11institutetext: Anton Pannekoek Institute for Astronomy, University of
Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: Astronomical Institute, Graduate
School of Science, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai 980-8578,
Japan

# How large are the monomers of dust aggregates in planet-forming disks?

Insights from quantitative optical and near-infrared polarimetry

R. Tazaki 1122 C. Dominik 11

###### Abstract

Context.  The size of the constituent particles (monomers) of dust aggregates
is one of the most uncertain parameters directly affecting collisional growth
of aggregates in planet-forming disks. Despite its importance, the monomer
size has not yet been meaningfully constrained by disk observations.

Aims.  We attempt to derive the monomer size from optical and near-infrared
(IR) polarimetric observations of planet-forming disks.

Methods.  We perform a comprehensive parameter survey on the degree of linear
polarization of light scattered by dust aggregates, using an exact numerical
method called the $T$-matrix method. We investigate the effect of the monomer
size, aggregate size, porosity, and composition on the degree of polarization.
The obtained results are then compared with observed polarization fractions of
several planet-forming disks at optical and near-IR wavelengths.

Results.  It is shown that the degree of polarization of aggregates depends
sensitively on the monomer size unless the monomer size parameter is smaller
than one or two. Comparing the simulation results with the disk observations,
we find that the monomer radius is no greater than $0.4\leavevmode\nobreak\
\mu$m. The inferred monomer size is therefore similar to subunit sizes of the
solar system dust aggregates and the maximum size of interstellar grains.

Conclusions.  Optical and near-IR quantitative polarimetry will provide
observational grounds on the initial conditions for dust coagulation and
thereby planetesimal formation in planet-forming disks.

###### Key Words.:

Scattering – Polarization – Protoplanetary disks

##  1 Introduction

Collisional growth of dust aggregates is the first step of planet formation.
The properties of the constituent particles (monomers) of an aggregate are
important as they directly affect the impact strength of the aggregate, which
intimately links to the maximum aggregate radius to be reached by collisional
growth (Birnstiel et al. 2012; Okuzumi et al. 2016; Pinilla et al. 2017;
Okuzumi & Tazaki 2019). In particular, the monomer size and surface materials
coating each monomer are of crucial relevance for the impact strength. For
example, larger monomers and/or less sticky materials make aggregates more
fragile (Chokshi et al. 1993; Dominik & Tielens 1997; Wada et al. 2009).
Although material dependence of the impact strength has been extensively
studied by laboratory experiments in the last decades (e.g., Poppe et al.
2000; Gundlach & Blum 2015; Musiolik et al. 2016a, b), the monomer size
remains uncertain.

Previous studies of dust coagulation commonly assume sub-micron-sized monomers
(Weidenschilling 1984, 1997; Dullemond & Dominik 2005; Ormel et al. 2007;
Okuzumi et al. 2012; Kataoka et al. 2013; Krijt et al. 2016; Kobayashi &
Tanaka 2021). This monomer size has been motivated by previous studies on the
subunit sizes of interplanetary dust particles (Brownlee 1985; Rietmeijer
1993; Wozniakiewicz et al. 2013) or the largest size of interstellar grains
(Mathis et al. 1977; Draine & Lee 1984; Jones et al. 2013). Recent in situ
measurements of dust aggregates of the comet 67P/Churyumov–Gerasimenko (C-G)
also suggest the presence of sub-micron-sized subunits (Bentley et al. 2016;
Mannel et al. 2016, 2019). In this way, the current estimates of the monomer
radius have relied on perspectives on the solar system or the interstellar
medium.

Despite its essential importance in collisional growth, the monomer radius has
not yet been meaningfully constrained by observations of planet-forming disks.
Graham et al. (2007) found that optical scattered light from the debris disk
around AU Mic is highly polarized, hinting at the presence of porous
aggregates of sub-micron monomers; however, the authors mainly focused on the
porosity of the aggregates rather than the size of the monomers. In contrast,
recent observations of younger disks start to question the presence of sub-
micron-sized monomers. The detections of millimeter-wave scattering
polarization seem to imply that collisional growth stalls at a small aggregate
size ($\sim 100\leavevmode\nobreak\ \mu\mathrm{m}$) (Kataoka et al. 2016;
Stephens et al. 2017), which may be attributed to fragile nature of
aggregates, such as due to micron-sized (or even larger) monomers and/or less
sticky materials (Okuzumi & Tazaki 2019; Arakawa & Krijt 2021).

Observations of polarized scattered light from planet-forming disks would be
crucial to clarify the monomer size. The scattering polarization properties of
large porous aggregates have been suggested to strongly reflect the properties
of the monomer, as reported in numerical simulations (West 1991; Kozasa et al.
1993; Lumme et al. 1997; Petrova et al. 2000, 2004; Kimura 2001; Kimura et al.
2006; Bertini et al. 2007; Kolokolova & Kimura 2010; Tazaki et al. 2016; Min
et al. 2016; Halder et al. 2018) and laboratory experiments (Zerull et al.
1993; Gustafson & Kolokolova 1999; Volten et al. 2007). In contrast, Shen et
al. (2009) pointed out that the effect of monomer size on polarization is
negligible for aggregates with a porosity of $\sim 60\%$. However, they only
investigated small monomer sizes, and hence, it is unclear whether their
conclusions remain valid for larger monomers. In the solar system, optical
polarimetry of comets has successfully revealed the presence of sub-micron-
sized monomers in cometary aggregates (Gustafson & Kolokolova 1999; Kimura et
al. 2003, 2006).

This study aims to investigate the monomer size from a new vantage point, that
is, polarimetric observations of planet-forming disks from optical to near-
infrared (IR) wavelengths. To our knowledge, this is the first attempt at
deriving the monomer radius from disk observations. As shown later in this
paper, optical and near-IR wavelengths are the optimal wavelengths for
distinguishing sub-micron- and micron-sized monomers.

Fig. 1 compiles the maximum values of the observed polarization fractions
($P_{\mathrm{max}}^{\mathrm{obs}}$) of several planet-forming disks, where the
polarization fraction means the degree of linear polarization of the observed
polarization signals from the disk. Except for an edge-on debris disk (AU
Mic), we generally have access to light reflected off the disk with a
scattering angle of 90 degrees at which the degree of linear polarization of
aggregates is often maximized (e.g., Kimura et al. 2006; Volten et al. 2007;
Min et al. 2016). Thus, the observed maximum polarization fraction should be
correlated with the maximum degree of polarization of light scattered by each
aggregate ($P_{\mathrm{max}}$) in those disks. We can also say that the
maximum degree of polarization of aggregates must be higher than the observed
maximum polarization fractions ($P_{\mathrm{max}}\gtrsim
P_{\mathrm{max}}^{\mathrm{obs}}$) because disk scattered light usually suffers
a depolarizing effect due to multiple scattering (for an optically thick
disk), line-of-sight integration (for an optically thin edge-on disk, such as
AU Mic), or limited spatial resolution. Considering the presence of such
depolarization effects, we can rule out the aggregate models that yield
$P_{\mathrm{max}}<P_{\mathrm{max}}^{\mathrm{obs}}$.

As the number of disks with quantitative polarimetry data is currently
limited, we shall adopt a working hypothesis that the monomer size and
composition are the same among those disks, but the aggregate size and
porosity may vary from one disk to another. With the hypothesis, aggregates
with a successful monomer model should be capable of producing the degree of
polarization of $\gtrsim 24-40\%$ and $\gtrsim 35-66\%$ at optical and near-IR
wavelengths, respectively, as well as a reddish polarization color, i.e.,
$P_{\mathrm{max}}^{\mathrm{obs}}$(optical)$\lesssim
P_{\mathrm{max}}^{\mathrm{obs}}$(near-IR).

Figure 1: Observed maximum polarization fractions
$P_{\mathrm{max}}^{\mathrm{obs}}$ of several planet-forming disks. References:
HD 142527 (Hunziker et al. 2021), HD 169142 (Tschudi & Schmid 2021), HD 34700
A (Monnier et al. 2019), GG Tau (Silber et al. 2000), AB Aur (Perrin et al.
2009), UX Tau A (Tanii et al. 2012), TW Hya (Poteet et al. 2018), HL Tau
(nebula region) (Murakawa et al. 2008) and AU Mic (Graham et al. 2007).

The main goal of this paper is to clarify what kind of aggregates and monomers
can meet these criteria. To this end, we calculate $P_{\mathrm{max}}$ values
of various kinds of aggregates by performing exact light scattering
simulations. Comparing the obtained $P_{\mathrm{max}}$ values with
$P_{\mathrm{max}}^{\mathrm{obs}}$, we demonstrate that aggregates are likely
consisting of monomers less than or comparable to 0.4$\leavevmode\nobreak\
\mu$m in radius, while a larger monomer model fails to explain the
observations.

