Unveiling Dust Aggregate Structure in Protoplanetary Disks by Millimeter-wave Scattering Polarization

Prior to presenting our dust particle models in Section 2.3, we briefly review the expected dust structure in protoplanetary disks.

The structure of dust aggregates depends on how they coagulate. In disks, coagulation begins with low-collision velocity: no restructuring of dust aggregates occurs upon impact at this stage. Two limiting cases for aggregation are often used: the Ballistic Cluster–Cluster Aggregation (BCCA) and Ballistic Particle–Cluster Aggregation (BPCA). Aggregates consist of unit particles (monomer) of the same radius for the sake of simplicity.

BCCA is successive collisions between similar-sized aggregates. This type of coagulation is expected to occur if the aggregate-size distribution is narrow. Initial dust coagulation in disks is thought to take place in this manner (Ormel et al. 2007; Okuzumi et al. 2009, 2012). Aggregates formed by collisions of similar-size aggregates tend to show fractal dimension 1.4 ≲ d_f ≲ 2 (Ossenkopf 1993; Weidenschilling & Cuzzi 1993; Wurm & Blum 1998; Kempf et al. 1999; Krause & Blum 2004). Because the volume filling factor f and radius of an aggregate a obey $f\propto {a}^{{d}_{f}-3}$, their volume filling factor decreases with increasing radius. For 100 μm sized BCCA aggregates consisting of 0.1 μm monomers, the volume filling factor is f = 3 × 10⁻⁴; thus, they are extremely fluffy.

BPCA is successive collisions of a monomer and an aggregate. This type of collision is realized when small and large aggregates coexist in disks, such as due to fragmentation of large aggregates. It has been suggested that fragmentation of dust aggregates seems to be necessary to explain disk infrared observations (Dullemond & Dominik 2005). The BPCA aggregates tend to have d_f ≈ 3, and hence, their volume filling factor is constant and is about f = 0.15 (Kozasa et al. 1992). Dominik et al. (2016) studied hierarchical coagulation (an aggregate of aggregates), where an aggregate consisting of some monomers is used as a projectile in BPCA instead of using a monomer particle. Hierarchical coagulation produces an approximately 10 times lower filling factor, f ≈ 0.01, yet still shows d_f ≈ 3 at the core of the aggregate.

In disks, dust structure might also be affected by some compaction events. For high-speed collisions such that the impact energy is high enough to rearrange the aggregate structure, but still insufficient to disrupt an aggregate, collisional compaction occurs (Dominik & Tielens 1997; Blum & Wurm 2000; Wada et al. 2007, 2008, 2009; Paszun & Dominik 2009). Wada et al. (2008) showed that the fractal dimension of BCCA aggregates can be increased up to ${d}_{f}\approx 2.5$ upon such an impact. Laboratory measurements have often observed bouncing of two colliding dust aggregates (Blum & Münch 1993; Weidling et al. 2012; Kothe et al. 2013; Brisset et al. 2017). Weidling et al. (2009) suggested that sequential bouncing collisions gradually increase the volume filling factor up to f ≈ 0.36. However, the conditions for bouncing collisions are still being debated (Wada et al. 2011; Kothe et al. 2013; Seizinger & Kley 2013). Kataoka et al. (2013a, 2013b) proposed gas and gravitational compaction of dust aggregates, which increases the volume filling factor. Dust structure under these compressions is characterized by a bi-fractal structure, where d_f = 3 on the larger scale and 2 on the smaller scale.

To summarize, dust aggregates in disks might be categorized into two limiting groups: compact dust aggregates showing fractal dimension d_f ≈ 3 with 0.01 ≲ f ≲ 0.4 and fluffy dust aggregates with d_f ≈ 2 with f ≪ 1.

At (sub)millimeter wavelengths, disk emission comes mostly from the outer cold regions where water ice condenses onto dust particles. Thus, we assume that a dust particle is a mixture of silicate, water ice, troilite, and amorphous carbon. Although silicate, troilite, and amorphous carbon are probably separate grain populations, we assume that these components are mixed into a single dust particle for the sake of simplicity. The mass fraction of each component is determined by a recipe described in Min et al. (2011), where we adopt the carbon partition parameter w = 0.5. The derived mass fractions (material density) of silicate, water ice, troilite, and amorphous carbon are 32% (3.30 g cm⁻³), 45% (0.92 g cm⁻³), 10% (4.83 g cm⁻³), and 13% (1.80 g cm⁻³), respectively. Water ice dominates the mass and volume of dust particles. The resulting mean material density is 1.48 g cm⁻³. Optical constants of silicate, water ice, troilite, and amorphous carbon are taken from Draine (2003), Warren & Brandt (2008), Henning & Stognienko (1996), and Zubko et al. (1996), respectively. Their optical constants are mixed by using the Bruggeman mixing rule (Bruggeman 1935). At λ = 1 mm, the mixed refractive index, m, is m = 2.6 + 0.074i.

