk with
the stellar luminosity of $L_{*}=0.45L_{\odot}$ (Herczeg & Hillenbrand 2014),
accretion luminosity of $L_{\rm acc}=0.85L_{\odot}$ (Gangi et al. 2022) and
accretion rate of $\dot{M}=3$–$10\times 10^{-8}M_{\odot}\leavevmode\nobreak\
{\rm yr^{-1}}$ (McClure 2019; Robinson et al. 2022; Gangi et al. 2022) which
is much higher than the typical value of the Class II disks (e.g., $\sim
3.6\times 10^{-9}M_{\odot}\leavevmode\nobreak\ {\rm yr}^{-1}$; Manara et al.
2017 see also Manara et al. 2022). Its high accretion rate and relatively low
stellar luminosity makes the CW Tau disk suitable for studying accretion
heating of the disk. In this work, we adopt $\dot{M}=4\times
10^{-8}M_{\odot}\leavevmode\nobreak\ {\rm yr}^{-1}$, which falls within the
range of observed values for CW Tau.

Figure 1:  Brightness temperature profile observed at ALMA Bands 4, 6, 7 and 8
obtained by Ueda et al. (2022). The transparent region denotes the uncertainty
arising from both flux calibration and thermal noise.

Figure 1 presents the observed brightness temperature at ALMA Bands 4, 6, 7
and 8. The observational data has a common angular resolution of
0$\aas@@fstack{\prime\prime}$1, corresponding to a spatial resolution of 13.2
au for the distance of CW Tau (132 pc; Gaia Collaboration et al. 2021). The
brightness temperature at ALMA Bands 4, 6, 7, and 8 shows a nearly identical
value of $\sim 35$ K at $\lesssim 10$ au. At the outer region, $\gtrsim 20$
au, the brightness temperature exhibits variations among the different ALMA
bands. In particular, the brightness temperature at Band 4 is lower compared
to the other bands.

###  2.2 Theoretical model

We construct two-dimensional (radial position $r$ and vertical height $z$)
models of the CW Tau disk in order to compare them with the ALMA observations.
In this section, we provide a detailed description of our theoretical models
of the disk structure.

####  2.2.1 Temperature structure

The temperature structure of the disk is determined by the combined effects of
stellar irradiation and accretion heating:

$\displaystyle T^{4}=T_{\rm irr}^{4}+T_{\rm acc}^{4},$ (1)

where $T_{\rm irr}$ and $T_{\rm acc}$ denote the disk temperature determined
by stellar irradiation and accretion heating, respectively. The temperature
structure of the passively irradiated disk is calculated as

$\displaystyle T_{\rm irr}^{4}=\frac{L_{*}+L_{\rm acc}}{16\pi\sigma
r^{2}}\left\\{\exp{\left(-\tau_{*}\right)}+2\psi\right\\},$ (2)

where $\sigma$ is the Stefan-Boltzmann constant, $\tau_{*}$ is the radial
optical depth for stellar irradiation. The angle $\psi$ represents the angle
between the ray from the central star and the disk surface. Its actual value
depends on the distance from the central star and the detailed disk surface
structure. We assume $\psi=0.02$ which corresponds to $h_{*}/r\sim 0.07$ with
$h_{*}$ being the height of the disk photosphere above the midplane (Dullemond
et al. 2001).

Equation (2) provides the black body temperature in the optically thin regime
when $\tau_{*}\ll 1$, whereas it yields temperature profile of the classical
optically thick passive disk in the limit of $\tau_{*}\gg 1$ (e.g., Dullemond
et al. 2001; see also Huang et al. 2018). In the optically thick interior
region ($\tau_{*}\gg 1$), the temperature structure is vertically isothermal.
It is worth noting that the temperature structure of the optically thin region
$(\tau_{*}\ll 1)$ has no significant impact on our analysis as the emission
observed by ALMA is dominated by the emission from the region where
$\tau_{*}\gg 1$. The temperature of optically thick region of the passively-
heated disk can be lower than our model because of scattering of the stellar
irradiation (Okuzumi et al. 2022), which is not included in our model. With
our dust model, this could reduce the disk temperature by 10%. We examine the
validity of Equation (2) in Appendix A.

The temperature structure determined by accretion heating is (e.g., Nakamoto &
Nakagawa 1994; Sierra & Lizano 2020)

$\displaystyle T_{\rm acc}^{4}=\frac{3}{4}\left(\tau_{\rm
z}+\frac{2}{3}\right)\frac{3\dot{M}\Omega_{\rm K}^{2}}{8\pi\sigma},$ (3)

where $\Omega_{\rm K}$ is the Keplerian frequency and $\tau_{\rm z}$ is the
vertical optical depth for the dust thermal emission integrated from each
position to $z=\infty$ (or $z=-\infty$ if $z<0$);

$\displaystyle\tau_{\rm z}(z)=\int_{z}^{\infty}\kappa_{\rm
R}(z^{\prime})\rho_{\rm d}(z^{\prime})dz^{\prime},$ (4)

with $\kappa_{\rm R}$ and $\rho_{\rm d}$ being the Rosseland-mean opacity and
dust density, respectively. Because $\tau_{\rm z}$ increases as $z$ approaches
the midplane, $T_{\rm acc}$ increases as $z$ approaches the midplane. We note
that the actual value of $\dot{M}$ can be lower/higher than our adopted value
by a factor of $\sim 2$ (see Section 2.1), which decrease/increase $T_{\rm
acc}$ by $\sim$20%. As $T_{\rm acc}$ is a function of the product of
$\tau_{\rm z}$ and $\dot{M}$, the disk temperature structure is almost
identical as long as $\tau_{\rm z}\dot{M}$ is identical.

In this study, we consider two scenarios: a passive disk where the temperature
structure is solely determined by stellar irradiation (i.e., $T^{4}=T_{\rm
irr}^{4}$), and an active disk where additional heating due to accretion is
taken into account ($T^{4}=T_{\rm irr}^{4}+T_{\rm acc}^{4}$). As shown in
Section 3, accretion heating dominates over stellar irradiation ($T\approx
T_{\rm acc}$) near the midplane within the inner few to few dozen au,
depending on the specific disk models.

####  2.2.2 Dust opacities

The dust opacity plays a crucial role in determining both the temperature
structure of the disk and the (sub-)millimeter emission observed with ALMA.
The dust composition is assumed to be that of the DSHARP model (Birnstiel et
al. 2018) which is a mixture of water ice (Warren & Brandt 2008), astronomical
silicates (Draine 2003), troilite and refractory organics (Henning &
Stognienko 1996). It is important to note that the dust composition remains
highly uncertain, and therefore, we also present modeling results using an
alternative dust model, the DIANA dust model (Woitke et al. 2016) in Appendix
B.

The dust size distribution ranges from $a_{\rm min}$ to $a_{\rm max}$ with a
power-law index of $-p_{\rm d}$. We assume $p_{\rm d}=3.5$ which is similar to
the so-called MRN distibution (Mathis et al. 1977) as a fiducial case, while
more flat profile ($p_{\rm d}=2.5$) is also applied to investigate its impact
on the brightness temperature. The latter corresponds to more top-heavy
distribution where the dust opacity is dominated by larger grains. The minimum
dust size $a_{\rm min}$ is set to be $0.05\leavevmode\nobreak\ {\rm\mu m}$. We
confirmed that our conclusions remain unchanged even if we adopt larger
$a_{\rm min}$, $0.5\leavevmode\nobreak\ {\rm\mu m}$. We consider two different
dust porosity values, namely $p=0$ (corresponding to compact dust) and $0.9$
(corresponding to a filling factor of $f=1$ and $0.1$, respectively). We do
not consider very porous dust ($p\gg 0.9$) because extremely porous dust may
not account for the high polarization degree of the dust thermal emission
observed toward the CW Tau disk (Bacciotti et al. 2018; Tazaki et al. 2019;
Kirchschlager et al. 2019). The opacities are computed with Optool (Dominik et
al. 2021).