The paper is organized as follows. Sect. 2 summarizes our aggregate models and
a method for solving light scattering. The outcomes of the light scattering
simulations are presented in Sect. 3. In Sect. 4, we apply the simulation
results to recent quantitative polarimetric observations of the disk around HD
142527. In Sect. 5, we discuss the results and finally in Sect. 6 we summarize
our findings.

##  2 Models and Methods

Dust coagulation in a planet-forming disk starts with hit-and-stick
collisions, which have often been modeled by ballistic cluster cluster
aggregation (BCCA) or ballistic particle cluster aggregation (BPCA). These
aggregation models tend to produce fluffy aggregates whose porosity is higher
than $\sim 85\%$. As they grow, the aggregates may experience bouncing
collision (Zsom et al. 2010; Lorek et al. 2018), although it is still a matter
of debate if fluffy aggregates are eligible for bouncing collision (Wada et
al. 2011; Seizinger & Kley 2013). If bounce collisions occur, the aggregates
suffer gradual compaction after each collision, and then the porosity will
decrease to $\sim 64\leavevmode\nobreak\ \%$ (Weidling et al. 2009).

With these in mind, we adopt three types of aggregates built from mono-
disperse spherical monomers 111The particle position data is available from B.
T. Draine’s Web site https://www.astro.princeton.edu/~draine/agglom.html.. The
first aggregation model is BPCA, where a single monomer particle is shot one-
by-one to the target aggregate. BPCA clusters typically have a fractal
dimension of $\sim 3$ and a porosity of $85$–$87\%$ at $N\geq 32$, where $N$
is the number of monomers. Since the degrees of polarization of BCCA and BPCA
clusters are more or less similar (Kolokolova et al. 2006; Kimura et al.
2006), we focus only on BPCA as a representative of hit-and-stick
agglomerates. To investigate aggregates with lower porosity, we also consider
aggregation models called BAM1 and BAM2 (Shen et al. 2008). BAM1 and BAM2 are
a modified version of BPCA. The first contact point between an arriving
monomer and the target aggregate was found similarly to BPCA, but BAM1 and
BAM2 allow the newly attached monomer to roll over the aggregate surface to
find the second and third contact points, respectively. In this way, the
resultant aggregates become less porous than BPCA clusters, i.e., a porosity
of $\sim 69$–$78\%$ for BAM1 and $\sim 47$–$68\%$ for BAM2 at $N\geq 32$. The
BAM2 clusters have a similar porosity to compressed aggregates with multiple
bouncing collisions (Weidling et al. 2009). The aggregate shapes are shown in
Fig. 2.

To measure the size of an aggregate, we use the volume-equivalent radius
$R_{\mathrm{V}}=R_{\mathrm{0}}N^{1/3}$, where $R_{\mathrm{0}}$ is the monomer
radius. For example, if two aggregates have the same $R_{\mathrm{V}}$ value,
they have the same material volume. As the volume-equivalent radius usually
does not represent an apparent size of an aggregate, we also calculate the
characteristic radius $R_{\mathrm{c}}=\sqrt{5/3}R_{g}$ as a measure of its
apparent size, where $R_{g}$ is the radius of gyration (Kozasa et al. 1992;
Mukai et al. 1992). Namely, $R_{\mathrm{V}}$ and $R_{\mathrm{c}}$ specify the
mass and apparent size of an aggregate, respectively. The size parameters of
the monomer and the aggregate are defined by $x_{\mathrm{0}}=2\pi
R_{\mathrm{0}}/\lambda$ and $x_{\mathrm{c}}=2\pi R_{\mathrm{c}}/\lambda$,
respectively, where $\lambda$ is a wavelength. Also, to measure the porosity
of an aggregate, we adopt a commonly used definition:
$\mathcal{P}=1-(R_{\mathrm{V}}/R_{\mathrm{c}})^{3}$.

This study adopts three different monomer radii:
$R_{\mathrm{0}}=0.1\leavevmode\nobreak\ \mu$m, $0.2\leavevmode\nobreak\ \mu$m,
and $0.4\leavevmode\nobreak\ \mu$m. The number of monomers is set as
$N=2^{i}$, where $i$ runs from $3$ to $i_{\mathrm{max}}$ with an increment of
one, and $i_{\mathrm{max}}=12$, $9$, $6$ for
$R_{\mathrm{0}}=0.1\leavevmode\nobreak\ \mu$m, $0.2\leavevmode\nobreak\ \mu$m,
and $0.4\leavevmode\nobreak\ \mu$m, respectively. Thus, the largest aggregates
we studied have $R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m, corresponding
to $R_{\mathrm{c}}=2.3$, $2.6$, and $3.1\leavevmode\nobreak\ \mu$m for BAM2,
BAM1, and BPCA clusters with $R_{\mathrm{0}}=0.1\leavevmode\nobreak\ \mu$m,
respectively. These sizes are large enough to study aggregates at the disk
surfaces probed by optical and near-IR observations, as larger aggregates tend
to settle down below the disk scattering surfaces.

As a monomer composition, we consider a mixture of pyroxene silicate
(Mg0.7Fe0.3SiO3) (Dorschner et al. 1995), water ice (Warren & Brandt 2008),
carbonaceous material (either organics or amorphous carbon), and troilite
(Henning & Stognienko 1996) with the mass fractions provided by Birnstiel et
al. (2018). The mass abundance of water ice adopted in Birnstiel et al. (2018)
was reduced from its base models (Pollack et al. 1994; D’Alessio et al. 2001).
Such a reduced water-ice abundance is in line with recent modelings of water-
ice features arising from the surface regions of the disks (Tazaki et al.
2021; Betti et al. 2022). Since the actual form of carbonaceous material is
rather uncertain, we also consider two carbonaceous materials: one is organics
(Henning & Stognienko 1996), and the other one is amorphous carbon (Zubko et
al. 1996).

We derive an effective refractive index for a mixture of the four materials
using the Bruggeman mixing rule. The resultant optical constants are
summarized in Table 1. The monomers comprising organics have refractive
indices lower than those comprising amorphous carbon. We will refer to each
model as the org model and amc model. To specify the monomer size and
composition, we will use a model name in a form of COMP-SIZE, where COMP and
SIZE specify the monomer composition (org or amc) and radius in unit of nm
(100, 200, or 400), respectively. For example, amc-200 indicates a monomer
with the amc composition and a radius of 200 nm. We also tested the effect of
ice-mantled monomers on the polarization characteristics of aggregates and
found no significant differences between ice-mantled and fully mixed cases.
Thus, in the following, we only consider homogeneous monomers for simplicity.

Table 1: Real ($n$) and imaginary ($k$) parts of the refractive index of each monomer made of a mixture of silicate, water ice, carbonaceous (organics or amorphous carbon), and troilite. model | org |  | amc
---|---|---|---
$\lambda$ ($\mu$m) | $n$ | $k$ |  | $n$ | $k$
3.78 | 1.53 | 0.0219 |  | 2.13 | 0.393
2.18 | 1.47 | 0.0134 |  | 1.98 | 0.385
1.63 | 1.48 | 0.0138 |  | 1.92 | 0.404
1.25 | 1.49 | 0.0104 |  | 1.86 | 0.420
1.04 | 1.49 | 0.0108 |  | 1.81 | 0.434
0.735 | 1.50 | 0.0119 |  | 1.70 | 0.468
0.554 | 1.51 | 0.0138 |  | 1.59 | 0.472

To solve light scattering by dust aggregates, we use an exact numerical
technique so-called the cluster $T$-matrix method (Mackowski & Mishchenko
1996, and references therein). We use a publicly available code for the
cluster $T$-matrix method MSTM v3.0 (Mackowski & Mishchenko 2011). The great
advantage of the $T$-matrix approach is its analyticity of mathematical
formulation, which allows analytical orientation averaging. All cluster
$T$-matrix simulations presented in this paper adopt analytical orientation
averaging. The results are also averaged over 4 realizations for each
aggregate model. Because the degree of linear polarization does not vary
significantly between realizations, increasing the number of realizations
furthermore would not change our conclusions.

##  3 Results

We present the results of cluster $T$-matrix simulations. First of all, we
study the effect of monomer size on the degree of linear polarization of light
scattered by aggregates in Sect. 3.1. In Sect. 3.2, we summarize the results
of our comprehensive parameter survey on the polarization properties, which
will then be compared with the observed polarization fractions in Sect. 3.3 to
derive the monomer radius in planet-forming disks.

Figure 2: BPCA (upper), BAM1 (middle), and BAM2 (lower) clusters with the
volume-equivalent radius of $R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m.
From left to right, the number of monomers and the monomer radius ($N$,
$R_{\mathrm{0}}$) are ($64$, 400 nm), ($512$, 200 nm), and ($4096$, 100 nm),
respectively.