We consider three types of dust structure as illustrated in Figure 1. The basic properties of each dust model are as follows.

• 1.  
  Solid spheres: Solid spheres have homogeneous structure, and the filling factor is unity. As solid spheres have been commonly used in previous studies, it is valuable to compare how their optical properties are different from those of lower density dust particles resulting from their structure and porosity. The important parameter of this model is the sphere's radius.
• 2.  
  Fluffy dust aggregates: We define fluffy dust aggregates as aggregates with d_f ≈ 2. We use the characteristic radius a_c to describe the aggregate radius (Kozasa et al. 1992; Mukai et al. 1992). We adopt a fractal dimension d_f = 1.9 and fractal prefactor k₀ = 1.03, which are the typical values for BCCA aggregates formed with oblique collisions (Tazaki et al. 2016). The fractal prefactor is defined by $N={k}_{0}{({a}_{g}/{a}_{0})}^{{d}_{f}};$ N is the number of monomers and ${a}_{g}=\sqrt{3/5}{a}_{c}$ is the radius of gyration. Because we fix the monomer radius as a₀ = 0.1 μm, the free parameter of the model is the aggregate radius a_c only. The volume filling factor is given by f = 3 × 10⁻⁴(a_c/100 μm)^−1.1. In this study, we only consider a_c > 4 μm so that the volume filling factor of fluffy aggregates is always less than 0.01.
• 3.  
  Compact dust aggregates: Compact dust aggregates are defined as aggregates with d_f = 3 and 0.01 ≤ f < 1 in this paper. This model is intended to mimic BPCA aggregates (with/without hierarchical effects). The aggregate radius and the volume filling factor are the important parameters. We use the term "compact" because these grains are relatively compact compared to the fluffy dust aggregate model with f < 0.01.

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In this paper, we use the term dust particles, or simply dust, in more general contexts, i.e., when we intend to mention more general cases rather than to indicate a specific model given above.

We mainly focus on dust particle radii from submillimeter (∼100 μm) to centimeter sizes (∼10⁴ μm). Because this study aims to understand the scattering polarization of disks at (sub)millimeter wavelengths, dust particle radii should be comparable to or larger than the observing wavelength in order to attain high scattering albedo.

Computing optical properties of compact/fluffy dust aggregates is not an easy task, whereas those of solid spheres can be readily computed by the Mie theory (Bohren & Huffman 1983). Powerful numerical techniques for directly solving their optical properties have been developed, such as the T-Matrix Method (Mackowski & Mishchenko 1996) and the Discrete Dipole Approximation (Draine & Flatau 1994). However, millimeter-sized BPCA aggregates of 0.1 μm monomers contain about 10¹¹ monomers, and solving for electromagnetic interactions between every pair of monomers is very time consuming. Therefore, computing those properties with numerical techniques is not a realistic choice given current computer capabilities.

In order to reduce computational cost, approximate methods are useful. For fluffy dust aggregates, we can approximate their optical properties by using the modified mean field (MMF) theory (Tazaki et al. 2016; Tazaki & Tanaka 2018). On the other hand, the approximation of the optical properties of compact aggregates is somewhat more difficult because multiple scattering comes into play. Hence, as our current best, conservative estimate, we adopt the effective medium theory (EMT). It was recently found that the optical properties of compact dust aggregates, in particular for scattering opacity, could be (not perfectly but roughly) approximated by EMT (see Figure 6 in Tazaki & Tanaka 2018). This is partly because compact aggregates have d_f = 3, and hence, their scattering behaviors, such as the interference of scattered waves, have some similarities to those of the Mie theory, which also deals with spherical particles (d_f = 3). Thus, we anticipate that qualitatively similar results can be obtained even if we use EMT. We adopt the Maxwell Garnett mixing rule (Maxwell Garnett 1904) to obtain the effective refractive index, m_eff, of a mixture of vacuum (the matrix component) and dust composition (the inclusion component) described in Section 2.2. Because the Maxwell Garnett mixing rule works well when the matrix component dominates, we only consider f ≤ 0.1 (Kolokolova & Gustafson 2001).

For solid spheres and compact aggregates, optical properties are averaged over the dust size distribution in order to suppress strong resonances of scattering properties. We assume a dust size distribution obeying n(a)da ∝ a^pda with maximum and minimum radii of a_max and a_min, respectively. n(a)da represents the number of dust particles in the size range [a, a + da]. We set p = −3.5 and a_min = 0.01 μm, although our results are insensitive to a_min as long as a_min ≪ a_max.

Figure 2 shows Pω (left panel) and Pω^eff (right panel) as a function of the maximum dust radius a_max for solid spheres (f = 1) and compact dust aggregates (f = 0.1, 0.01) at wavelength λ = 1 mm. In order to understand the behavior of Pω^eff against a_max, we start to discuss Pω and then Pω^eff for each dust model. It is also useful to define the size parameter x = 2πa_max/λ. Because we adopt λ = 1 mm, the size parameter of unity corresponds to a_max = 160 μm.

