Light Scattering by Fractal Dust Aggregates. II. Opacity and Asymmetry Parameter

TL;DR

This study evaluates various approximation methods for the optical properties of fractal dust aggregates, introduces a modified mean field theory (MMF), and highlights its effectiveness in modeling astrophysical dust opacity and scattering.

Contribution

We develop and validate a modified mean field theory (MMF) that accurately models the optical properties of fractal dust aggregates, improving upon existing methods.

Findings

RGD theory fails with multiple scattering

EMT and DHS are inaccurate for fractal aggregates at short wavelengths

MMF accurately reproduces extinction opacity of dust aggregates

Abstract

Optical properties of dust aggregates are important at various astrophysical environments. To find a reliable approximation method for optical properties of dust aggregates, we calculate the opacity and the asymmetry parameter of dust aggregates by using a rigorous numerical method, the T-Matrix Method (TMM), and then the results are compared to those obtained by approximate methods; the Rayleigh-Gans-Debye (RGD) theory, the effective medium theory (EMT), and the distribution of hollow spheres method (DHS). First of all, we confirm that the RGD theory breaks down when multiple scattering is important. In addition, we find that both EMT and DHS fail to reproduce the optical properties of dust aggregates with fractal dimension of 2 when the incident wavelength is shorter than the aggregate radius. In order to solve these problems, we test the mean field theory (MFT), where multiple scattering can be taken into account. We show that the extinction opacity of dust aggregates can be well reproduced by MFT. However, it is also shown that MFT is not able to reproduce the scattering and absorption opacities when multiple scattering is important. We successfully resolve this weak point of MFT, by newly developing a modified mean field theory (MMF). Hence, we conclude that MMF can be a useful tool to investigate radiative transfer properties of various astrophysical environments. We also point out an enhancement of the absorption opacity of dust aggregates in the Rayleigh domain, which would be important to explain the large millimeter-wave opacity inferred from observations of protoplanetary disks.

Figures

Tables

Table 1. Table 1: Computational methods used in this paper

Acronym	Name	Relevant paper(s)
TMM	T-Matrix Method	Mackowski & Mishchenko (1996)
RGD	Rayleigh–Gans–Debye theory	Paper I
MFT	Mean Field Theory	Berry & Percival (1986); Botet et al. (1997)
MMF	Modified Mean Field Theory	This study
EMT (MG-Mie)	Mie theory with Maxwell-Garnett mixing rule	Mukai et al. (1992); Kataoka et al. (2014)
EMT (APMR-Mie)	Mie theory with aggregate polarizability mixing rule	Min et al. (2008)
DHS	Distribution of hollow spheres	Min et al. (2005, 2016)
CDE	Continuous distribution of ellipsoids	Bohren & Huffman (1983)

Table 2. Table 2: Cross sections for a two-sphere cluster in the Rayleigh domain

	$C_{\mathrm{abs}}/N{C}_{\mathrm{abs}}^0$777A two-sphere cluster is elongated along the $\hat{y}$ direction.
model	unpolarized	$𝐄\left|\right|\widehat{𝐲}$	$𝐄⟂\widehat{𝐲}$
contact ($N_O=10$)	$1.345$	$1.835$	0.856
contact ($N_O=1$)	$1.188$	$1.546$	0.830
separated ($3R_0/2$)888This separation distance is center-to-center distance of a two-sphere cluster.	1.084	1.256	0.911
separated ($2R_0$)	1.038	1.131	0.945

Table 3. Table 3: Coefficients of the mean field for the first 5 orders of the RCCA model

$n$	${\overline{d}}_{1,n}^{\left(1\right)}$	${\overline{d}}_{1,n}^{\left(2\right)}$
1	$0.352+0.245i$	$0.396+0.124i$
2	$0.419+0.119i$	$0.408+0.179i$
3	$0.388-0.039i$	$0.448+0.093i$
4	$0.123-0.111i$	$0.067-0.079i$
5	$0.014-0.024i$	$0.005-0.009i$