Figure 3: Effect of monomer size for aggregates with a porosity of $\sim
87\%$. The upper panel shows the degree of linear polarization of light
scattered by aggregates with the org composition at
$\lambda=1.04\leavevmode\nobreak\ \mu$m. The blue, orange, and green lines are
the results for $R_{\mathrm{0}}=100$, $200$, $400$ nm, respectively. The lower
panel shows the wavelength dependence of the maximum polarization. The
observed maximum polarization fractions are plotted as lower limits, since the
amount of depolarization that the observed polarized light suffered is
unknown.

###  3.1 The effect of monomer size on scattering polarization

When we attempt to derive the monomer radius from polarized disk scattered
light, one of the fundamental questions is: If the size, porosity, and
composition (refractive index) of large aggregates ($x_{\mathrm{c}}>1$) are
the same, how does the degree of linear polarization depend on the monomer
radius? To answer this, we first consider BPCA clusters with the org
composition and $R_{\mathrm{c}}=3.1\leavevmode\nobreak\ \mu$m, as shown in
Fig. 2 (upper panel).

To assess the effect of monomer radius on scattering polarization, we need to
fix the parameters of aggregates other than the monomer radius, such as
porosity and characteristic radius. Since BPCA clusters have a fractal
dimension of 3, the characteristic radius scales as $N^{1/3}$. To fix
$R_{\mathrm{c}}$, the number of monomers should be decreased by a factor of 8
if the monomer radius is increased by a factor of 2. Therefore, we consider
three aggregates having the following sets of parameters ($N$,
$R_{\mathrm{0}}$)=($4096$, 100 nm), ($512$, 200 nm), and ($64$, 400 nm). All
of them have nearly the same characteristic radii and porosities
($R_{\mathrm{c}}$, $\mathcal{P}$)=($3.14\leavevmode\nobreak\ \mu$m, $86.8\%$),
($3.12\leavevmode\nobreak\ \mu$m, $86.4\%$), and ($3.12\leavevmode\nobreak\
\mu$m, $86.5\%)$ for $R_{\mathrm{0}}=100$, $200$, and $400$ nm, respectively.
Furthermore, the three aggregates have the same material volume, as they have
$R_{\mathrm{V}}=R_{\mathrm{0}}N^{1/3}=1.6\leavevmode\nobreak\ \mu$m.

Fig. 3 shows the degree of linear polarization of scattered light for
unpolarized incoming light ($-S_{12}/S_{11}$) for the three aggregates, where
$S_{ij}$ represents a scattering matrix element (Bohren & Huffman 1983). Since
we have fixed the parameters other than the monomer radius, the difference in
the polarization can be attributed to a monomer-size effect.

Fig. 3 (upper panel) shows the angular dependence of polarization at
$\lambda=1.04\leavevmode\nobreak\ \mu$m. At this wavelength, the size
parameters of the three aggregates are $x_{\mathrm{c}}\simeq 19$, and the size
parameters of the monomers are $x_{\mathrm{0}}=0.6$, $1.2$, and $2.4$ for
$R_{\mathrm{0}}=100$, $200$, and $400$ nm, respectively. For
$x_{\mathrm{0}}=0.6$ ($R_{\mathrm{0}}=100$ nm), the polarization curve appears
to be very similar to that of Rayleigh scattering despite the large aggregate
size ($x_{\mathrm{c}}\gg 1$). In contrast, for $x_{\mathrm{0}}>1$
($R_{\mathrm{0}}=200$ and $400$ nm), the degree of polarization drops rapidly
and eventually oscillates around zero. In this way, the monomer radius has a
significant impact on the degree of polarization of aggregates.

Fig. 3 (lower panel) shows the wavelength dependence of the maximum degree of
polarization $P_{\mathrm{max}}$, which is defined as the maximum value of
$-S_{12}/S_{11}$ for all scattering angles. For the case of Fig. 3 (upper
panel), $P_{\mathrm{max}}$ are $89\%$, 73%, 9.7% for $R_{\mathrm{0}}=100$ nm,
200 nm, and 400 nm, respectively. The maximum polarization is found to be
almost monotonically decreasing with decreasing $\lambda$ for all cases. At
$\lambda=3.78\leavevmode\nobreak\ \mu$m, the size parameters of the three
monomers are all lie in $x_{\mathrm{0}}<0.67$. In this case, the difference in
$P_{\mathrm{max}}$ between different monomer sizes is less than $5\%$ points,
and thus, changing the monomer radius does not result in changing the maximum
polarization. The polarization for $R_{\mathrm{0}}=400$ nm starts to decrease
at $\lambda=2.18\leavevmode\nobreak\ \mu$m, which corresponds to
$x_{\mathrm{0}}\simeq 1.2$. The difference in $P_{\mathrm{max}}$ between
different monomer sizes now reaches $\sim 10\%$ points, and becomes more
prominent for a shorter wavelength. Therefore, the effect of monomer size
seems to appear only when the monomer is sufficiently large, i.e.,
$x_{\mathrm{0}}\gtrsim 1$.

Fig. 3 suggests that the aggregates consisting of 400-nm monomers fail to
explain a high observed polarization fraction at $\lambda\lesssim
1.25\leavevmode\nobreak\ \mu$m for several disks. Although a more detailed
parameter study and the comparison with observations will be presented later
(see Sects. 3.2 and 3.3), this result already hints at the presence of
relatively small monomers in disks.

Figure 4: Effect of monomer size for the aggregates with a porosity of $\sim
59\%$. The orange and green lines are the results for $R_{\mathrm{0}}=200$ nm
(BAM2 clusters of 128 monomers) and 400 nm (BAM1 clusters of 16 monomers),
respectively. The porosity of each aggregate is indicated in the legend. The
inset panel shows the angular dependence of the degree of polarization
($-S_{12}/S_{11}$) at two wavelengths (solid lines:
$\lambda=1.63\leavevmode\nobreak\ \mu$m, dotted lines:
$\lambda=1.04\leavevmode\nobreak\ \mu$m).

Shen et al. (2009) claimed that the monomer size has negligible effect on
polarization for moderately porous aggregates ($\mathcal{P}\approx 60\%$),
although they only considered relatively small monomers
($x_{\mathrm{0}}\lesssim 1.6$). To compare our results with theirs, we
selected two aggregates with a porosity of $\sim 60\%$ as follows. The first
selected aggregate is BAM2 clusters with 128 monomers with a radius of 200 nm.
The other aggregate is BAM1 clusters with 16 monomers with a radius of 400 nm.
The two aggregates have the same material volume
($R_{\mathrm{V}}=R_{\mathrm{0}}N^{1/3}\simeq 1.01\leavevmode\nobreak\ \mu$m).
Their characteristic radii and porosities are also very similar:
($R_{\mathrm{c}}$,$\mathcal{P}$)=($1.35\leavevmode\nobreak\ \mu$m, $58.4\%$)
and ($1.37\leavevmode\nobreak\ \mu$m, $60.1\%$). Although there is still a
subtle difference in porosity, we think that such a tiny difference does not
significantly alter our conclusions.

Fig. 4 shows the degree of polarization for the two low-porosity aggregates
($\sim 59\%$). We found that the wavelength dependence of $P_{\mathrm{max}}$
is surprisingly similar between the two when $\lambda\gtrsim
1.63\leavevmode\nobreak\ \mu$m, which corresponds to $x_{\mathrm{0}}\lesssim
1.54$. Thus, it appears to reaffirm the conclusion made by Shen et al. (2009).
However, we also found a significant difference in $P_{\mathrm{max}}$ at
$\lambda\lesssim 1.25\leavevmode\nobreak\ \mu$m, which corresponds to
$x_{\mathrm{0}}\gtrsim 2.01$. Therefore, a manifestation of the monomer size
effect needs a slightly larger $x_{\mathrm{0}}$ value for low-porosity
aggregates, that is, $x_{\mathrm{0}}\gtrsim 2$. In other words, the absence of
the monomer size effect in Shen et al. (2009) is likely due to its limited
range of size parameters ($x_{\mathrm{0}}\lesssim 1.6$).

To summarize, even if we fix the aggregate size, porosity, and composition,
the degree of polarization depends sensitively on the monomer size unless
$x_{\mathrm{0}}\lesssim 1$–$2$. This tendency has been confirmed for a wide
range of aggregate porosity ($59$–$87\%$). Conversely, if aggregates consist
of small monomers such that $x_{\mathrm{0}}\lesssim 1$, the polarization is
insensitive to the monomer radius and is determined by the porosity, size, and
composition of aggregates (Fig. 4, see also Shen et al. 2009). This is in
agreement with the findings from laboratory experiments (Zerull et al. 1993;
Gustafson & Kolokolova 1999). Our results suggest that, if we try to
distinguish between sub-micron- and micron-sized monomers, the optical and
near-IR wavelengths are the optimal wavelengths, because the monomer size
parameter is close to or larger than unity at these wavelengths.