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For solid spheres (f = 1), the maximum value of Pω appears at around x ≈ 1. The single-scattering albedo ω is usually much less than unity when x ≪ 1 and becomes of order unity when x ≳ 1. On the other hand, the degree of linear polarization P is high as long as x ≪ 1, but it suddenly decreases for x ≳ 1. As a result, Pω is maximized at x ≈ 1. The Pω^eff for solid spheres is almost the same as Pω because, as the maximum dust radius increases, the degree of polarization drops much faster than the term 1 − g. Thus, considering Pω^eff instead of Pω does not significantly change the previous interpretation by Kataoka et al. (2015), where solid spherical particles are assumed.

Next, we focus on compact dust aggregates (f = 0.1 and 0.01). First of all, decreasing the volume filling factor leads to a reduction of the single-scattering albedo at x ≤ 1 because the scattering opacity is proportional to the aggregate mass, whereas the absorption opacity is not. A notable impact of decreasing f is that the degree of polarization becomes high even if x > 1. This is due to the fact that aggregates with f ≪ 1 have an effective refractive index close to that of vacuum and are regarded as optically thin particles. Because multiple scattering is suppressed inside optically thin particles, a high degree of polarization is obtained. Pω increases with increasing a_max until $x| {m}_{\mathrm{eff}}-1|$ exceeds unity, at which point aggregates become optically thick. Because $x| {m}_{\mathrm{eff}}-1|$ is proportional to a_maxf (Kataoka et al. 2014), the maximum value of Pω approximately occurs at a_maxf ≈ λ/2π ≈ 160 μm. For example, when f = 0.1, Pω is maximized at approximately a_max = 1.6 mm. The maximum Pω value is larger for a lower filling factor because a lower filling factor makes aggregates less absorbing and also suppresses depolarization due to multiple scattering.

Unlike the case of solid spheres, the effective single-scattering albedo plays a significant role for compact aggregates. As shown in Figure 2 (right), the maximum value of Pω^eff is significantly attenuated at x > 1, in particular for the case of f = 0.01.

As a result, as the volume filling factor decreases, the maximum value of Pω^eff decreases. This suggests that compact aggregates with a lower filling factor produce fainter polarized-scattered light than solid spheres. We will confirm this by performing radiative transfer simulations in Section 4.

The scattering properties of fluffy dust aggregates are significantly different from those of spherical particles. We use the MMF theory to compute optical properties of fluffy aggregates. In this section, we use d_f = 2.0 and k₀ = 1.0 in order to attain clear physical insights into the optical properties of fluffy dust aggregates.

First of all, the degree of linear polarization of fluffy dust aggregates is as high as 100% because multiple scattering is suppressed when x₀ ≪ 1 and d_f ≤ 2 (Tazaki et al. 2016), where x₀ = 2πa₀/λ is the size parameter of the monomer particle. Here, the wavelength of interest is the (sub)millimeter wavelength and the monomer radius is of submicron size, and therefore, this condition is satisfied. Even if a_c ≫ λ/2π, the highest degree of polarization remains 100%. Thus, we obtain Pω^eff = ω^eff.

In Figure 3, we show the opacities of fluffy dust aggregates as a function of the characteristic radius a_c. The absorption opacity does not depend on a_c because fluffy aggregates with different a_c have the same mass-to-area ratio, and the absorption opacity remains the same (Kataoka et al. 2014).⁵ At a_c ≲ λ/2π, the scattering opacity increases with increasing aggregate radius (mass). On the other hand, once the aggregate radius exceeds λ/2π, the effective scattering opacity saturates. An analytical solution to the upper limit on the effective scattering opacity can be found under the conditions of x₀ ≪ 1 and d_f = 2 (see the Appendix for the derivation). The analytical solution to the effective scattering opacity of fluffy dust aggregates at a_c > λ/2π is

$$\begin{eqnarray}&&{\kappa }_{\mathrm{sca}}^{\mathrm{eff}}=\displaystyle \frac{{\kappa }_{\mathrm{sca},\mathrm{mono}}}{2{x}_{0}^{2}},\end{eqnarray} \tag{ 1 }$$

where κ_sca,mono is the scattering opacity of the single monomer. It is clear from Equation (1) that the effective scattering opacity of fluffy aggregates with d_f = 2 does not depend on the aggregate radius or the number of monomers.

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The physical interpretation of Equation (1) can be captured when the differential scattering cross section per unit mass Z₁₁ is plotted as a function of scattering angle (Figure 4). For an aggregate smaller than the wavelength, Rayleigh scattering occurs. This means that scattering is coherent for all scattering angles. Once the aggregate radius becomes larger than the wavelength, Z₁₁ saturates at intermediate and backward scattering angles. This is because scattered waves from a pair of monomers separated by a distance larger than (approximately) the wavelength are out of phase. On the other hand, scattered waves from particles separated by smaller distances can be in phase for such scattering angles. Therefore, scattered light at these scattering angles is dominated by coherent scattered light from the small-scale structure of the aggregate rather than the large-scale structure. As a consequence, Z₁₁ becomes irrelevant to a_c at these scattering angles, and then ${\kappa }_{\mathrm{sca}}^{\mathrm{eff}}$ also becomes independent of how large the aggregate is. The above explanation is also illustrated schematically in Figure 7 in Tazaki et al. (2019).