Figure 2: Effective extinction opacity (top), effective albedo (middle) and
Rosseland-mean opacity (bottom) of compact ($p=0$; left) and porous ($p=0.9$;
right) dust with different $a_{\rm max}$. The power-law index of the dust size
distribution is set to $p_{\rm d}=3.5$. The vertical gray dotted lines denote
the wavelength of ALMA Bands 4, 6, 7 and 8.

Figure 2 shows the effective extinction opacity $\kappa_{\rm ext}$, effective
albedo $\omega_{\rm eff}$ and Rosseland-mean opacity $\kappa_{\rm R}$ of our
dust model. The effective extinction opacity is defined as $\kappa_{\rm
ext}=\kappa_{\rm abs}+\kappa_{\rm sca}(1-g)$, where $\kappa_{\rm
abs},\kappa_{\rm sca}$ and $g$ are the absorption opacity, scattering opacity
and anisotropic scattering parameter, respectively. The effective albedo is
given as $\omega_{\rm eff}=\kappa_{\rm sca}(1-g)/\kappa_{\rm ext}$. The
extinction opacity has a slope of $\kappa_{\rm ext}\propto\lambda^{-1.7}$ when
$a_{\rm max}\ll\lambda/2\pi$, whereas it is flatter when $a_{\rm
max}\gg\lambda/2\pi$. In the intermediate regime ($a_{\rm
max}\sim\lambda/2\pi$), the opacity slope is steeper than $-1.7$ because of
the Mie interference. The interference is significant only for compact grains,
leading to a higher opacity compared to porous grains (Kataoka et al. 2014).

The effective scattering albedo at the ALMA wavelengths ($\sim$0.8–2 mm) is
$\lesssim 0.2$ for $a_{\rm max}\ll\lambda/2\pi\sim 300\leavevmode\nobreak\
{\rm\mu m}$. However, for $a_{\rm max}\gtrsim 300\leavevmode\nobreak\ {\rm\mu
m}$, the scattering albedo reaches 0.8 or even higher at the ALMA wavelengths.
The high scattering albedo efficiently reduces the emergent intensity at the
ALMA wavelengths, which makes the disk fainter than that without scattering
(e.g., Liu 2019; Zhu et al. 2019). At the ALMA wavelengths, compact dust has
slightly higher $\omega_{\rm eff}$ compared to porous dust. This is because
porous dust scatters incident radiation more efficiently in the forward
direction, resulting in a lower effective albedo. We note that the absorption
and scattering opacity can be scaled by $a_{\rm max}f$ except for the behavior
of the Mie interference (Kataoka et al. 2014). However, the asymmetry
parameter $g$ depends on the porosity, which makes the difference in the
optical properties of compact and porous dust for a given $a_{\rm max}f$
(Zhang et al. 2023).

####  2.2.3 Dust disk

The dust surface density is given by

$\displaystyle\Sigma_{\rm d}=\frac{\tau_{\rm 10,B4}}{\kappa_{\rm
ext,B4}}\left(\frac{r}{\rm 10\leavevmode\nobreak\ au}\right)^{-1},$ (5)

where $\tau_{\rm 10,B4}$ is the vertical extinction optical depth at ALMA Band
4 at 10 au. Although the true dust surface density is highly uncertain, the
wavelength-independent brightness temperature implies that the inner disk
region is at least moderately optically thick at these wavelengths (see Ueda
et al. 2022). In our model, we adopt $\tau_{\rm 10,B4}=3$ for all models. In
Appendix C, we show the effect of $\tau_{\rm 10,B4}$ on the brightness
temperature. We confirmed that $\tau_{\rm 10,B4}$ needs to be larger than
unity in order to match the observed brightness temperature at ALMA Band 4. We
will discuss the estimated dust and gas surface densities as well as the
corresponding accretion efficiency parameter $\alpha_{\rm acc}$ in Section
4.2.

In the vertical direction, the dust density distribution is given by:

$\displaystyle\rho_{\rm d}=\frac{\Sigma_{\rm d}}{\sqrt{2\pi}h_{\rm
d}}\exp{\left(-\frac{z^{2}}{2h_{\rm d}^{2}}\right)},$ (6)

where $h_{\rm d}$ represents the scale height of the dust disk. In our
fiducial model, we assume that the dust is well-coupled with the gas, i.e.,
$h_{\rm d}=h_{\rm g}$ with $h_{\rm g}$ being the gas scale height. The gas
scale height is calculated by $h_{\rm g}=c_{\rm s}/\Omega_{K}$ with $c_{\rm
s}$ being the sound speed of the gas at the midplane. We also investigate the
effect of dust settling. In incorporate dust settling, we define the Stokes
number as

$\displaystyle{\rm St}=\frac{\pi}{2}\frac{\rho_{\rm m}fa}{\Sigma_{\rm g}},$
(7)

where $\rho_{\rm m}=1.675\leavevmode\nobreak\ {\rm g\leavevmode\nobreak\
cm^{-3}}$ is the material density, $f=1-p$ is the dust filling factor and
$\Sigma_{\rm g}$ is the gas surface density and $a$ is the dust radius. For
simplicity, we assume that the gas surface density is 100 times larger than
the dust surface density. The dust scale height is assumed to be in mixing-
settling equilibrium (Dubrulle et al. 1995; Youdin & Lithwick 2007);

$\displaystyle h_{\rm d}=\left(1+\frac{\rm St}{\alpha_{\rm t}}\frac{1+2{\rm
St}}{1+{\rm St}}\right)^{-1/2}h_{\rm g},$ (8)

where $\alpha_{\rm t}$ is the turbulence strength for vertical dust mixing. In
our model with settling, we assume relatively weak vertical mixing with
$\alpha_{\rm t}=10^{-4}$ in order to investigate the difference from the no-
settling limit.

The computational domain extends radially from $r=0.5$ to 50 au,
logarithmically divided into 100 cells. For the polar direction, the
calculation domain covers from $\theta=-\pi/6$ to $\pi/6$ (where $\theta=0$
gives disk midplane) with 400 uniformly spaced cells.

To investigate the impact of different physical mechanisms on the brightness
temperature, we constructed models with various setups. The details of these
models are summarized in Table 1.

Table 1: Summary of our models Model name | Accretion | Dust | $p$ | $p_{\rm d}$
---|---|---|---|---
| heating | settling |  |
Passive disk | off | off | 0 | 3.5
Fiducial active disk | on | off | 0 | 3.5
Settling | on | on | 0 | 3.5
Top-heavy | on | off | 0 | 2.5
Porous | on | off | 0.9 | 3.5
Figure 3:  Brightness temperature profile of the passive disk model ($p=0$,
$\tau_{\rm 10,B4}=3$ and $p_{\rm d}=3.5$) with (top) and without (bottom)
scattering with different $a_{\rm max}$. The transparent shaded region shows
the observed brightness temperature. The model brightness temperature is
calculated after the intensity is convolved with the beam size of
0$\aas@@fstack{\prime\prime}$1 which corresponds to the physical distance of
13.2 au. The black dotted line denote the midplane temperature profile
convolved with a gaussian with a full width at half maximum (FWHM) of 13.2 au.

###  2.3 Imaging simulation

Given the disk structure obtained in Section 2.2, we perform Monte Carlo
radiative transfer calculations using RADMC-3D (Dullemond et al. 2012) to
generate synthetic images. The dust opacities are the same as those described
in Section 2.2.2. We use $10^{7}$ photon packages for each simulation. To
compare the synthetic images with the observations, we smooth the images with
the observing beam size of $0\aas@@fstack{\prime\prime}1$, which corresponds
to a physical distance of 13.2 au for CW Tau ($d=132$ pc). The disk
inclination is assumed to be $58^{\circ}$ (Ueda et al. 2022) and the intensity
is averaged over the azimuthal direction to obtain the radial intensity
profile. The radial intensity profile is then converted into the brightness
temperature using the full Planck function. To account for anisotropic
scattering, we employ the Henyey-Greenstein approximation (Henyey & Greenstein
1941).