###  3.2 The maximum polarization for various aggregates

Figure 5: Wavelength dependence of $P_{\mathrm{max}}$ for BPCA with the org
(upper panels) and amc (lower panels) compositions. From left to right: the
monomer radius of the aggregates is $R_{\mathrm{0}}=100$, $200$, and $400$ nm,
respectively. The gray solid line in each panel represents $P_{\mathrm{max}}$
for a single spherical monomer.

We showed that the monomer size is important for the polarization
characteristics of aggregates. Next, we study how the maximum polarization
$P_{\mathrm{max}}$ depends on aggregate radius, porosity, composition, and
monomer radius.

Fig. 5 shows the effect of aggregate radius on the maximum polarization for
the case of BPCA. A similar plot for BAM1 and BAM2 is available in Appendix A.
Apart from the porosity effect on the maximum polarization, the overall
dependencies for BAM1 and BAM2 are similar to the case of BPCA. Fig. 6 shows
the effect of porosity on the maximum polarization. In those figures, we also
plot the observed maximum polarization as a lower limit because the amount of
depolarization that the observed polarized light suffered is unknown. To
estimate $P_{\mathrm{max}}$ values from $P_{\mathrm{max}}^{\mathrm{obs}}$, we
need a more detailed modeling, i.e., radiative transfer calculations for each
disk, which is beyond the scope of this paper.

The key dependencies seen in Figs. 5 and 6 are summarized below.

#### Dependence on aggregate radius

Each panel of Fig. 5 shows the wavelength dependence of $P_{\mathrm{max}}$ for
various aggregate radii. For comparison, we also plot the wavelength
dependence for the single monomer (gray solid line). For org-100, each monomer
obeys Rayleigh scattering, and the maximum polarization is nearly 100% at all
wavelengths. As with a single monomer, the maximum polarization of aggregates
remains high at all wavelengths even if $R_{\mathrm{V}}$ increases to
$1.6\leavevmode\nobreak\ \mu$m (the characteristic radius of
3.1$\leavevmode\nobreak\ \mu$m). In contrast, for org-200 and -400, as
$P_{\mathrm{max}}$ of the monomer drops, $P_{\mathrm{max}}$ of the aggregate
does as well. The above results nicely illustrate the fact that the
polarization for aggregates tends to reflect the polarization characteristics
of the monomers, as known since the early 1990s (West 1991; Kozasa et al.
1993).

The resemblance of the polarization between aggregates and monomers is
physically reasonable as multiple scattering is often subdominant (not
negligible though) for highly porous aggregates ($x_{\mathrm{c}}>1$ and
$x_{\mathrm{0}}\ll 1$) (Berry & Percival 1986; Botet et al. 1997; Tazaki et
al. 2016; Tazaki & Tanaka 2018). In the single scattering limit, every nonzero
scattering matrix element is equally affected due to the constructive and
destructive interference of light waves from aggregates. Consequently, each
scattering matrix element does depend on the aggregate structure, but the
ratio of them, i.e., the degree of linear polarization, is independent of the
aggregate structure and size and is solely determined by the properties of
monomers (see Eq. (9) in Tazaki et al. 2016).

The deviation of $P_{\mathrm{max}}$ of the aggregates from that of the monomer
is due to multiple scattering within each aggregate. Larger aggregates provide
a higher probability of multiple scattering, resulting in lower
$P_{\mathrm{max}}$ values. However, the dependence on the aggregate radius is
generally weak compared to that of the monomer radius. For example, even if we
increase $R_{\mathrm{V}}$ by more than an order of magnitude (from
$0.1\leavevmode\nobreak\ \mu$m to $1.6\leavevmode\nobreak\ \mu$m), the
reduction of $P_{\mathrm{max}}$ is less than a factor of two, whereas
increasing the monomer radius only by a factor of two considerably affects the
maximum polarization (Fig. 3). Such weak aggregate size dependencies have also
been seen in previous numerical simulations (Kozasa et al. 1993; Kimura 2001;
Kimura et al. 2003, 2006; Petrova et al. 2004; Bertini et al. 2007) and micro-
wave analog experiments (Zerull et al. 1993; Gustafson & Kolokolova 1999).

#### Dependence on monomer composition

Top and bottom panels of Fig. 5 show the maximum polarization for aggregates
made of the org and amc composition, respectively. In general, the monomer
composition has a strong impact on the maximum polarization.

For all monomer radii, the maximum polarizations of the amc model are lower
and higher than those of the org model at near-IR and optical wavelengths,
respectively. Higher $n$ values or lower $k$ values tend to decrease the
maximum polarization of aggregates (Petrova et al. 2004; Kimura et al. 2006;
Lumme & Penttilä 2011), because such a change in refractive index tends to
induce more multiple scattering within each aggregate. As shown in Table 1,
the amc model has a higher refractive index in both the real and imaginary
parts than the org model. Because of the higher real part, $P_{\mathrm{max}}$
is smaller at near-infrared wavelengths, and because of the higher imaginary
part, $P_{\mathrm{max}}$ is larger at optical wavelengths. As a result, the
wavelength dependence of the maximum polarization of the amc model is
shallower than that of the org model.

For $R_{\mathrm{0}}=400$ nm, the wavelength dependence is complex. For both
compositions, the maximum polarization drops rapidly at $\lambda\sim
1-2\leavevmode\nobreak\ \mu$m. This drop in polarization is triggered by a
rapid reduction in the degree of polarization of the monomer itself (see gray
line), not by multiple scattering within the aggregate. We initially suspected
that such a sharp drop is an artifact arising from the assumption of perfectly
spherical monomers in our aggregate models and might be mitigated by
considering more realistic, nonspherical monomers. However, comparing the
degree of polarization of spherical and nonspherical particles, we found that
a similar sharp drop can occur for nonspherical particles as well (see
Appendix B for more detailed discussion). Thus, we suppose that the sharp drop
is a realistic property, although a more detailed study on the polarization of
aggregates of nonspherical monomers is desirable.

#### Dependence on aggregate porosity

Figure 6: Effect of porosity on the maximum polarization for the aggregates
with $R_{\mathrm{c}}=2.0\leavevmode\nobreak\ \mu$m with the org (upper panels)
and amc (lower panels) compositions. From left to right: the monomer radius of
the aggregates is $R_{\mathrm{0}}=100$, $200$, and $400$ nm, respectively. The
yellow, brown, and cyan lines represent the results for BPCA, BAM1, and BAM2.
The porosity of each aggregate model, $\mathcal{P}$, is indicated in each
legend.

Fig. 6 shows the effect of porosity on the maximum polarization. To isolate
the effect of porosity, we fixed the characteristic radius of aggregates to
$R_{\mathrm{c}}=2.0\leavevmode\nobreak\ \mu$m by interpolating the simulation
data.

For the org case, the maximum polarization monotonically decreases with
decreasing $\lambda$ for all porosities. Decreasing the porosity also results
in decreasing the maximum polarization, because a denser aggregate results in
more multiple scattering (Kolokolova & Kimura 2010). For the amc case, lower
porosity aggregates (BAM1 and BAM2) exhibit an inverted wavelength dependence
of the polarization: increasing $P_{\mathrm{max}}$ with decreasing $\lambda$.
Shen et al. (2009) also found a similar inverted trend for BAM2. Because of
this inverted trend, the maximum polarization of BAM2 will be similar to or
even slightly higher than that of BPCA at optical wavelengths.

We interpret this inverted behavior as follows. The wavelength at which the
inversion sets in found to be the wavelength at which the radiation field in
the aggregate begins to be non-uniform. The non-uniform radiation field is due
to attenuation of the incoming radiation at the surface of the aggregate. With
decreasing wavelength from this, the successive orders of scattering get fewer
(less multiple scattering), which results in increasing $P_{\mathrm{max}}$.
The above interpretation is further augmented by the following points. We
confirmed that the radiation field for org aggregates is almost uniform for
all porosities and wavelengths we investigated, and therefore they are not
eligible for inversion. Although BPCA clusters with amc composition exhibit a
non-uniform radiation field, they are less efficient multiple scatterers in
the first place, and hence the inversion effect does not appear clearly.

Another intriguing feature seen in Fig. 6 is that the porosity dependence
diminishes for shorter wavelengths for org aggregates. This effect is
reminiscent of the neighborhood effect (Kimura & Mann 2004; Kolokolova &
Kimura 2010); the degree of polarization is characterized by a local structure
of a length scale of $\sim\lambda/2\pi$ or so within an aggregate. Once the
size parameter of the monomers exceeds around unity, the electromagnetic
interactions between neighboring monomers, rather than the overall aggregate
structure, characterize the degree of polarization. In our cases, the porosity
dependence diminishes for $\lambda\leq 0.735\leavevmode\nobreak\ \mu$m for
$R_{\mathrm{0}}=200$ nm and $\leq 1.25\leavevmode\nobreak\ \mu$m for
$R_{\mathrm{0}}=400$ nm, which translate into $x_{\mathrm{0}}\geq 1.71$ and
$2.01$, respectively. In other words, the diminished porosity dependence
occurs when the size parameter of monomers $x_{\mathrm{0}}\gtrsim 1.71-2.01$
for the org composition.