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By using Equation (1) and formulae for opacities in the Rayleigh limit (Bohren & Huffman 1983), the effective single-scattering albedo of fluffy aggregates, with characteristic radius larger than the wavelength, is

$$\begin{eqnarray}&&{\omega }^{\mathrm{eff}}=\displaystyle \frac{{x}_{0}}{{x}_{0}+3{ \mathcal R }},\,{ \mathcal R }=\mathrm{Im}\left\{\displaystyle \frac{{m}^{2}-1}{{m}^{2}+2}\right\}/{\left|\displaystyle \frac{{m}^{2}-1}{{m}^{2}+2}\right|}^{2}.\end{eqnarray} \tag{ 2 }$$

By substituting values into Equation (2), we obtain Pω^eff = 6 × 10⁻³, where we have assumed m = 2.6 + 0.074i and P = 1. As a result, it is found that fluffy aggregates are inefficient scatterers of millimeter-wave radiation. Even if the monomer radius is a few microns, the effective albedo is still ≲0.1. Therefore, fluffy aggregates of (sub)micron-sized monomers are not likely to contribute millimeter-wave scattering in protoplanetary disks.

Finally, we address how a different choice of fractal dimension affects scattering properties, as the fractal dimension of dust aggregates in disks might be 2 ≲ d_f ≲ 3. Although the detailed angular dependence of scattering matrix elements is not easy to compute, the qualitative wavelength dependence of opacities might be obtained by using the MMF theory (Tazaki & Tanaka 2018). In Figure 5, we show the effective scattering albedo of dust aggregates with various fractal dimensions having a_c = 160 μm and a₀ = 0.1 μm. Because the MMF theory is incapable of predicting the degree of polarization P for d_f > 2 due to the importance of multiple scattering, we only show ω^eff in Figure 5.

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At λ > 2πa_c = 1 mm (Rayleigh limit), the effective albedo increases with fractal dimension. Because we fixed both a_c and a₀, increasing the fractal dimension results in increasing the aggregate mass. Therefore, the effective albedo increases with d_f. At λ ≪ 2πa_c, the effective albedo for higher fractal dimensions becomes smaller due to forward scattering. Figure 5 predicts that aggregates with a higher fractal dimension produce faint and reddish scattered light in the short wavelength domain, e.g., infrared wavelengths, whereas those with d_f = 2 produce bright and blue scattered light. This tendency has already been confirmed by rigorous computations of optical properties as well as radiative transfer simulations (Mulders et al. 2013; Min et al. 2016; Tazaki et al. 2019).

As a result, at (sub)millimeter wavelengths, increasing the fractal dimension of aggregates gradually increases the effective scattering albedo. Thus, aggregates with higher fractal dimension are more likely to scatter at millimeter wavelength.

In order to simulate millimeter-wave scattering polarization of a protoplanetary disk, we use a publicly available 3D Monte Carlo radiative transfer code RADMC-3D (Dullemond et al. 2012). We assume a vertically isothermal disk, and the radial temperature profile approximately obeys T(r) = 86 K(r/10 au)^−0.5, where T is the dust temperature and r is the distance from the central star. The inner and outer disk radii are 10 au and 100 au, respectively, and the dust surface density is set as Σ_d = 0.157 g cm⁻² (r/10 au)⁻¹, leading to 10⁻⁴ M_⊙ total disk dust mass. We assume the dust model is the same everywhere within the disk. In addition, we ignore polarized thermal emission from aligned grains. The number of photon packets used in Monte Carlo scattering simulations is 10⁹. Because our primary focus is on how dust properties affect polarization, we fix both the disk model and disk inclination angle. Because moderate disk inclination angles are favored for scattering polarization (Yang et al. 2016a), we assume that the inclination angle is 45°. The imaging wavelength is set as λ = 1 mm.

Dust models used in simulations are solid spheres and compact/fluffy dust aggregates. In Table 1, we summarize the optical properties used in simulations for solid spheres and compact dust aggregates (a_maxf = 160 μm with f = 1, 0.1, 0.01) as well as for fluffy dust aggregates (d_f = 1.9, k₀ = 1.03, a_c = 160 μm). It is worth reminding the reader that the value of 160 μm is chosen so that the dust particle radius is equal to λ/2π, where λ = 1 mm. We will also discuss how different choices of a_max affect the results.