##  3 Comparison of models and observations

In this section, we compare our theoretical models with the observations.

###  3.1 Passive disk models

Let us start the comparison from the simplest cases: passive disks, where the
temperature structure is determined solely by stellar irradiation.

Figure 3 shows the brightness temperature profile of the passive disk model
for different $a_{\rm max}$. In Figure 3, we show the brightness temperature
computed with and without scattering. When scattering is considered, the
emergent intensity is reduced, leading to lower brightness temperatures
compared to the actual dust temperature, even in optically thick disks (Liu
2019; Zhu et al. 2019; Ueda et al. 2020).

For $a_{\rm max}=10\leavevmode\nobreak\ {\rm\mu m}$, the model brightness
temperatures at $r\lesssim 10$ au are identical for all wavelengths and are
$\sim 1.5$ times higher than observed. Scattering has little effects on the
brightness temperatures in this case because of the low albedo of the
$10\leavevmode\nobreak\ {\rm\mu m}$-sized grains (see the center panel of
Figure 2). At $\gtrsim 15$ au, the brightness temperature at Band 4 is
slightly lower than those at the other bands because that region is marginally
optically thin at wavelengths $\gtrsim$ 2 mm.

For $a_{\rm max}\gtrsim 100\leavevmode\nobreak\ {\rm\mu m}$, the brightness
temperatures are comparable to or lower than those for $a_{\rm
max}=10\leavevmode\nobreak\ {\rm\mu m}$. The observed brightness temperature
in our models without scattering is affected by the optical depth of the disk.
In our model, the dust surface density is adjusted such that the extinction
optical depth at ALMA Band 4 reaches a value of 3 at a radial distance of 10
au ($\tau_{\rm 10,B4}=3$). In the models without scattering, the effect of
scattering is taken into account in the calculation of the dust surface
density. This means that the dust surface density remains the same for models
with and without scattering. However, when calculating the emergent intensity,
scattering is ignored. Because of this setup, the vertical optical depth is
effectively smaller for the models without scattering if scattering dominates
over absorption. For instance, dust with $a_{\rm max}=300\leavevmode\nobreak\
{\rm\mu m}$ has the effective albedo of $\sim 0.9$ at ALMA Band 4, which
corresponds to the absorption optical depth of $\tau_{\rm abs,10,B4}=\tau_{\rm
ext,10,B4}(1-\omega_{\rm eff})\sim 0.3$. Therefore, the no-scattering model
with $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$ is optically thin at
ALMA Band 4 and hence shows lower brightness temperature. For the models with
scattering, the model disks are always optically thick at $<10$ au at all
wavelengths we considered.

We see from Figure 3 that the brightness temperatures from the passive disk
models without scattering exceed those from the observations. This implies
that the scattering-induced intensity reduction takes place in the inner
region of the CW Tau disk. For $a_{\rm max}\sim 150\leavevmode\nobreak\
{\rm\mu m}$, the brightness temperatures are reduced by $\sim$30% by
scattering, which makes the passive disk with $150\leavevmode\nobreak\ {\rm\mu
m}$-sized dust consistent with the observations. The model with $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$ has a high effective albedo ($\gtrsim
0.8$) at all wavelengths considered, yielding brightness temperatures lower
than observed. For $a_{\rm max}\gtrsim 300\leavevmode\nobreak\ {\rm\mu m}$,
the effective albedo at ALMA Bands 7-8 decreases as dust size increases, which
makes the models with large grains are consistent with the observations at
ALMA Bands 7-8. However, at Band 4, dust with sizes in the range of
$300\leavevmode\nobreak\ {\rm\mu m}\lesssim a_{\rm max}\lesssim
1\leavevmode\nobreak\ {\rm mm}$ still exhibit a high albedo, leading to
brightness temperatures that are too low compared to the observed values.
Because the scattering albedo at ALMA Band 4 decreases for $a_{\rm max}\gtrsim
1\leavevmode\nobreak\ {\rm mm}$, very large dust with $a_{\rm max}\gtrsim
10\leavevmode\nobreak\ {\rm cm}$ may account for the observations.

Figure 4:  Brightness temperature at the center of model images normalized by
the observed brightness temperature. The gray shaded region denotes the region
where the model explains the observations with an accuracy of 10%, which is
the typical ALMA calibration uncertainty.

Figure 4 presents the brightness temperature at the center of the model images
normalized by the observed value, providing insight into the temperature
structure within the central region with diameter of $\sim 13$ au, which
corresponds to the size of one beam. We see that the brightness temperature of
models with $a_{\rm max}\lesssim 100\leavevmode\nobreak\ {\rm\mu m}$ exceeds
the observed values by $\sim 30\%$. In contrast, the brightness temperatures
are significantly lower than the observations if $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$. The intermediate size, $a_{\rm
max}\sim 150\leavevmode\nobreak\ {\rm\mu m}$, reasonably explains the
observations. If dust size is larger than millimeter, the model brightness
temperatures are more similar to the observed values as dust size increases,
showing that very large grains ($\gtrsim$ few cm) are roughly consistent with
the observations.

In summary, when considering a disk heated solely by stellar irradiation, two
possible ranges for the maximum dust radius emerge to explain the observed
brightness temperatures: $\sim 150\leavevmode\nobreak\ {\rm\mu m}$ or
$\gtrsim$ few cm. Very small dust grains, $a_{\rm max}\ll
150\leavevmode\nobreak\ {\rm\mu m}$, are unlikely to account for the
observations due to their low effective albedo at millimeter wavelengths,
which results in higher brightness temperatures than observed. On the other
hand, millimeter-sized dust grains are also improbable as their high effective
albedo at ALMA Band 4 leads to significantly lower model brightness
temperatures compared to the observed values.

###  3.2 Active disk models

The passive disk models considered in the previous subsection ignore any
internal heating sources. However, the inner region of the disk can be heated
by gas accretion, and hence, the disk may have a vertical temperature gradient
that affects the brightness temperature. In this subsection, we will show how
accretion heating affects the brightness temperature of the CW Tau disk.

####  3.2.1 Case of $a_{\rm max}=10\leavevmode\nobreak\ {\rm\mu m}$

To clarify the impact of the vertical temperature structure on the ALMA
brightness temperatures, we start the analysis by considering the scenario of
$a_{\rm max}=10\leavevmode\nobreak\ {\rm\mu m}$, wherein scattering of dust
thermal emission is negligible. The most notable factor in this case is the
substantial vertical optical depth $\tau_{\rm z}$, which establishes the
accretion heating as the primary determinant of the inner disk temperature.

Figure 5:  Two-dimensional temperature structure (top), dust temperature at
the ALMA surfaces (middle) and synthetic brightness temperature profiles
(bottom). The maximum dust size is $10\leavevmode\nobreak\ {\rm\mu m}$. The
left panels show the passive disk model where the disk is heated only by the
stellar irradiation, whereas the right panels show the active disk model where
the disk is heated by both stellar irradiation and disk accretion. The light
green lines in the top panels denote the height where the vertical extinction
optical depth reaches unity (ALMA surfaces) at ALMA Bands 4 (solid), 6
(dashed), 7 (dash-dotted) and 8 (dotted). The dust temperatures shown in the
middle panels are not beam-convolded, while the brightness temperatures shown
in the bottom panels are taken from the synthetic images convolved with a
gaussian with a FWHM of 13.2 au. The transparent regions denote the observed
brightness temperatures.