Kolokolova & Kimura (2010) point out that the neighborhood effect appears
easily for more absorbing aggregates. We confirmed that the converging
behavior already starts to appear for amc-100 at optical wavelengths. However,
the convergent behavior is missing for amc-200 and -400. We speculate that
this is perhaps due to (i) artifacts arising from ripple patterns in
polarization curves and/or (ii) different coordination number of monomers,
although further clarification seems difficult within the limit of our
simulation data. As discussed in Appendix B, strong ripple patterns were
confirmed in the polarization curves for amc-200 and -400 at an optical
wavelength, while it is absent for org-200 and -400. Such a ripple pattern
would not be present for realistic monomers, as the monomers must not be mono-
disperse and perfectly spherical particles, and hence, their poly-dispersity
and non-sphericity will smear them out. More detailed light scattering
simulations taking into account aggregates of nonspherical monomers is
necessary to clarify this point.

###  3.3 How large are the monomers in planet-forming disks?

In Sects. 3.1 and 3.2, we investigated the wavelength dependence of the
maximum polarization for various aggregates and clarified the impact of the
monomer size on the polarization. In particular, as we discussed in Sect. 3.1,
the optical and near-IR wavelengths are the optimal wavelengths for
distinguishing sub-micron- and micron-sized monomers because the size
parameter of the monomers becomes close to or larger than unity at these
wavelengths. Comparing the simulation results with the observed maximum
polarization fraction, i.e., Fig. 1, we can assess the monomer radius of
aggregates in planet-forming disks for the first time.

Currently available disk measurements favor the presence of relatively small
monomers. As shown in Figs. 5 and 6, the aggregates with monomers of
$R_{\mathrm{0}}=400$ nm fail to reproduce the combination of a high
polarization fraction and a reddish polarization color at optical and near-IR
wavelengths regardless of the aggregate size, composition, and porosity. For
the nebula scattered light around HL Tau, the observed maximum polarization
fractions may be marginally explained by aggregates with amc-400 unless the
observed signals are depolarized by multiple scattering. This is in harmony
with the findings of Murakawa et al. (2008). Using a one-dimensional single
scattering model with the Mie theory, Murakawa et al. (2008) modeled polarized
scattered light from the HL Tau nebulosity and then estimated the maximum
grain radius to be $0.4\leavevmode\nobreak\ \mu$m. Since we have shown that
the degree of polarization of aggregates reflects the monomer property, the
maximum grain radius they derived may be interpreted as the monomer radius
rather than the aggregate size.

Monomers larger than $R_{\mathrm{0}}=400$ nm would be unfavorable because
their polarization will be even lower and/or bluish at an optical wavelength.
As a result, we arrive at the conclusion that the monomers of aggregates in
planet-forming disks are likely no greater than $400$ nm
($=0.4\leavevmode\nobreak\ \mu$m). The inferred monomer size is similar to the
one seen in cometary dust aggregates and the maximum size of interstellar
grains (see Sect. 5.1).

Although a monomer radius of 100 or 200 nm is consistent with the
observations, we cannot exclude the presence of monomers much smaller than 100
nm. This point will be clarified by studying the presence/absence of the
variation in $P_{\mathrm{max}}^{\mathrm{obs}}$ at optical wavelengths across
various disks. In Fig. 6, we showed that the maximum polarization is
insensitive to aggregate porosity when $x_{\mathrm{0}}\gtrsim 1.71$–$2.01$ at
least for the org composition. This may imply that if aggregate porosity is
diverse from one disk to another, the near-IR maximum polarization fractions
should vary accordingly, but such a variation is suppressed for optical
wavelengths when $R_{\mathrm{0}}>100$ nm. Conversely, if the variation at the
optical wavelength is similar to the near-IR counterpart, the monomer radius
is likely less than 100 nm. This strategy of deriving the monomer radius has
been employed in the cometary field (Gustafson & Kolokolova 1999; Kimura et
al. 2003, 2006).

Fig. 1 shows that the variation in optical and near-IR polarizations are
around $\sim 16\%$ points and $\sim 30\%$ points, hinting at less variation
for shorter wavelengths, and this may indicate the presence of monomers larger
than 100 nm. However, the number of current observational data points is
rather limited, and therefore the time is not yet ripe to draw a robust
conclusion about the lower size limit. A future survey on optical polarization
will be crucial to strengthen this argument.

Fig. 1 also points to a red polarization color. Within the limits of our
composition models, dust aggregates in disks are unlikely to be dark and dense
aggregates, i.e., BAM2 with amc composition, because such aggregates tend to
exhibit an inverted, or blueish, wavelength dependence. In other words, such
an inverted trend is smoking gun evidence of dark and dense aggregates (or can
be dark monolithic grains). However, it is worth keeping in mind that the
wavelength dependence could be affected by the assumption of a refractive
index. If one allows more conducting materials (i.e., amorphous carbon or
metals) or less dielectric materials (i.e., water ice), $|m|$ and
$d|m|/d\lambda$ will be larger than our amc model. In this case, the inverted
trend might be alleviated, although it is unclear whether such a composition
model is reasonable for protoplanetary dust. In any case, the wavelength
dependence of polarization is useful for diagnosing not only the monomer
radius but also the monomer composition.

##  4 Application to the HD 142527 disk

Our argument so far has been based on the premise of the working hypothesis,
that is, the monomer radius and its composition are the same for various
disks. However, this may not be always true, and hence, each aggregate model
does not necessarily satisfy all lower limits shown in Figs. 5 and 6.
Therefore, multiwavelength quantitative polarimetry for a specific planet-
forming disk is crucial for a monomer size study. Here, as a case study, we
focus on the disk around HD 142527, because polarization measurements have
been made with the widest wavelength coverage among the disks shown in Fig. 1.

HD 142527 is a binary star surrounded by a large circumbinary disk, whose
optical/near-IR scattered light has been extensively investigated (Fukagawa et
al. 2006; Rameau et al. 2012; Avenhaus et al. 2014, 2017; Rodigas et al. 2014;
Hunziker et al. 2021). Hunziker et al. (2021) found that a high polarization
fraction toward this disk, hinting at aggregate nature of dust particles.
Tazaki et al. (2021) found that the observed disk color is consistent with
dust particles of a radius of $\sim 3\leavevmode\nobreak\ \mu$m. Also, the
reddish scattered light is more likely explained by compact aggregates222The
meaning of ‘compact aggregates’ here is those having a higher fractal
dimension (see Fig. 10 in Tazaki et al. 2019). Thus, this statement does not
necessarily mean low porosity aggregates. For example, BPCA clusters have a
fractal dimension of 3 and are thought to be eligible for producing a reddish
color, but their porosity of $\sim 85\%$ is still a class of highly porous.
rather than fractal aggregates (Min et al. 2016; Tazaki et al. 2019). From
these results, compact dust aggregates of $\sim 3\leavevmode\nobreak\
\mu\mathrm{m}$ in radius are thought to be responsible for the scattered light
of the disk. Such aggregates are quite similar to those presented in Fig. 2,
and therefore we may apply our simulation results to interpret the
observations.

Hunziker et al. (2021) conducted a quantitative polarimetric measurement of
the disk around HD 142527 at optical (the VBB) and near-IR (the $H$-band)
wavelengths. They accurately derived disk polarization fractions at the far
side: $28.0\pm 0.9\%$ at the VBB and $35.1\pm 2.1\%$ at the $H$-band. They
also argued that the maximum polarization fraction may be as high as $30\pm
5\%$ for the VBB and $40\pm 10\%$ for the $H$-band. In this study, we adopt
$28.0\pm 0.9\%$ and $35.1\pm 2.1\%$ for the maximum polarization fractions at
optical and near-IR wavelengths for this disk because the disk far side is
close to a scattering angle of 90 degrees at which the degree of polarization
is often maximized (see Fig. 3) and better accuracy of the data.

Since the observed polarization fractions suffer a depolarization effect due
to multiple scattering at the disk surface, we converted the observed maximum
polarization fraction to $P_{\mathrm{max}}$ by using a simple plane-parallel
model developed by Ma & Schmid (2022). The depolarization effect depends on
the single scattering albedo of dust aggregates. The single scattering albedo
of our aggregates with $R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m ranges
from 0.5–0.95. With this range, the plane-parallel model predicts that
$P_{\mathrm{max}}\sim 42-86\%$ for the VBB and $51-100\%$ for the $H$-band.
The estimated range agrees with the $P_{\mathrm{max}}$ values estimated in
Hunziker et al. (2021), where they derived
$P_{\mathrm{max}}=50\leavevmode\nobreak\ \%$ and $70\leavevmode\nobreak\ \%$
at VBB and the $H$-band, respectively.