Table 1.  Optical Properties of Dust Models at Wavelength λ = 1 mm

Dust Model	κabs (cm2g−1)	${\kappa }_{\mathrm{sca}}^{\mathrm{eff}}\ ({\mathrm{cm}}^{2}\ {{\rm{g}}}^{-1})$	g	ωeff	P(θ = 90°)	Pωeff
Solid Spheres (amax = 160 μm)	3.37	5.54	0.30	0.62	0.82	0.51
Compact aggregates (amax = 1. 6 mm, f = 0.1)	2.17	2.09	0.89	0.49	0.92	0.45
Compact aggregates (amax = 1. 6 cm, f = 0.01)	1.95	0.11	0.99	0.05	1.00	0.05
Fluffy aggregate (ac = 160μm)	1.94	4.29 × 10−3	0.14	2.21 × 10−3	1.00	2.21 × 10−3

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The quantity a_maxf is useful to characterize particle properties because the mass-to-area ratio of dust particles is proportional to af, and thus, both absorption and dynamical properties are characterized by af (Kataoka et al. 2014). Indeed, as pointed out by Kataoka et al. (2014), the mass absorption opacity of aggregates with the same a_maxf value becomes very similar (Table 1).

Figure 6 shows a polarization map of a disk containing solid spheres (f = 1) and compact aggregates (f = 0.1, 0.01). In each image, polarization angles (E-vector direction) are indicated by bars with length proportional to the polarization fraction.

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As the volume filling factor decreases, the polarized intensity from the disk diminishes. This can be interpreted as a consequence of low effective scattering albedo as shown in Figure 2. For the cases of f = 1 and 0.1, polarization angles (E-vector direction) tend to be oriented parallel to the disk minor axis at the inner regions of disks. This is characteristic of scattering polarization from an optically thin disk (Kataoka et al. 2016a; Yang et al. 2016a).

For f = 0.1, near- and far-side asymmetry of the polarization pattern can be seen, whereas, for f = 1, the pattern is symmetric. For f = 0.1, the polarization orientation at the disk near side tends to be oriented in the azimuthal direction, whereas that of the far-side is oriented parallel to the minor axis. The dip in polarized intensity at the near side is the location where the polarization angle changes its direction from radial at the inner disk to azimuthal at the outer disk. This asymmetric polarization pattern is caused by anisotropic scattering. For f = 1, the assumed particle radius is comparable to λ/2π, and hence, scattering is close to isotropic. However, for f = 0.1, the assumed particle radius (a_max = 1. 6 mm for the f = 0.1 model) is larger than λ/2π ≈ 160 μm, and thus, forward scattering occurs. Because forward-scattered light amplifies a scattered-light component with azimuthal polarization at the disk near side, near–far-side asymmetry of the polarization pattern appears.

Next, we study the dust particle radius dependence of the average polarization fraction obtained by radiative transfer simulations. Figure 7 shows the average polarization fraction $\langle {P}_{\mathrm{disk}}\rangle$, which is a ratio of the polarized flux to the total flux for the entire disk, for various a_max. We compare the particle radius dependence of Pω^eff as well as Pω, where we use $\langle {P}_{\mathrm{disk}}\rangle ={CP}{\omega }^{\mathrm{eff}}$ to compare simulation results; the C-value is a numerical factor calibrated at a_max = 160 μm for each model.

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As shown in Figure 7, the solid sphere model (f = 1) shows strong dependence on a_max. This means that a high polarization fraction occurs at the vicinity of a_max ≈ λ/2π. On the other hand, for the case of f = 0.1, dependence on dust radius becomes much weaker. Therefore, when dust porosity is taken into account, scattering polarization can be detected for a wider range of dust particle radius compared to the solid sphere model, although if the porosity is too high (f = 0.01) insufficient polarized intensity is produced. At a_max = 160 μm, decreasing the volume filling factor makes the polarization fraction of scattered light small as a consequence of lower albedo for lower filling factor particles. This is consistent with the results in Figure 2.

As shown in Figure 7, Pω^eff can reproduce the overall dependence of polarization fraction on a_max observed in simulations. It is also shown that Pω fails to reproduce the dust particle radius dependence, indicating the importance of considering ω^eff instead of ω.

As a result, we conclude that dust aggregates with a lower filling factor produce smaller polarization fractions, due to the effective reduction of their scattering opacity caused by strong forward scattering. In other words, higher polarization fraction is more likely to be produced by relatively compact dust particles (f ≳ 0.1).

Next, we perform radiative transfer simulations with fluffy dust aggregates with a_c = 160 μm, and the result is shown in Figure 8. It is found that fluffy aggregates show very faint polarized intensity. This is mainly due to low scattering albedo as can be seen in Table 1. What happens when the aggregate radius is further increased? We perform additional radiative transfer simulations for fluffy aggregates with a_c = 1.6 mm and 1.6 cm, and we find that these larger aggregates give rise to almost the same results as found with a_c = 160 μm. This is due to the saturation of the effective scattering opacity as can be seen in Figure 3. As a result, it is found that a fluffy aggregate of submicron monomers with any characteristic radius is unlikely to produce millimeter-wave scattering polarization.