Figure 5 shows the two-dimensional temperature structure of the disk with
$a_{\rm max}=10\leavevmode\nobreak\ {\rm\mu m}$, both with and without
considering the effects of accretion heating. It is evident that within a
radius of approximately 20 au, the temperature structure is primarily
influenced by accretion heating when $a_{\rm max}=10\leavevmode\nobreak\
{\rm\mu m}$. Additionally, Figure 5 presents the dust temperatures at the
surfaces where the extinction optical depths at ALMA wavelengths reach unity
(hereafter ALMA surfaces). It is worth noting that despite having identical
total vertical optical depths in the passive and active disk models, the ALMA
surfaces are higher in the active disk model due to the elevated gas scale
height compared to the passive model.

The dust temperature at the ALMA surfaces remains constant regardless of the
observing wavelengths in the case of a passively heated disk. However, in the
presence of accretion heating, the dust temperature at the ALMA wavelengths
becomes wavelength-dependent. This dependence arises because longer observing
wavelengths probe regions closer to the midplane, where accretion heating is
more efficient. Consequently, longer observing wavelengths result in higher
dust temperatures. For instance, at 10 au, the dust temperature observed with
ALMA Band 8 is $\sim 100\leavevmode\nobreak\ {\rm K}$, while the dust
temperature observed with ALMA Band 4 is $\sim 180$ K. In comparison, the
midplane temperature is $\sim 200$ K.

Figure 6:  Two-dimensional temperature structure and dust temperature at ALMA
surfaces for different values of $a_{\rm max}$. Top: two-dimensional
temperature structure of the fiducial active disk. The light green lines
denote the height where the vertical extinction optical depth reaches unity at
ALMA Bands 4 (solid), 6 (dashed), 7 (dash-dotted) and 8 (dotted). bottom: dust
temperature at the height where the vertical extinction optical depth at the
ALMA wavelengths reaches unity. The radial profile are not beam-convolved. The
dust settling is not taken into account. The dust is assumed to be compact
with a size distribution of $p_{\rm d}=3.5$. The dust surface density is set
so that $\tau_{\rm 10,B4}=3$.

Figure 5 (bottom panels) also compares the brightness temperature profile
obtained from the radiative transfer simulation with the observed values. The
bottom-left panel of Figure 5 corresponds to the top-left panel in Figure 3.
The brightness temperature of the fiducial active disk is more than two times
higher than that of the passive disk. Furthermore, a wavelength-dependent
brightness temperature is seen within $\sim 10$ au, which is not seen in the
observations. This is because of the vertical temperature gradient shown in
the top and middle panels in Figure 5. The brightness temperature is
independent of the inclusion of scattering because of the negligible
scattering albedo. Based on these, we conclude that the maximum dust size of
10 ${\rm\mu m}$ is unlikely regardless of the efficiency of accretion heating.

####  3.2.2 Fiducial active disks

Figure 6 shows the two-dimensional temperature structure of the fiducial
active disk, considering different values of $a_{\rm max}$. We see that the
temperature of the inner disk region is higher for smaller $a_{\rm max}$
because of the higher vertical optical depth for dust thermal emission
($\tau_{z}$). For $a_{\rm max}\lesssim 300\leavevmode\nobreak\ {\rm\mu m}$,
the dust temperature at the ALMA surfaces is higher at longer wavelengths, as
shown in Section 3.2.1. In contrast, For $a_{\rm max}\gtrsim
1\leavevmode\nobreak\ {\rm mm}$, the dust temperature is not sensitive to the
observing wavelengths. This is because the extinction opacity of large dust
($\gtrsim 1$ mm) is less sensitive to the ALMA wavelength than that of small
dust (see Fig 2), and hence all ALMA observations trace similar disk heights.
The difference in the dust temperature is the most significant at $a_{\rm
max}\sim 100\leavevmode\nobreak\ {\rm\mu m}$ because of its steep slope in the
extinction opacity. Although the dust temperature is nearly independent on the
observing wavelengths when $a_{\rm max}\gtrsim 1\leavevmode\nobreak\ {\rm
mm}$, it remains lower than the midplane temperature at $\lesssim 6$ au.

The temperature structure is almost identical for models with $a_{\rm
max}=1\leavevmode\nobreak\ {\rm mm}$ and $1\leavevmode\nobreak\ {\rm cm}$.
This similarity arises because we ensure that the vertical extinction optical
depth at ALMA Band 4 ($\tau_{\rm 10,B4}$) is constant across all models. In
the regime of $a_{\rm max}\gtrsim 1\leavevmode\nobreak\ {\rm mm}$, both the
millimeter extinction opacity and the Rosseland-mean opacity decrease with
increasing dust radius in the same manner ($\propto a_{\rm max}^{-1/2}$). To
keep $\tau_{\rm 10,B4}$ to be constant, the dust surface density is inversely
proportional to the millimeter extinction opacity. Consequently, despite the
decrease in the Rosseland-mean opacity as the dust radius increases, the
infrared optical depth $\tau_{\rm z}$, which governs the magnitude of
accretion heating, remains unchanged when $\tau_{\rm 10,B4}$ is held constant.

Figure 7:  Same as Figure 3 but for the fiducial active disk.

Figure 7 shows the brightness temperature profile of the fiducial active disk
model. In contrast to the case of $a_{\rm max}=10\leavevmode\nobreak\ {\rm\mu
m}$, when $a_{\rm max}\gtrsim 100\leavevmode\nobreak\ {\rm\mu m}$, scattering
is no longer negligible and reduces the brightness temperature which makes the
interpretation of the brightness temperature more complicated.

For $a_{\rm max}=155\leavevmode\nobreak\ {\rm\mu m}$, the model brightness
temperatures are significantly higher than the observed values. The
millimeter-wave scattering makes the brightness temperatures lower than that
of no-scattering limit. The model with $a_{\rm max}=300\leavevmode\nobreak\
{\rm\mu m}$ shows similar brightness temperatures with the observations if
scattering is included. The brightness temperatures with scattering are $\sim
2$ times lower than that without scattering. Furthermore, the midplane
temperature is $\sim 3$ time higher than the brightness temperatures. This
clearly demonstrates that, even if the disk is fully optically thick, the
observed brightness temperature can be much lower than the true midplane
temperature because of the combined effect of scattering and vertical
temperature gradient (see also Sierra & Lizano 2020).

When $a_{\rm max}$ is of the order of 1 mm, the brightness temperature at Band
4 is significantly lower than the observed value because of scattering even if
the accretion heating takes place (see also Figure 8). In contrast, when
$a_{\rm max}$ is greater than $\sim$ 1 cm, the model brightness temperatures
are similar to the observations as seen for $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$. This is because the effective albedo
at ALMA Band 4 decreases as $a_{\rm max}$ increases in the regime of $a_{\rm
max}\gtrsim 1\leavevmode\nobreak\ {\rm mm}$. In this case, the midplane
temperature is $\sim 2$ times higher than the observed brightness
temperatures.

Figure 8:  Same as Figure 4 but for the fiducial active disk.

The central brightness temperature of the fiducial active disks is shown in
Figure 8. In the small dust regime ($<300\leavevmode\nobreak\ {\rm\mu m}$),
the model brightness temperature are too high to account for the observations
because of the efficient accretion heating. While the passive disk model with
$a_{\rm max}\sim 300\leavevmode\nobreak\ {\rm\mu m}$ underestimates the
brightness temperature at ALMA Bands 4-7, the fiducial active disk model with
$a_{\rm max}\sim 300\leavevmode\nobreak\ {\rm\mu m}$ aligns well with the
observations. This is attributed to the enhancement of brightness temperature
by accretion heating, particularly at longer wavelengths. Large grain models
($\gtrsim$ few cm) are roughly consistent with the observations, although they
tend to overestimate the brightness temperature at Band 8.