Fig. 7 summarizes $P_{\mathrm{max}}$ values at the VBB and the $H$-band for
various aggregate porosities, monomer radii, and compositions. We assumed the
aggregate radius of $R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m, and then
characteristic radii are around $2.3$–$3.1\leavevmode\nobreak\ \mu$m, which
approximately agree with the sizes inferred from a previous modeling (Tazaki
et al. 2021). The inferred observational range of $P_{\mathrm{max}}$ is shown
with the gray-hatched square in the plot. The inferred range falls onto the
region where $R_{\mathrm{0}}=100$–$200$ nm, while aggregates with
$R_{\mathrm{0}}=400$ nm produce either too small optical or near-IR
polarization fraction. One may wonder if an intermediate composition between
amc and org yields a closer fit to the observations. However, we found that
even if we consider an intermediate composition, the large monomer still fails
to reproduce the estimated range of maximum polarization (Appendix C).

Therefore, the wavelength dependence of the maximum polarization fractions of
the HD 142527 disk is consistent with large aggregates ($\sim
3\leavevmode\nobreak\ \mu$m) with relatively small monomers
($R_{\mathrm{0}}=100$–$200$ nm). In particular, the $P_{\mathrm{max}}$ values
estimated by Hunziker et al. (2021) are similar to the model of BPCA clusters
with amc–200 monomers or BAM2 clusters with org–100 monomers (Fig. 8). To
distinguish between the two possibilities, a further detailed study is
necessary, which is left for our future tasks.

Figure 7: Maximum polarization of dust aggregates at the VBB (optical) and the
$H$-band (near-IR) with $R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m.
Different line styles correspond to different aggregate models (BPCA: solid,
BAM1: dashed, BAM2: dotted lines), and different symbols correspond to
different composition models (circles: org , squares: amc). The number beside
each point denotes the monomer radius in unit of $\mu$m. The dark gray hatched
area is the maximum polarization estimated by Hunziker et al. (2021), while
the light gray hatched area is the estimated $P_{\mathrm{max}}$ values for an
assumed range of the single scattering albedo of $0.5$–$0.95$. Figure 8:
Wavelength dependence of the maximum polarization toward the disk around HD
142527. The orange and blue lines show BPCA clusters with amc-200 monomers and
BAM2 clusters with org-100 monomers, respectively. Both aggregate models have
$R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m.

##  5 Discussion

###  5.1 Comparison with previous estimates on the monomer radius

We have demonstrated that the monomer radius is likely no greater than 400 nm
to explain the observed polarization fractions. We are aware, however, that,
our results may be biased toward aggregates with smaller monomers because
aggregates with larger monomers, if present, may sink below the disk
scattering surface probed by optical and near-IR observations. This issue is
inevitable as the younger disks are generally optically thick at shorter
wavelengths.

One possibility to overcome this issue is to observe debris disks, which are
optically thin. Graham et al. (2007) showed that scattered light around AU Mic
may be explained by fluffy, presumably primordial, aggregates. At a wavelength
of $\lambda=0.590\leavevmode\nobreak\ \mu$m, the observed polarization
fraction is as high as $0.4$, and the authors estimated the maximum
polarization of each aggregate to be $P_{\mathrm{max}}>0.5$. Although a high
polarization at an optical wavelength could be explained by large and dark
compact aggregates (see Fig. 6 lower, right panel), these aggregates tend to
make the disk color reddish (Min et al. 2016), while the fluffy aggregates
makes it gray or blueish (Tazaki et al. 2019). Given bluish scattered light
around AU Mic (Fitzgerald et al. 2007), fluffy aggregates with sub-micron
monomers seems favorable. A similarity of the monomer size between optically
thick younger disks and a debris disk lends further credibility to that sub-
micron-sized monomers are a good representative of a dominant building block
of aggregates.

It is worth mentioning that there are some debris disks that show
qualitatively different scattering polarization properties from the AU Mic
disk, such as HR 4796 A. For HR 4796 A, the total intensity phase function
exhibits enhanced back scattering (Milli et al. 2017), and the degree of
polarization peaks at a scattering angle of $40$–$50$ degrees (Perrin et al.
2015; Arriaga et al. 2020). None of our simulation results satisfies these
properties. However, very large ($\gtrsim 100\leavevmode\nobreak\
\mu\mathrm{m}$), densely packed aggregates of sub-micron- and micron-sized
monomers may explain such observations (Markkanen et al. 2018). Thus, light
scattering simulations for very large aggregates would be crucial to constrain
the monomer sizes for such disks.

The monomer sizes inferred in this study are smaller than those needed to
explain the disk observations of millimeter-wave scattering polarization
($R_{\mathrm{0}}\gtrsim\mu$m) (Okuzumi & Tazaki 2019; Arakawa & Krijt 2021)
(see Sect. 5.2). On the other hand, the inferred sizes are reminiscent of the
maximum sizes of interstellar grains or typical subunits sizes seen in solar
system dust aggregates. For the interstellar grain model, the maximum grain
radius of $250$ nm has been often employed (Mathis et al. 1977; Draine & Lee
1984). A more recent model also adopts a grain size distribution peaked at
$\sim 200$ nm (Jones et al. 2013).

In the solar system, the chondritic porous interplanetary dust particle (CP
IDP) typically exhibits a cluster-of-grapes morphology with an average subunit
size of 300 nm in diameter (Brownlee 1985). Rietmeijer (1993) measured the
subunit-size distribution of one CP IDP and found that the subunit size spans
from 64 to 7580 nm with a mean of 585 nm in diameter and follows a log-normal
distribution. One of the major constituents of CP IDP is GEMS (glass with
embedded metal and sulfide). Wozniakiewicz et al. (2013) measured the size
distribution of GEMS in four CP IDPs. The geometric mean subunit sizes are
ranging from 76–138 nm in radius for the four CP IDPs. Despite its similarity
to the interstellar grain sizes, GEMS has been considered to be the
condensates at the early solar nebula rather than the one directly inherited
from the interstellar medium (Keller & Messenger 2011).

Recent in situ measurements of dust aggregates of the comet 67P/C-G by the
Rosetta/MIDAS instrument have also revealed the presence of sub-micron-sized
subunits. Bentley et al. (2016) identified seven subunits in particle D ($\sim
1.09^{+0.01}_{-0.25}\leavevmode\nobreak\ \mu$m in size), using a 80-nm-scan
resolution, which are ranging from $0.260^{+0.050}_{-0.120}$ $\mu$m to
$0.540^{+0.020}_{-0.250}$ $\mu$m in diameter. Mannel et al. (2016) also
derived a subunit distribution of larger particles E and F and obtained the
mean subunit diameter of $1.48^{+0.13}_{-0.59}\leavevmode\nobreak\ \mu$m and
$1.36^{+0.15}_{-0.59}\leavevmode\nobreak\ \mu$m, respectively, although the
scan resolutions of particles E and F are 210 nm and 195 nm, respectively, and
thereby are worse than those used to analyze particle D. Mannel et al. (2019)
analyzed the subunits of particle G with a high-resolution scan (8 nm
resolution) and derived the arithmetic mean subunit diameter of
$99.49^{+0.89}_{-6.41}$ nm. They also hypothesized that particles D, E, and F,
which were analyzed with a lower scan resolution in earlier studies, also
consist of $\sim 100$ nm subunits in diameter.

The tensile strength of the comet 67P/C-G provides another way to study the
cometary monomer size. Based on a formula derived from numerical simulations,
Tatsuuma et al. (2019) estimated a monomer radius to be
$3.3$–$220\leavevmode\nobreak\ \mu\mathrm{m}$ to explain the tensile
strengths. Kimura et al. (2020b) claims, however, that sub-micron sized
monomers can still explain the observation if the volume effect in tensile
strengths plays a role. In this way, the monomer radius derived from the
tensile strength of the comet 67P/C-G seems still inconclusive and further
study is necessary.

Although our results do not allow to distinguish the origin of monomers, such
as interstellar origin and condensation at the disk forming epoch, the common
statement that the planet formation begins with coagulation of sub-micron
monomers has been confirmed from the vantage of disk observations for the
first time.