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As a result, we conclude that higher fractal dimension and higher filling factor are more favorable to producing millimeter-wave scattering polarization.

5.1.1. Dust Structure and Porosity Estimate

5.1.2. Dust Size Estimate

Hereafter, we assume that dust particles in disks have d_f = 3, as higher fractal dimension is favored. For dust particles with d_f = 3, the polarization fraction depends on dust radius as we showed in Figure 7. Conversely, the observed scattering-polarization fraction should contain information from which we can constrain the dust radius in disks.

Because previous studies have relied on the assumption of f = 1 (solid sphere), scattering polarization occurs efficiently at the vicinity of a_max ≈ λ/2π (Figure 7). Thus, the derived maximum dust radius is about a few times 100 μm because the observing wavelengths of ALMA are in the (sub)millimeter domain. However, the maximum dust radius is thought to depend on the physical properties of disks, such as age, gas, and dust surface density (Birnstiel et al. 2012). Hence, if the derived dust constraints are true, we need to explain how the a_max of various disks is fine-tuned at about this radius.

By considering particle porosity, this fine-tuning issue may be relaxed. Figure 9 shows Pω^eff as a function of a_maxf for various values of the volume filling factor. For the case of f = 1, Pω^eff is a sharp function of a_max, where the peak appears at a_max = λ/2π. As the volume filling factor decreases, the width of Pω^eff broadens, while the peak value is attenuated (see also Figure 7). Therefore, for particles with moderate porosity (0.1 ≲ f ≲ 1), scattering polarization can be expected for a wider range of a_max, while for extremely high porosity (f ≲ 0.01), it is hard to produce scattering polarization. Hence, moderate porosity dust particles have a role to relax the tight constraint on a_max.

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There is another issue concerning dust size constraints, that is, the inconsistency between opacity index and scattering polarization (Kataoka et al. 2016a; Yang et al. 2016a). If the disk is optically thin at observing wavelengths, the spectral slope of the observed flux density depends on the opacity index β, where β is the spectral slope of the absorption opacity, i.e., κ_abs ∝ λ^−β. In Figure 9, the opacity index is shown as a function of a_maxf, where the opacity index is defined at wavelengths between 870 μm and 1.3 mm. As shown in Figure 9, the opacity index at which scattering polarization can be anticipated is typically about β ≳ 1. However, the opacity index of disks is typically equal to or less than unity (Testi et al. 2014), implying the presence of millimeter- to centimeter-size dust particles in the disk (Draine 2006). Recently, Dent et al. (2019) clearly showed that scattering polarization is detected at disk regions where β ≲ 1. This inconsistency also seems to occur for another disk (e.g., Stephens et al. 2017; Hull et al. 2018).

Dust porosity has been suggested as a solution for this inconsistency. However, the opacity index of moderately porous particles is almost at the Rayleigh-limit value, that is, β ≈ β_ISM ≈ 1.6 (Planck Collaboration et al. 2014). Thus, this inconsistency may not be simply solved by considering particle porosity. Perhaps, other parameters, such as the functional shape of the dust size distribution and the optical depth of the disks, may be important to solve this issue. Recently, it has been pointed out that the spectral index could be affected by scattering if the disk is optically thick (Liu 2019; Zhu et al. 2019).

5.1.3. Wavelength Dependence

The wavelength dependence of scattering polarization is another important point. In Figure 10, we show Pω^eff as a function of wavelength. For the case of f = 1, scattering polarization efficiently occurs at λ ≈ 2πa_max (Kataoka et al. 2015). If dust porosity is taken into account, the wavelength dependence becomes weaker than that predicted from the solid sphere model as already expected from Figure 7. Therefore, the wavelength dependence of scattering polarization is useful to constrain dust porosity. For example, if we detect scattering polarization both at Bands 3 and 7 of ALMA, porous dust models, rather than the solid sphere model, seem to be favored as the origin of scattering polarization.

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Stephens et al. (2017) found that the polarization pattern of the HL Tau disk is wavelength dependent. At λ = 0.87 mm, the polarization pattern of HL Tau's disk is consistent with a scattering origin, whereas at λ = 3 mm, the polarization pattern becomes circularly symmetric, indicating that another origin for the polarization is important, such as grain alignment (Kataoka et al. 2017; Tazaki et al. 2017; Yang et al. 2019). DG Tau's disk also shows a wavelength-dependent polarization pattern between λ = 0.87 mm (Bacciotti et al. 2018) and λ = 3 mm (Harrison et al. 2019), where the polarization at λ = 0.87 mm is partially explained by scattering polarization and the polarization at λ = 3 mm is perhaps caused by grain alignment (Harrison et al. 2019).