###  3.3 Effect of dust settling, size distribution and porosity

In this section, we explore the influence of dust settling, dust size
distribution, and dust porosity on the brightness temperatures of the active
disk. Figure 9 shows the two-dimensional temperature structure of the fiducial
active disk (same as the second right column of Figure 6), settling model,
top-heavy model and porous dust model. The maximum dust size is set to 1 mm.
We discuss the effect of each component in the following subsections.

Figure 9:  Two-dimensional temperature structure and dust temperature at ALMA
surfaces for different disk models. Top: two-dimensional temperature structure
of the fiducial active disk model (left), Settling model ($\alpha_{\rm
t}=10^{-4}$, second left), Top-heavy model (second right) and Porous model
(right) . The light green lines denote the height where the vertical
extinction optical depth reaches unity at ALMA Bands 4 (solid), 6 (dashed), 7
(dash-dotted) and 8 (dotted). The maximum dust size $a_{\rm max}$ is 1 mm in
the all models. Bottom: dust temperature at the height where the vertical
extinction optical depth at the ALMA wavelengths reaches unity. The radial
profile is not convolved.

####  3.3.1 Effect of dust settling

As dust grains grow larger, they are expected to settle down towards the
midplane of the disk. If dust settling takes place, the effective dust size
that contributes to the millimeter emission decreases because larger grains
can be hidden within the optically thick midplane layer (Ueda et al. 2021a).

Figure 9 compares the dust temperature at the ALMA surfaces of the models
without (left) and with (second left) dust settling with $\alpha_{\rm
t}=10^{-4}$. We find that dust settling effectively increases the temperature
of dust observed with ALMA. This behavior can be explained as follows: The
efficiency of accretion heating is primarily determined by the distribution of
small grains, whereas the height of the ALMA surfaces is predominantly
influenced by the distribution of large grains. Due to settling, larger grains
tend to selectively settle towards the midplane, causing the ALMA surfaces to
descend more rapidly towards the midplane compared to the isotherm height of
the active disk (as shown in the top rows of Figure 9). This allows ALMA to
observe a hotter layer.

Figure 10:  Same as Figure 4 but for the settling model with $\alpha_{\rm
t}=10^{-4}$.

Figure 10 shows the central brightness temperature of the beam-convolved
images of the settling model. The radial brightness temperature profile of
each model is shown in the appendix D (see Figure 23). For $a_{\rm
max}\lesssim 3\leavevmode\nobreak\ {\rm mm}$, the brightness temperature is
systematically higher than that of the fiducial active disk model (i.e.,
without settling). This is because of the effect explained above. Particularly
for $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$, the settling model
overpredicts the brightness temperatures because of the higher temperature of
the dust responsible for the emission. We note that the settling model with
$a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$ reasonably explains the
observations if $\alpha_{\rm t}=10^{-3}$ because dust settling is less
effective compared to $\alpha_{\rm t}=10^{-4}$. We also note that if the disk
is active, the turbulence strength for dust mixing may not be significantly
lower than that for accretion ($\alpha_{\rm acc}$; Equation (10)), implying
that $\alpha_{\rm t}=10^{-3}$ may be more likely for the active disk model
(see also Section 4.2).

For $a_{\rm max}\gtrsim 3\leavevmode\nobreak\ {\rm mm}$, the brightness
temperatures of the settling model are not simply higher than those of the
model without settling. The temperature of dust observed with ALMA is higher
for the models with settling compared to those without settling. However, the
differential settling makes effective dust size smaller, which increases the
scattering albedo and hence reduces the emergent intensity at the ALMA
wavelengths. The emergent intensity, and hence the brightness temperature, is
determined by the balance between these two effects.

Overall, large grain models ($a_{\rm max}\gtrsim 3\leavevmode\nobreak\ {\rm
mm}$) predict the brightness temperatures similar to the observations.
Compared to the no-settling models (Figure 8), the settling model is more
consistent with the observations if $a_{\rm max}\gtrsim 3\leavevmode\nobreak\
{\rm mm}$. We note that if $\alpha_{\rm t}=10^{-3}$, dust settling has no
significant impact on the brightness temperatures when $a_{\rm max}\lesssim
1{\rm mm}$ (see also Section 4.2).

####  3.3.2 Effect of dust size distribution

The dust size distribution plays a critical role in both the millimeter
thermal emission and the efficiency of accretion heating. In our fiducial
case, we assume a slope of $p_{\rm d}=3.5$ for the dust size distribution.
However, it is important to note that the dust size distribution in
protoplanetary disks is uncertain and is influenced by dust coagulation and
fragmentation processes. In particular, if dust fragmentation is not
efficient, the dust size distribution can exhibit a more top-heavy nature,
meaning it is dominated by larger grains, rather than following a slope of
$p_{\rm d}=3.5$ (e.g., Birnstiel et al. 2011).

Figure 11:  Same as Figure 4 but for the top-heavy dust model.

In the top-heavy model, we adopt a dust size distribution with a slope of
$p_{\rm d}=2.5$ instead of 3.5. As shown in Figure 9, the top-heavy dust size
distribution leads to a cooler disk region near the midplane compared to the
fiducial active disk. This is because the dust mass is more dominated by large
grains which have lower surface-to-mass ratio and hence accretion heating is
less efficient.

Figure 11 shows the central brightness temperature of the beam-convolved
images of the top-heavy model. The radial brightness temperature profile of
each model is shown in the appendix D (see Figure 24). Although the dust
temperature is lower than that of fiducial active disk model, the brightness
temperature does not simply follow the dust temperature. In the regime of
$a_{\rm max}\lesssim 1\leavevmode\nobreak\ {\rm mm}$, scattering is more
efficient than the case of $p_{\rm d}=3.5$ because mm-sized grains have higher
scattering albedo than micron-sized grains at ALMA wavelengths. Therefore, for
$a_{\rm max}\lesssim 1\leavevmode\nobreak\ {\rm mm}$, the brightness
temperatures of the top-heavy model are lower compared to the fiducial active
disk model. In contrast, in the regime of $a_{\rm max}\gtrsim
1\leavevmode\nobreak\ {\rm mm}$, scattering is less efficient than the case of
$p_{\rm d}=3.5$ because cm-sized or larger grains have lower scattering albedo
than mm-sized grains at ALMA wavelengths. Therefore, for $a_{\rm max}\gtrsim
1\leavevmode\nobreak\ {\rm mm}$, the brightness temperatures of the top-heavy
model are higher compared to the fiducial active disk model. In the middle of
these two regimes ($a_{\rm max}\sim 1\leavevmode\nobreak\ {\rm mm}$), the
brightness temperature is not sensitive to $p_{\rm d}$. Overall, the top-heavy
models do not appear to be consistent with the observations.

####  3.3.3 Effect of porosity

In the previous sections, we focused on studying the brightness temperature of
disks with compact dust grains. However, it is important to consider that dust
in protoplanetary disks can be porous due to the pairwise collisional growth
(e.g., Okuzumi et al. 2012).

In Figure 9, we compare the dust temperature between the fiducial active disk
model (left) and the porous dust model (right). We see that the accretion
heating is more efficient in the porous dust model. This is because, for a
given optical depth at ALMA wavelengths (e.g., $\tau_{\rm B4}$), the optical
depth for the dust thermal emission ($\tau_{\rm z}$) is higher for porous
dust. Figure 12 shows the ratio between the Rosseland-mean opacity (at 100K)
and the extinction opacity at ALMA Band 4. For $a_{\rm max}\gtrsim
100\leavevmode\nobreak\ {\rm\mu m}$, porous dust has higher $\kappa_{\rm
R,100K}/\kappa_{\rm ext,B4}$. This means that, for a given optical depth at
ALMA wavelength, porous dust has larger vertical optical depth for accretion
heating and hence the accretion heating is more efficient compared to compact
dust.