###  5.2 Are icy aggregates in planet-forming disks sticky?

Recent detections of millimeter-wave scattering polarization provide new
insight into the aggregate sizes in planet-forming disks (Kataoka et al. 2016;
Stephens et al. 2017; Hull et al. 2018; Ohashi et al. 2018; Bacciotti et al.
2018; Dent et al. 2019; Ohashi & Kataoka 2019; Ohashi et al. 2020; Mori &
Kataoka 2021). To reproduce the millimeter-wave polarization of the HL Tau
disk, Okuzumi & Tazaki (2019) argued that the critical fragmentation velocity
should be as low as $0.1-1$ m s-1. However, the origin of such a low critical
fragmentation velocity is still unknown.

The critical fragmentation velocity is known to scale as (Dominik & Tielens
1997)

$u_{\mathrm{f}}=u_{\mathrm{f0}}\left(\frac{R_{\mathrm{0}}}{100\leavevmode\nobreak\
\mathrm{nm}}\right)^{-5/6},$ (1)

where $u_{\mathrm{f0}}$ is the fragmentation velocity of aggregates consist of
0.1-$\mu$m-sized monomers. In general, $u_{\mathrm{f0}}$ depends on surface
materials coating each monomer and aggregate structure. To explain the
suggested low $u_{\mathrm{f}}$ value, two possibilities exist (i) larger
monomer radii ($R_{\mathrm{0}}\gtrsim\mu$m) and/or (ii) lower adhesion forces
(a low $u_{\mathrm{f0}}$ value). As we found that the monomer radius in the HL
Tau nebula may be around 0.4 $\mu$m (Sec. 3.3), the second possibility appears
to be more likely. When $R_{\mathrm{0}}=400$ nm, we need
$u_{\mathrm{f0}}\lesssim 3\leavevmode\nobreak\ \mathrm{m\leavevmode\nobreak\
s}^{-1}$.

For H2O-ice aggregates, by performing numerical collisional simulations, Wada
et al. (2009) derived $u_{\mathrm{f0}}=50\leavevmode\nobreak\
\mathrm{m\leavevmode\nobreak\ s}^{-1}$ for collisions between equal-sized
aggregates. The velocity may vary from $u_{\mathrm{f0}}\sim 25$–$80$ m s-1
depending on the mass ratio of colliding aggregates (Wada et al. 2013;
Hasegawa et al. 2021). Laboratory experiments have also supported this highly
sticky properties of H2O ice (Gundlach & Blum 2015). Although recent
experiments suggest a poorer stickiness of H2O ice at a lower temperature
(Musiolik & Wurm 2019; Gundlach et al. 2018), the interpretation of
temperature dependence is still a matter of discussion (Kimura et al. 2020a,
b). If H2O ice is as sticky as previously thought, the values of
$u_{\mathrm{f0}}$ are an order of magnitude higher than our necessary value,
$u_{\mathrm{f0}}\lesssim 3\leavevmode\nobreak\ \mathrm{m\leavevmode\nobreak\
s}^{-1}$. Therefore, H2O ice would not be the primary surface composition of
monomers, at least in outer disk regions we can observe.

One possible explanation is the presence of CO2 ice on the monomer surface.
Laboratory experiments have shown that CO2 ice is much less sticky than H2O
ice (Musiolik et al. 2016a, b; Fritscher & Teiser 2021), i.e.,
$u_{\mathrm{f0}}\sim 5\leavevmode\nobreak\ \mathrm{m\leavevmode\nobreak\
s}^{-1}$ (Okuzumi & Tazaki 2019). Arakawa & Krijt (2021) attributed the higher
stickiness of H2O ice than CO2 to viscoelastic dissipation upon collisions.
Due to its poor stickiness, the presence of CO2 ice has significant impact on
dust coagulation outside the CO2 snowline (Pinilla et al. 2017; Okuzumi &
Tazaki 2019). Although the value ($u_{\mathrm{f0}}\sim 5\leavevmode\nobreak\
\mathrm{m\leavevmode\nobreak\ s}^{-1}$) is still higher than the necessary
values, it has a similar order of magnitude. Thus, we speculate that monomers
at the outer disk regions are covered by non-sticky ice, such as CO2, and then
efficient collisional fragmentation gives rise to relatively small aggregate
sizes, as inferred from observations.

###  5.3 Can coagulation trigger Streaming instability?

One scenario for planetesimal formation is the streaming instability (Youdin &
Goodman 2005) followed by a strong dust clumping (Johansen et al. 2007, 2009;
Bai & Stone 2010a, b; Sekiya & Onishi 2018; Carrera et al. 2021). In the
absence of external turbulence, dust settling will set the maximum dust-to-gas
ratio, which is proportional to the metallicity. The threshold metallicity
required to trigger the strong clumping rapidly increases with decreasing the
particle Stokes number (Carrera et al. 2015; Yang et al. 2017; Li & Youdin
2021). One key question is whether collisional growth results in a Stokes
number large enough to trigger the strong clumping (Drażkowska & Dullemond
2014; Lorek et al. 2018).

If dust coagulation in the disks is limited by collisional fragmentation, the
Stokes number of the maximum grain radius will be given by (Birnstiel et al.
2012)

$\mathrm{St}_{\mathrm{frag}}=\frac{f_{\mathrm{f}}}{3}\frac{u_{\mathrm{f}}^{2}}{\alpha_{\mathrm{t}}c_{\mathrm{s}}^{2}},$
(2)

where $\alpha_{\mathrm{t}}$ is the turbulence parameter, $c_{\mathrm{s}}$ is
the sound speed, and $f_{\mathrm{f}}=0.37$ is a numerical factor. It is
important to notice the fact that $\mathrm{St}_{\mathrm{frag}}$ no longer
depends on the gas surface density and the material density. By substituting
Eq. (1) into (2), we obtain

$\mathrm{St}_{\mathrm{frag}}\simeq
0.02\left(\frac{u_{\mathrm{f0}}}{5\leavevmode\nobreak\
\mathrm{m\leavevmode\nobreak\
s}^{-1}}\right)^{2}\left(\frac{R_{\mathrm{0}}}{100\leavevmode\nobreak\
\mathrm{nm}}\right)^{-5/3}\left(\frac{\alpha_{\mathrm{t}}}{10^{-3}}\right)^{-1},$
(3)

where we have used $c_{\mathrm{s}}=3.8\times 10^{4}$ cm s-1.

We estimate the metallicity $Z$ needed to trigger strong clumping by using a
model by Li & Youdin (2021). Assuming a typical pressure gradient parameter
$\Pi=0.05$, we obtain $Z\geq 0.1$ for $\alpha=10^{-3}$ and
$\mathrm{St}_{\mathrm{frag}}=0.02$, and $Z\geq 0.008$ for $\alpha=10^{-4}$ and
$\mathrm{St}_{\mathrm{frag}}=0.2$. In the moderate turbulence case
($\alpha_{\mathrm{t}}=10^{-3}$), particles should accumulate approximately 10
times with respect to the gas.

Therefore, even in the outer region of the disk, efficient particle
accumulation seems necessary to trigger strong clumping unless the turbulence
is extremely low. In the above estimate, the role of turbulence is to just
determine the thickness of the dust layer. However, a role for turbulence
would not be that simple. Johansen et al. (2007) found that turbulence driven
by magneto-rotational instability may create a temporary overdense region,
which promotes the subsequent clumping by the streaming instability. In this
way, on the one hand, external turbulence renders clumping challenging. On the
other hand, it promotes clumping. Therefore, clarifying the interplay between
external turbulence and the streaming instability would be essential to draw
conclusions over the question if it works out in a turbulent disk.

##  6 Summary

We have studied the effect of monomer size and composition on scattering
polarization of dust aggregates by using an exact light scattering technique.
Applying the simulation results to disk observations, we have estimated the
monomer radius of aggregates in planet-forming disks for the first time. The
main findings of this paper are as follows.

  * •

The maximum degree of polarization of aggregates sensitively depends on the
monomer size when the monomer size parameter $x_{\mathrm{0}}\gtrsim 1$–$2$.
Contrary to the previous study, we showed that the effect of monomer size
appears not only for highly porous aggregates ($\mathcal{P}\sim 87\%$) but
also for less porous aggregates ($\mathcal{P}\sim 59\%$), although less porous
aggregates need a slightly larger monomer size for this effect to appear
(Sect. 3.1).

  * •

In contrast, the maximum degree of polarization of aggregates becomes
insensitive to monomer size when $x_{\mathrm{0}}\lesssim 1$. In this case, the
maximum polarization depends mainly on aggregate size, porosity, and
composition. In general, aggregates with a larger size, lower porosity, or a
higher real part of the refractive index tend to yield a lower maximum
polarization fraction at near-IR wavelengths (Sects. 3.1 and 3.2).

  * •

Since the effect of monomer size is noticeable when $x_{\mathrm{0}}\gtrsim
1-2$, the optical or near-IR wavelengths are the optimal wavelengths to
distinguish between sub-micron-sized and micron-sized monomers because the
monomer size parameter becomes close to or larger than unity at these
wavelengths (Sect. 3.1).