The observed wavelength dependence of disks around HL Tau and DG Tau might be reproduced either with or without dust porosity (e.g., see the blue solid line and the dashed line in Figure 10). Although wavelength dependences between ALMA bands are similar to each other, they have completely different wavelength dependence at far-infrared wavelengths. Therefore, far-infrared wavelength polarimetry, such as by the SOFIA telescope and also the future SPICA telescope, is essential to distinguish between these models. The SOFIA and SPICA telescopes are insufficient to spatially resolve most disks, and hence, axisymmetric polarization patterns will be canceled out. However, scattering polarization tends to have a unidirectional polarization pattern for inclined disks, and therefore, we may expect to detect polarized waves even if the polarization pattern is integrated over the entire disk. It is worth keeping in mind that at far-infrared wavelengths, disks may become optically thick. Yang et al. (2017) pointed out that large optical depths of disks may reduce the scattering-polarization fraction.

The porosity evolution of dust aggregates in disks has been simulated by a number of authors (Ormel et al. 2007; Okuzumi et al. 2009, 2012; Zsom et al. 2010; Kataoka et al. 2013a; Krijt et al. 2015, 2016; Lorek et al. 2018). However, how dust aggregates grow to form planetesimals is a matter of debate. Here, we discuss implications for dust evolution and planetesimal formation by assuming that particles responsible for polarized-scattered light dominate the entire population in disks.

A possible scenario for forming a planetesimal is direct coagulation via fluffy dust aggregates. Dust aggregates consisting of submicron-size icy monomers are found to be very sticky, and they are resistant to compaction and fragmentation upon high-speed collisions (Suyama et al. 2008; Wada et al. 2008, 2009). By assuming perfect sticking, fluffy dust aggregates (f ≈ 10⁻⁴) form in disks, and they finally become planetesimals by direct coagulation (Okuzumi et al. 2012; Kataoka et al. 2013a). These results have also been confirmed by more recent studies (Krijt et al. 2015, 2016; Lorek et al. 2018). However, as we mentioned in Section 5.1.1, these aggregates are not favored from observations of millimeter-wave scattering polarization. Although the presence of these aggregates cannot be ruled out only by scattering-polarization observations, the presence of relatively compact particles is not anticipated in this model.

Therefore, a dust evolution model should answer how relatively compact particles could be formed in disks in order to explain millimeter-wave scattering polarization.

The first possibility is dust compaction due to bouncing collisions. Laboratory experiments suggest that sequential bouncing collisions lead to the gradual compaction of a dust aggregate (Weidling et al. 2009). If bouncing collisions occur, Zsom et al. (2010) suggested that the volume filling factor is increased up to 0.36 within 10⁴ orbital timescales, which is consistent with scattering-polarization constraints. Windmark et al. (2012a) showed that dust coagulation stalls at about 100 μm, which is also similar to the dust size expected from scattering polarization, although the dust size where coagulation stalls depends on disk properties, such as gas density, temperature, and turbulent strength. The dust evolution from a bouncing scenario seems to be consistent with that inferred from scattering-polarization observations; however, the onset of bouncing collisions is still a matter of debate (Wada et al. 2011; Kothe et al. 2013; Seizinger & Kley 2013; Brisset et al. 2017), and hence, further studies are necessary to draw more robust conclusions.

A second possibility is the increase of the internal density of dust aggregates caused by dust collisions with high mass ratio (Okuzumi et al. 2009; Dominik et al. 2016). For example, monomer–aggregate collisions without restructuring produces aggregates with d_f = 3 with f ≈ 0.15 which is known as BPCA. Such collisions are expected when fragmentation of dust aggregates occurs efficiently and produces tiny fragments as small as monomer particles (Wada et al. 2008; Paszun & Dominik 2009). It was suggested that fragmentation of dust aggregates seems to be necessary to explain disk infrared observations (Dullemond & Dominik 2005).

In addition, recent laboratory measurements suggested that the adhesion energy of icy dust might be smaller than previous estimates, implying the importance of dust fragmentation in disks. Musiolik et al. (2016a, 2016b) showed that CO₂ ice, which is also expected to condense onto dust particles in outer disk regions, does not show a high adhesion energy. Hence, dust aggregates of CO₂-ice-coated particles are thought to be much more fragile than those of H₂O ice. It is worth mentioning that recent experiments also question a high adhesion energy of H₂O ice at low temperatures (T ≲ 150–200 K; Gundlach et al. 2018; Musiolik & Wurm 2019), although earlier works by Gundlach & Blum (2015) confirmed high adhesion energy even at temperatures down to 100 K.

Motivated by these laboratory experiments, Okuzumi & Tazaki (2019) performed a dust coagulation simulation taking the nonsticky properties of the CO₂-ice-coated particles. Because lower adhesion energy predicts lower critical fragmentation velocity (Dominik & Tielens 1997), the maximum dust radius is also reduced if the fragmentation limits coagulation (Birnstiel et al. 2011, 2012). As a result, Okuzumi & Tazaki (2019) showed that scattering-like polarization of HL Tau's disk can be successfully explained by considering fragmentation of CO₂-ice-coated particles.