Figure 12:  Ratio between the Rosseland-mean opacity (at 100K) and the
extinction opacity at ALMA Band 4. The red solid and blue dashed lines denote
the compact and porous dust, respectively.  Figure 13:  Same as Figure 4 but
for the porous dust model.

Figure 13 shows the central brightness temperature of the convolved images of
the porous dust model. The radial brightness temperature profile of each model
is shown in the appendix D (see Figure 25). If dust is porous, the brightness
temperature is higher than that of compact dust model regardless of the dust
size. We see that the porous dust models predict $\gtrsim 1.5$ times higher
brightness temperatures than the observations. We also confirm that the
brightness temperatures of porous dust model are still higher even if $p_{\rm
d}=2.5$.

##  4 Discussion

###  4.1 Dust size in the inner region of the CW Tau disk

Our study reveals that the maximum dust size in the inner region ($\lesssim
10$ au) of the CW Tau disk is preferred to be $\sim$150–300 ${\rm\mu m}$
(small grain solution) or larger than a few centimeters (large grain
solution). Distinguishing between these two scenarios is essential for gaining
insights into grain growth and temperature structure of the disk.

The CW Tau disk shows the strong scattering-induced polarization at ALMA Band
7 with a polarization degree of $\gtrsim 1$% (Bacciotti et al. 2018). This
indicates that the thermal emission at ALMA Band 7 is primarily contributed by
dust grains with a radius of $\lambda/2\pi\sim 140\leavevmode\nobreak\ {\rm\mu
m}$ (Kataoka et al. 2015), although the angular resolution of the polarimetric
observations ($\sim$0$\aas@@fstack{\prime\prime}$2) is worse than ours
(0$\aas@@fstack{\prime\prime}$1). This implies that the small grain solution
is more consistent with the polarimetric observations than the large grain
solution.

The effective observed-dust size can be smaller than the true maximum dust
size if differential dust settling takes place (Sierra & Lizano 2020; Ueda et
al. 2021a), which potentially makes the large grain solution matches with the
polarimetric observation. However, it is important to note that achieving the
observed high polarization degree ($\gtrsim 1$%) with $a_{\rm max}$ larger
than $\sim 1$ mm is unlikely, even under the assumption of extremely low
turbulence strength (Ueda et al. 2021a).

The polarization degree also depends on the detailed structure of dust grains.
Recent laboratory experiment suggests that non-spherical irregular shape dust
can produce strong scattering-induced polarization even if the dust size is
significantly larger than the observing wavelength (Lin et al. 2023). If this
is the case, the polarization degree of non-spherical grains is less sensitive
to the observing wavelength than the compact spherical grains. Furthermore,
inclusion of small amount of porosity ($p\lesssim 0.9$) also makes the
polarization degree less sensitive to the dust size (Tazaki et al. 2019; Zhang
et al. 2023). However, we note that, if the porosity is as high as 0.9,
accretion heating is so efficient that the model brightness temperatures
exceed the observed values. Therefore, if $p\gtrsim 0.9$, some sort of
reduction of accretion heating due to, e.g., wind-driven accretion (Mori et
al. 2019; Kondo et al. 2022), is required. Future multi-wavelength
polarimetric observations would help us constrain the detailed dust structure.

###  4.2 Surface densities and effective accretion $\alpha$

In this section, we discuss the expected dust and gas surface densities, as
well as the effective turbulence strength for the accretion of the CW Tau
disk.

Figure 14:  Gas surface density (hundred times of the dust; top) and effective
turbulence strength for the disk gas accretion (bottom) estimated from our
fiducial active disk models. The gray dashed line denotes the gas surface
density of the minimum mass solar nebula (MMSN; Hayashi 1981). The dotted
lines denote the criteria for the disk to be gravitationally unstable ($Q=1$).

Figure 14 shows the gas surface density of our fiducial active disk
models(assuming hundred times of the dust). The gas surface density of the
minimum mass solar nebula (MMSN), $1700(r/{\rm au})^{-1.5}\leavevmode\nobreak\
{\rm g\leavevmode\nobreak\ cm^{-2}}$ (Hayashi 1981), is also shown for
comparison. We also plot the criterion for the disk to be graivationally
unstable (Toomre 1964):

$\displaystyle Q\equiv\frac{\Omega_{\rm K}c_{\rm s}}{\pi G\Sigma_{\rm g}}=1.$
(9)

In general, our model disks are expected to be gravitationally stable within
the region of focus, approximately $\lesssim 10$ au. If $a_{\rm
max}=100\leavevmode\nobreak\ {\rm\mu m}$, the gas surface density reaches
$Q=1$ around $\sim 10$ au. If $a_{\rm max}$ were $10\leavevmode\nobreak\
{\rm\mu m}$, the disk mass would be $\sim 1.5$ times higher; however, as
already shown in Section 3, the model with $a_{\rm max}=10\leavevmode\nobreak\
{\rm\mu m}$ fails to reproduce the observed brightness temperatures. On the
other hand, for $a_{\rm max}\gtrsim 300\leavevmode\nobreak\ {\rm\mu m}$, the
gas surface density is estimated to be much lower than the $Q=1$ line within
20 au. Moreover, the estimated gas surface density aligns closely with the
MMSN model within 10 au for $a_{\rm max}\gtrsim 300\leavevmode\nobreak\
{\rm\mu m}$. This indicates that the CW Tau disk has enough capability to form
a planetary system similar to our solar system.

Figure 14 also shows the effective turbulence strength for the disk accretion;

$\displaystyle\alpha_{\rm acc}=\frac{\dot{M}}{3\pi\Sigma_{\rm g}c_{\rm
s}h_{\rm g}}.$ (10)

For the given accretion rate of $4\times 10^{-8}M_{\odot}\leavevmode\nobreak\
{\rm yr^{-1}}$, the effective turbulence strength accounting for the disk
accretion needs to be $\sim 10^{-4}$ for $a_{\rm max}=100\leavevmode\nobreak\
{\rm\mu m}$, while it is $\sim 10^{-3}$–$10^{-2}$ for $a_{\rm max}\gtrsim
300\leavevmode\nobreak\ {\rm\mu m}$. The estimated $\alpha_{\rm acc}$ falls
into a reasonable range of $\alpha_{\rm acc}=10^{-4}$–$10^{-2}$, although the
high $\alpha_{\rm acc}$ ($\sim 10^{-2}$) found for the models of $a_{\rm
max}=1$–10 mm is higher than the typical value estimated from the observed
disk lifetime in various star-forming regions (Manara et al. 2022, but see
also Hartmann et al. 1998).

The model with $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$ yields
$\alpha_{\rm acc}\sim 10^{-3}$. As shown in Section 3.2.2 and 3.3.1, the model
with $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$ can account for the
observations if vertical dust settling is ignored but cannot if dust settling
takes place with $\alpha_{\rm t}=10^{-4}$. To further investigate this, we
conducted a simulation of the settling model with $\alpha_{\rm t}=10^{-3}$ and
confirm that the model with $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$
reasonably matches the observations if $\alpha_{\rm t}=10^{-3}$. This suggests
that the effective turbulence strength for vertical dust mixing, $\alpha_{\rm
t}$, is not much lower than the effective turbulence strength for disk
accretion, $\alpha_{\rm acc}$, if $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu
m}$.

###  4.3 Predictions for future observations

Our results show that the observed brightness temperatures at ALMA Bands 4-8
can be explained by either the passive or active disk models. It is crucial to
distinguish these models for understanding the heating mechanisms in the inner
region of disks.

Figure 15:  Brightness temperature profile at $\lambda=3.1$, 4.1 and 7.0 mm
predicted by our active disk models (solid lines) and passive disk models
(dotted lines). The angular resolution is assumed to be
0$\aas@@fstack{\prime\prime}$05\. The scattering-induced intensity reduction
is included.