  * •

By comparing our results with the observations, we found that the monomer
radius appears to be no greater than $0.4\leavevmode\nobreak\ \mu$m for
several planet-forming disks (Sect. 3.3). We also found that a monomer radius
of $0.1$–$0.2\leavevmode\nobreak\ \mu$m seems favorable to explain the recent
polarimetric observations of the disk around HD 142527 (Sect. 4).

  * •

It may be possible that the the monomers are much smaller than
$0.1\leavevmode\nobreak\ \mu$m. In this case, our results predict that the
variation in the maximum polarization fractions of various disks at optical
wavelengths will be similar to that at near-IR wavelengths. In contrast, the
variation in optical polarization would be small compared to the near-IR
variation when the monomer are larger than 0.1 $\mu$m. A large optical
polarization survey of planet-forming disks is therefore useful to test the
monomer size (Sect. 3.3).

  * •

Within the limit of our composition model, we can rule out dark densely-packed
aggregates at the disk surfaces, as they exhibit a bluish color of the maximum
degree of polarization at optical and near-IR wavelengths, which is
inconsistent with the observed reddish colors (Sect. 3.3).

Optical and near-IR quantitative polarimetry will provide the observational
grounds on the initial conditions for dust coagulation, and thereby,
planetesimal formation in disks.

###### Acknowledgements.

R.T. acknowledges the JSPS overseas research fellowship. We thank Daniel
Mackowski and Maxim A. Yurkin for making the MSTM and ADDA codes publicly
available, respectively. We also thank Bruce Draine for the availability of
particle data of BA, BAM1, and BAM2. We thank J. Ma and H. M. Schmid for a
fruitful discussion.

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##  Appendix A Maximum polarization of BAM1 and BAM2 clusters

Fig. 9 shows the maximum polarization for BAM1 and BAM2 clusters.

Figure 9: Same as Figure 5, but for BAM1 (upper panel) and for BAM2 (lower
panel).

##  Appendix B Effect of particle shapes on the maximum polarization

This study assumes spherical monomers. One of the artifacts arising from the
assumption of spherical monomers is appearance of oscillatory patterns,
sometimes referred to as ripples, in polarization curves when
$x_{\mathrm{0}}>1$ (Bohren & Huffman 1983). Irregular particle shape usually
results in smoother polarization curves, particularly for absorbing materials
(Yanamandra-Fisher & Hanner 1999; Muinonen & Pieniluoma 2011). Although the
presence of touching spherical monomers in an aggregate tends to suppress the
oscillation (Xing & Hanner 1997), it is still possible that the perfect sphere
assumption might introduce some artifact in wavelength dependence of
polarization, particularly when the monomer size parameter exceeds unity.

Previous studies show that the maximum polarization of aggregates of
nonspherical monomers tends to be similar or slightly lowered to those of
spherical monomers. Xing & Hanner (1997) compared optical properties of
aggregates of spheres and tetrahedral monomers with $m=1.88+0.71i$. Although
the scattering polarization by a single sphere and tetrahedron differs
significantly, once the monomers are aggregated and touched each other, the
resultant maximum polarizations from aggregates of 10-spheres and
-tetrahedrons are surprisingly similar to each other (see, e.g., Fig. 9ab in
Xing & Hanner 1997). Meanwhile, Moreno et al. (2007) investigated aggregates
of 256-spheres and -cubes with an absorbing composition and found that
$P_{\mathrm{max}}$ of aggregates of cubic monomers show a lower polarization
degree by about 20% points. In their comparison, the side length of the cube
and sphere diameter were taken to be the same. However, as pointed out by
Kimura et al. (2016), their comparison allows cubic monomers to be larger than
spherical monomers in terms of volume, and thus it is unclear to what extent
the difference in $P_{\mathrm{max}}$ can be purely attributed to the shape
effect of the monomers. Lumme & Penttilä (2011) performed simulations using
discrete dipole approximation (DDA) for aggregates consisting of Gaussian
random sphere (GRS) particles with a less absorbing composition ($k=0.01$).
Although the maximum polarization degree is lowered for aggregates of
nonspherical monomers, the effect is found to be non-significant.

Figure 10: Effect of non-sphericity on polarization curves. Scattered light of
a single sphere (dashed line) and GRS particles (solid line) with amc-400 at
$\lambda=0.554\leavevmode\nobreak\ \mu$m. The volume-equivalent size parameter
is 4.54. The results for GRS particles are averaged over 10 realizations. A
realization of GRS particles and a sphere are also shown. For comparison, the
polarization curve for aggregates with 64-spherical monomers with amc-400 is
shown as well.

Figure 11: Effect of particle shape on the degree of polarization for the org
(upper panel) and amc (lower panel) compositions. The blue and orange lines
represent the results for $R_{\mathrm{V}}=0.2\leavevmode\nobreak\ \mu$m and
$0.4\leavevmode\nobreak\ \mu$m, respectively. The solid and dashed lines
represent the results for GRS particles and spherical particles, respectively.

To assess the effect of monomer shape on the degree of polarization, we
compare scattering polarization of a single sphere and a nonspherical
particle. As a nonspherical particle, we generated GRS particles (Muinonen et
al. 1996). As a Legendre expansion coefficient for the autocorrelation
function, we adopt a power-law function $c_{l}\propto l^{-\nu}$ ($l\geq 2$)
(Nousiainen et al. 2003). We adopt $\nu=3.4$ and the relative standard
deviation of radius $\sigma=0.2$. With these parameters, we can nicely mimic
irregularity of solid particles in nature, i.e., saharan dust particles (Muñoz
et al. 2007). We generated 10 realizations of GRS particles. One realization
of GRS particles is shown in Fig. 10. We adopt a DDA technique to solve the
light scattering by GRS particles, using a publicly available code ADDA
(Yurkin & Hoekstra 2011). Since the impact of non-sphericity on the degree of
polarization appears mainly for $x_{\mathrm{0}}\gtrsim 1$, we consider
particles with $200$ nm and $400$ nm in volume-equivalent radius.

The results are shown in Fig. 11. At near-IR wavelengths, the maximum degree
of polarization of the GRS particles is almost identical to that of spherical
particles for all cases. On the other hand, at optical wavelengths, the
maximum polarization of the GRS particles deviates significantly from that of
the spherical particles for the amc composition. This is mainly because the
nonspherical particles smear out ripple patterns in the polarization curves
(Fig. 10).

Because such a strong ripple can be similarly suppressed by the presence of
touching spherical monomers (see the polarization curve of the BPCA clusters
in Fig. 10), the differences in the maximum polarization between aggregates
made of GRS and spherical particles would be less noticeable than those seen
in Fig. 11 (Xing & Hanner 1997). However, some difference may possibly remain.
Therefore, the assumption of spherical monomers would be robust at least
around near-IR wavelengths, but might introduce some artifacts at optical
wavelengths, particularly for amc-200 and -400.

##  Appendix C Effect of an intermediate composition on the maximum
polarization

Figure 12: Left panel: BPCA clusters with
$R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m and $R_{\mathrm{0}}=400$ nm with
5-composite materials (silicate, water ice, amorphous carbon, organics, and
troilite). $f_{\mathrm{org}}$ represents a mass fraction of organics with
respect to total carbonaceous materials (organics + amorphous).
$f_{\mathrm{org}}=0.0$ and $1.0$ mean the carbonaceous component is pure
amorphous carbon and organics, respectively. Right panel: Same as Fig. 7, but
for BPCA clusters with $R_{\mathrm{V}}=1.6\leavevmode\nobreak\ \mu$m and
$R_{\mathrm{0}}=400$ nm with 5-composite materials.

In Sect. 4, we argued that the observed maximum polarization fractions are
inconsistent with aggregates with $R_{\mathrm{0}}=400$ nm. However, it is
unclear to what extent this conclusion remains true if we relax the assumption
of refractive indices. In particular, it follows from Fig. 7 that an
intermediate composition between org and amc copmosition may yield a closer
fit to the observations.

To test this possibility, we performed additional $T$-matrix simulations for
BPCA clusters with an intermediate dust composition. We now consider a mixture
of organics and amorphous carbon as a carbonaceous component so that each
monomer is made of 5 different materials: silicate, water ice, organics,
amorphous carbon, and troilite. By changing the mass ratio of organics to
total carbon (organics + amorphous carbon), we calculated the maximum degree
of polarization.

The results are shown in Fig. 12. It is found that even if we consider an
intermediate composition, the results are still located outside the
observationally inferred region. Although models with
$f_{\mathrm{org}}=0.4$–$0.6$ may be close to the observation, these models
predict a significantly low polarization fraction at the $J$-band (Fig. 12,
left). The presence/absence of such a dip in the polarization fraction can be
testable by future observations.


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