Although laboratory experiments are still inconclusive, efficient dust fragmentation and subsequent dust collisions with high mass ratio may potentially explain the origin of the relatively compact dust aggregates suggested by millimeter-wave scattering polarization.

A third possibility is that they are produced as fragments of differentiated planetesimals. If planetesimals are large enough to be molten, that is, differentiated, its fragments can be very compact particles (f ≈ 1). Melting events of icy dust particles, such as a more gentle version of chondrule-forming events, may also explain compact particles. However, in these cases, it is necessary to explain why a_max is adjusted to λ/2π.

Although the origin of compact (sub)millimeter-size particles is an open question, these particles are expected to form planetesimals via either coagulation with mass transfer (Windmark et al. 2012a, 2012b) or the streaming instability (Youdin & Goodman 2005; Johansen et al. 2007; Bai & Stone 2010a, 2010b). Windmark et al. (2012a) showed that if dust coagulation stalls at submillimeter size and if a small amount of larger seed particles are present, i.e., centimeter size, the seed particles grow to form planetesimals via mass transfer. Meanwhile, if compact dust particles have St > 10⁻², where St is the Stokes number and describes dynamical coupling between a dust particle and gas, they are subjected to streaming instability (Drażkowska & Dullemond 2014). Thus, the gravitational collapse of dust clumps led by the streaming instability is also another feasible pathway toward planetesimals.

Comets in our solar system are thought to be primitive objects, and hence, they are regarded as a living fossil of icy planetesimals. Cometary dust particles provide useful insights into how they form in the early solar nebula.

Recently, the Rosetta orbiter followed the comet 67P/Churyumov–Gerasimenko (hereafter 67P) and conducted in situ measurements of cometary dust particles. Comet 67P is a kilometer-size comet, and it is considered to be a primordial rubble pile object (Davidsson et al. 2016) with bulk density of 0.533 ± 0.006 g cm⁻³ (Pätzold et al. 2016). Three instruments on board Rosetta are dedicated to analyzing dust particles: the Micro-Imaging Dust Analysis System (MIDAS), Grain Impact Analyser and Dust Accumulator (GIADA), and Cometary Second Ion Mass Analyzer (COSIMA).

These instruments revealed that cometary dust seems to have two different families in terms of morphology. MIDAS uses an the atomic force microscope and provides 3D tomographic images of cometary dust particles. MIDAS found that cometary dust particles seem to have two morphological populations: compact aggregate and fluffy aggregates (Bentley et al. 2016; Mannel et al. 2016, 2019). The dust aggregates analyzed by the MIDAS have sizes from a few to 10 μm, and their subunit diameter is typically about from 0.1 μm (Mannel et al. 2019) to 1 μm (Bentley et al. 2016; Mannel et al. 2016). In addition, Bentley et al. (2016) measured fractal dimensions of fluffy aggregates and found d_f = 1.7 ± 0.1. GIADA measures the cross section, momentum, and mass of each dust particle and is mainly sensitive to (sub)millimeter-sized dust particles. GIADA also measured two populations of dust particles (Della Corte et al. 2015; Fulle et al. 2015, 2016a, 2016b). Fulle et al. (2016a, 2016b) show that fluffy aggregates detected by GIADA seem to have fractal dimension d_f = 1.87. Also, Fulle et al. (2016b) derived the volume filling factor of compact dust particles as f = 0.48 ± 0.08. Measurements by COSIMA also support two morphological populations of dust particles (Langevin et al. 2016; Lasue et al. 2019).

Fluffy dust aggregates (d_f ≲ 2) detected by Rosetta seem to be primordial aggregates formed in the early solar nebula. However, Fulle et al. (2016b) estimated the mass fraction of fluffy aggregates contained in the nucleus of comet 67P, and it is only about 0.015%. Hence, comet 67P likely consists of compact dust aggregates with f = 0.48 ± 0.08. If compact aggregates with f ≈ 0.5 are distributed in disks, they are sufficient to produce millimeter-wave scattering in disks as we showed in Sections 3 and 4. Hence, dust particles seen in scattering polarization might be the building blocks of cometary objects.

In addition to the three dust analyzers, OSIRIS (Optical, Spectroscopic, and Infrared Remote Imaging System) also provides images of the nucleus as well as dust particles. Based on the OSIRIS images, the tensile strength of comet 67P is estimated (Groussin et al. 2015; Basilevsky et al. 2016). Recently, Tatsuuma et al. (2019) studied the tensile strength of dust aggregates and found that the tensile strength of comet 67P is reproduced when the monomer radius is between 3.3–220 μm. Hence, submillimeter-sized solid spheres are candidates to explain the measured tensile strength.

To summarize, dust particles seen in scattering polarization might be precursors of cometary objects. However, it is worth keeping in mind that present-day cometary dust particles are biased to more compact dust structures because cometary particles are the end product of a long series of compaction events. Also, dust outflow from a cometary coma may also selectively remove fluffy dust aggregates.