Figure 15 shows the brightness temperatures at $\lambda=3.1$ (ALMA Band 3),
4.1 (ALMA Band 2) and 7.0 mm (ALMA Band 1 or VLA Q band) computed from our
models. The angular resolution is assumed to be
0$\aas@@fstack{\prime\prime}$05\. The active disk with $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$ has $\sim$1.5–2 times higher
brightness temperatures at $\lambda=3.1$ and 4.1 mm than the other models. The
peak brightness temperature reaches $\sim 80$ K at $\lambda=3.1$ and 4.1 mm.
In contrast, at $\lambda=7.0$ mm, the brightness temperature of the active
disk model with $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$ is not
significantly different from that with $a_{\rm max}=10\leavevmode\nobreak\
{\rm cm}$. This is because the small grain solution has higher temperature but
lower optical depth, resulting in the cancellation of these effects.

The passive disk with $a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$ can be
distinguished from the other models by using $\lambda=7.0$ mm because it has
significantly lower brightness temperature. This is because the passive disk
has lower temperature than the active disk and the optical depth is lower than
the large grain model.

If $a_{\rm max}=10\leavevmode\nobreak\ {\rm cm}$, the accretion heating is not
so significant that both the passive and active disk predict similar
brightness temperature. However, the difference in those models is potentially
distinguishable by ALMA because typical uncertainty in the ALMA flux is 5-10%
which is smaller than the difference in the brightness temperatures predicted
by the models.

Even though the finest angular resolution of current facilities (ALMA and VLA)
at $\lambda\gtrsim 3\leavevmode\nobreak\ {\rm mm}$ is limited to $\sim$
0$\aas@@fstack{\prime\prime}$05, it can achieve $\sim$
0$\aas@@fstack{\prime\prime}$02 at shorter wavelengths. Figure 16 shows the
brightness temperature at $\lambda=0.75$ (ALMA Band 8), 0.89 (Band 7) and 1.34
mm (Band 6) computed from our models with the angular resolution of
0$\aas@@fstack{\prime\prime}$02\. We can see that the higher spatial
resolution observations at shorter wavelengths can also distinguish between
the active and passive disk models. The higher angular resolution allows us to
detect a steeper brightness temperature profile in the active disk model.
However, the difference in the brightness temperatures of the active and
passive disks decreases with observing wavelength decreases because the
shorter wavelength traces a more upper layer where the accretion heating is
less effective. The difference in the brightness temperatures of the two
passive disk models is maximized at $\lambda\sim 1.34$ mm because
$300\leavevmode\nobreak\ {\rm\mu m}$-sized grain has a maximum effective
albedo at that wavelength. In contrast, the active disk models with $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$ and 10 cm have similar brightness
temperature profile because the $300\leavevmode\nobreak\ {\rm\mu m}$ dust
model has a higher dust temperature, which compensates the efficient intensity
reduction by scattering.

Figure 16:  Brightness temperature profile at $\lambda=0.75$, 0.89 and 1.34 mm
predicted by our active disk models (solid lines) and passive disk models
(dotted lines). The angular resolution is assumed to be
0$\aas@@fstack{\prime\prime}$02\. The scattering-induced intensity reduction
is included.

###  4.4 Spectral energy distribution

Although our focus is on comparing our models with observations at ALMA
wavelengths, it is worthwhile to compare them at infrared wavelengths as well.
Figure 17 shows the comparison between our model spectral energy distributions
(SEDs) and the observed data. The observed SED is taken from Andrews et al.
(2013) (see also references therein). In order to evaluate the SEDs, we
extrapolate the disk model down to 0.01 au and remove the dust with
temperature higher than 1400 K which mimics the sublimation of silicate dust.
This modification is necessary because the dust responsible for infrared
emission could be located within 0.5 au, which is the inner boundary used for
the calculations in the comparison at ALMA wavelengths.

We find that the active disk models have higher luminosity at infrared
wavelengths ($\sim$1–20 ${\rm\mu m}$) than the passive disk models (see also,
e.g., D’Alessio et al. 1998; Dullemond et al. 2007). The passive disk models
tend to underestimate the fluxes observed at infrared wavelengths. For the
active disk models, the large grain models ($a_{\rm max}=1$ mm and 1 cm) still
underestimate the infrared fluxes, while the model with $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$ overestimates them. Although neither
of these models fully explains the observed SED at infrared wavelengths, the
active disk models are more resemble to the observations compared to the
passive disk models.

However, it is important to note that the infrared SED is highly influenced by
the intricate structure of the innermost region of the disk, specifically in
the vicinity of the silicate sublimation radius (see Dullemond & Monnier 2010
and Kraus 2015 for review), which is beyond the scope of our study. For
instance, the emission in the near-infrared range ($\sim 2\leavevmode\nobreak\
{\rm\mu m}$) is expected to be predominantly attributed to the scattering of
stellar light at the inner rim of the disk (e.g., Monnier & Millan-Gabet
2002), rather than the thermal emission from the region we focus on.
Additionally, the presence of a dust halo surrounding the inner disk,
potentially induced by the inner disk wind, is anticipated to enhance the
near-infrared flux (e.g., Vinković & Jurkić 2007). Therefore, detailed
modeling of the innermost region of the disk is necessary to obtain accurate
fluxes at near-infrared wavelengths. While significant uncertainties remain in
the near-infrared range, the observed high fluxes at mid-infrared wavelengths
may suggest a higher likelihood of active heating within the CW Tau disk.

Figure 17:  Comparison between our model spectral energy distributions and
observed SED. The observed values are from Andrews et al. (2013) and
references therein.

##  5 Summary

We investigate the impact of accretion heating on the brightness temperatures
of the inner region of the CW Tau disk. The key findings are as follows:

  * •

If $a_{\rm max}\lesssim 100\leavevmode\nobreak\ {\rm\mu m}$, the model
brightness temperatures are too high to explain the observations regardless of
the efficiency of accretion heating.

  * •

If the disk is passively heated, the maximum dust size needs to be either
$\sim 150\leavevmode\nobreak\ {\rm\mu m}$ or $\gtrsim$ few cm.

  * •

If the disk is actively heated, small grain models significantly overpredict
the brightness temperatures, and hence the maximum dust size needs to be
either $\sim 300\leavevmode\nobreak\ {\rm\mu m}$ or $\gtrsim$ few cm.

  * •

The midplane temperature would be $\sim$1.5–3 times higher than the observed
brightness temperatures because of combined effect of scattering and accretion
heating.

  * •

If dust settling occurs in the active disk, the ALMA observations trace deeper
disk layers compared to the disk with no-settling. Therefore, the dust
settling effectively increases the temperature of the dust responsible for the
millimeter emission from the active disk.

  * •

If turbulence strength parameter for vertical dust mixing $\alpha_{\rm t}$ is
$10^{-4}$, the active disk model with $a_{\rm max}=300\leavevmode\nobreak\
{\rm\mu m}$ overpredicts the brightness temperatures, suggesting that
$\alpha_{\rm t}>10^{-4}$.

  * •

The estimated effective accretion efficiency $\alpha_{\rm acc}$ falls into a
reasonable range of $10^{-4}$–$10^{-2}$, depending on the dust size. If
$a_{\rm max}=300\leavevmode\nobreak\ {\rm\mu m}$, $\alpha_{\rm acc}$ is
expected to be $\sim 10^{-3}$, indicating that $\alpha_{\rm t}$ is not much
smaller than $\alpha_{\rm acc}$.

  * •

The efficiency of accretion heating needs to be reduced if dust is porous
($p=0.9$) because the accretion heating is too efficient to explain the
observations regardless of the dust size.

Future observations using longer wavelengths, such as ALMA Bands 1–3 and VLA,
will play a crucial role in distinguishing between the active and passive disk
models. The results of our work highlight the importance of multi-wavelength
observations in improving our understanding of the physical properties of the
inner regions of protoplanetary disks.

###### Acknowledgements.

We thank the anonymous referee for useful comments. We also thank Sean Andrews
for providing the observed SED of CW Tau. T.U. acknowledges the support of the
DFG-Grant ”Inside: inner regions of protoplanetary disks: simulations and
observations” (FL 909/5-1). S. O. is supported by JSPS KAKENHI Grant Numbers
JP18H05438, JP20H00182, and JP20H01948. M.F. acknowledges funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (grant agreement No. 757957).

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##  Appendix A Midplane temperature of the passive disks

In this section, we examine the accuracy of our temperature of the passive
disk models. In our temperature model (Equation 2), we do not consider the
effect of scattering of the stellar irradiation and adopt constant value for
the incident angle of the stellar irradiation $\psi$ (0.02). To see the
validity of our simplified model, we perform thermal Monte Carlo radiative
transfer simulations with RADMC3D (Dullemond et al. 2012) with takeing the
anisotropic scattering into account.

Figure 18 compares the midplane temperature profiles obtained from the Monte
Carlo radiative transfer simulations and analytical model (Equation 2).
Although our analytical model slightly overestimates the midplane temperature,
the difference from the radiative transfer model is $\lesssim 10$% which is
comparable to the uncertainty of the ALMA flux. Therefore, we think that the
analytical model has sufficient accuracy to estimate the brightness
temperatures at the ALMA wavelengths.

Figure 18:  Comparison of the midplane temperature profiles of the passive
disks obtained from the Monte Carlo radiative transfer simulations and
analytical prescription (Equation 2).

##  Appendix B Effect of the dust composition

In this section, we present the modeling results using the DIANA dust, which
is an alternative to the DSHARP dust used previously. The DIANA dust consists
of pyroxene (${\rm Mg_{0.7}Fe_{0.3}SiO_{3}}$; Dorschner et al. 1995) and
amorphous carbon (Zubko et al. 1996) , with a mass ratio of 0.87 for pyroxene
and 0.13 for carbon, and has a porosity of 0.25 (Woitke et al. 2016). While
the DSHARP dust contains carbonaceous material in the form of organics, the
DIANA dust consists of amorphous carbon as the carbonaceous material, which
efficiently absorbs incident irradiation.

Figure 19 shows the effective albedo of the DIANA dust. The DIANA dust has
lower effective albedo than the DSHARP dust because of the high absorbing
efficiency of carbon. Specifically, the effective albedo of the DIANA dust
remains below 0.8 at the wavelengths of interest, while that of the DSHARP
dust is as high as $\sim$ 0.9 (Figure 2).

Figure 19:  Effective albedo of the DIANA dust model.

Figure 20 shows the normalized brightness temperature at the center of the
passive disk model using the DIANA dust model. Due to its smaller effective
albedo, the brightness temperatures consistently exceed those of the DSHARP
model (Figure 4). Particularly at ALMA Band 8, the brightness temperature of
the DIANA dust model is higher than the observed value, regardless of the
calue of $a_{\rm max}$.

Figure 20:  Same as Figure 4 but for the DIANA dust model.

The central brightness temperature of the active disk model using the DIANA
dust is shown in Figure 21. When taking into account the effect of accretion
heating, the brightness temperature is higher compared to the passive disk
model, thereby making the DIANA dust model less consistent with the
observations.

Figure 21:  Same as Figure 8 but for the DIANA dust model.

In general, the DIANA dust models predict higher brightness temperature than
the DSHARP dust models. Consequently, the DSHARP dust model appears to be more
in line with the observed data.

##  Appendix C Dependence of the surface density

Figure 22:  Same as Figure 8 but for different values of $\tau_{\rm 10,B4}$.

In this appendix, we discuss the effect of the dust surface density on the
brightness temperature. Figure 22 shows the brightness temperature at the
center of the synthetic images of different $\tau_{\rm 10,B4}$. The brightness
temperature at longer wavelength is more sensitive to the dust surface density
(i.e., $\tau_{\rm 10,B4}$) because it traces the region closer to the midplane
where the accretion heating dominates the irradiation heating. In addition to
that, smaller grain model is more sensitive to $\tau_{\rm 10,B4}$ because of
efficient accretion heating (see Figure 12).

If $\tau_{\rm 10,B4}=1$, the brightness temperature at ALMA Band 4 is always
lower than the observed value when $a_{\rm max}\gtrsim 300\leavevmode\nobreak\
{\rm\mu m}$. This suggests that $\tau_{\rm 10,B4}$ needs to be sufficiently
larger than unity. The brightness temperature at Band 4 increases as
$\tau_{\rm 10,B4}$ increases. If $\tau_{\rm 10,B4}=3$, the brightness
temperature at Band 4 is consistent with the observation when $a_{\rm
max}=300\leavevmode\nobreak\ {\rm\mu m}$, as shown in the main text. If
$\tau_{\rm 10,B4}>3$, the brightness temperature at Band 4 can be consistent
with the observation even when $a_{\rm max}>300\leavevmode\nobreak\ {\rm\mu
m}$. However, in this regime, the brightness temperature at ALMA Band 8 is
higher than the observed value.

##  Appendix D Radial brightness temperature profile of different models

Figure 23 shows the brightness temperature profile of models with dust
settling. The midplane temperature is independent on the presence of dust
settling (see Figure 7 and 23) as the vertical optical depth at the midplane
is identical. The dust differential settling effectively increase the
temperature of dust observed with ALMA as explained in Section 3.3.1. This can
be confirmed by comparing the brightness temperature shown in Figure 7 and 23
in the limit of no-scattering; The brightness temperature in the no-scattering
limit is higher for the model with settling. When scattering is taken into
account, the brightness temperature of the settling model is not simply higher
than the model without settling because the magnitude of scattering-induced
intensity reduction is affected by the dust settling.

Figure 23:  Same as Figure 7 but with dust settling.

In Figure 24, we show the brightness temperature of models with $p_{\rm
d}=2.5$ instead of 3.5. The smaller $p_{\rm d}$ corresponds to the more top-
heavy dust size distribution in which larger grains is more dominant in the
opacity. Because of the top-heavy dust size distribution, the dust temperature
is lower than that of the fiducial active disk. This can be confirmed by
comparing the brightness temperature in Figure 7 and 24 in the limit of no-
scattering (dashed lines); The brightness temperature in the no-scattering
limit is lower for the model with $p_{\rm d}=2.5$. When scattering is taken
into account, the brightness temperature of the top-heavy model is lower than
that of the fiducial active disk in the regime of $a_{\rm max}\lesssim
1\leavevmode\nobreak\ {\rm mm}$, because large grains have higher scattering
albedo at ALMA wavelengths. In contrast, in the regime of $a_{\rm max}\gtrsim
1\leavevmode\nobreak\ {\rm mm}$, the brightness temperature of the top-heavy
model is higher than the fiducial active disk model because of low scattering
albedo.

Figure 24:  Same as Figure 7 but with $p_{\rm d}=2.5$.

Figure 25 shows the brightness temperature profile of models with porous
grains ($p=0.9$; filling factor of 0.1).

The dust temperature of the active disk with porous dust is higher than that
of compact dust as explained in Section 3.3.3. This can be confirmed by
comparing the brightness temperature in Figure 7 and 25 in the limit of no-
scattering (dashed lines); The brightness temperature in the no-scattering
limit is higher for the model with porous dust. Furthermore, the effective
scattering albedo of the porous dust is slightly lower than that of compact
dust, which result in inefficient scattering-induced intensity reduction.
Because of these reasons, the model brightness temperature is significantly
higher than the observations.

Figure 25:  Same as Figure 7 but with $p=0.9$.