Equations

$$a_n$$

$$b_n$$

$$C_{\mathrm{ext}}^0=\frac{2\pi }{k^2}\sum _{n=1}^{\infty }(2n+1)\mathrm{Re}(a_n+b_n),$$

$$S_1^0$$

$$S_2^0$$

$$S_{11}^0=\frac{1}{2}(|S_1^0{|}^2+|S_2^0{|}^2).$$

$$|m-1|$$

$$\Delta \varphi$$

$$\Delta \varphi =\max (2x|m-1|,2x_c|m_{\mathrm{eff}}-1|),$$

$$C_{\mathrm{abs}}^{(\mathrm{RGD})}=N{C}_{\mathrm{abs}}^0$$

$$C_{\mathrm{sca}}=\frac{2\pi }{k^2}{\int }_0^{\pi }{S}_{11,\mathrm{agg}}(\theta )\sin \theta d\theta$$

$$S_{11,\mathrm{agg}}^{(\mathrm{RGD})}(\theta )=N^2{S}_{11}^0(\theta )\mathcal{S}(q),$$

$$(a_n,b_n)\to ({\bar{d}}_{1,n}^{(1)},{\bar{d}}_{1,n}^{(2)})$$

$${\bar{d}}_{1,n}^{(1)}$$

$${\bar{d}}_{1,n}^{(2)}$$

$$C_{\mathrm{ext}}^{(\mathrm{MFT})}=\frac{2\pi N}{k^2}\sum _{n=1}^{\infty }(2n+1)\mathrm{Re}({\bar{d}}_{1,n}^{(1)}+{\bar{d}}_{1,n}^{(2)}),$$

$$\mathcal{S}(q)={\int }_0^{\infty }g(u)4\pi u^2\frac{\sin (qu)}{qu}du.$$

$$g(u)=\frac{1}{4\pi R_g^3}{\left(\frac{u}{R_g}\right)}^{d_f-3}{f}_c\left(\frac{u}{R_g}\right)+\frac{1}{N}\delta (u),$$

$$f_c\left(\frac{u}{R_g}\right)=$$

$$\displaystyle\left\{\begin{array}[]{lll}\frac{2^{1-d_{f}}d_{f}^{d_{f}/2}}{\Gamma(d_{f}/2)}\exp{\left[-\frac{d_{f}}{4}(u/R_{g})^{2}\right]},&(\mathrm{GAUSS})\\ \frac{\{d_{f}(d_{f}+1)/2\}^{d_{f}/2}}{\Gamma(d_{f})}\exp{\left[-(\frac{d_{f}(d_{f}+1)}{2})^{1/2}(u/R_{g})\right]},&(\mathrm{EXPTL})\\ \frac{d_{f}}{2}\exp[{-\frac{1}{2}(u/R_{g})^{d_{f}}}].&(\mathrm{FLDIM})\end{array}\right.$$

$$C_{\mathrm{abs}}^{(\mathrm{MMF})}$$

$$\tau$$

$$⟨A{⟩}_{\mathrm{DHS}}$$

$$r_{\mathrm{out}}$$

$$\displaystyle n(f_{\mathrm{v}})=\left\{\begin{array}[]{ll}1/f_{\mathrm{max}}&(0\leq f_{\mathrm{v}}<f_{\mathrm{max}}),\\ 0&(f_{\mathrm{v}}\geq f_{\mathrm{max}}),\\ \end{array}\right.$$

$${ϵ}_{\mathrm{eff}}^{\mathrm{APMR}}=\frac{3({ϵ}_{\mathrm{i}}-1)+4f[{ϵ}_{\mathrm{i}}\ln {ϵ}_{\mathrm{i}}-({ϵ}_{\mathrm{i}}-1)]}{3({ϵ}_{\mathrm{i}}-1)-2f[{ϵ}_{\mathrm{i}}\ln {ϵ}_{\mathrm{i}}-({ϵ}_{\mathrm{i}}-1)]},$$

$$\alpha =3V_{\mathrm{m}}\frac{(ϵ-1)(2ϵ+1)}{(ϵ+2)(2ϵ+1)-2f_{\mathrm{v}}(ϵ-1{)}^2},$$

$${\bar{A}}_{1,n}^{1,\nu }$$

$${\bar{B}}_{1,n}^{1,\nu }$$

$$a(\nu ,n,p)$$

$$b(\nu ,n,p)$$

$$s_p(k{R}_g)=\frac{{\pi }^2}{k^3}{\int }_0^{\infty }u{J}_{p+1/2}(u)H_{p+1/2}^{(1)}(u)g(u/k)du$$

$$c=[1-(N-1)\tau Q{]}^{-1},$$

$$|N\tau Q|$$

Full text

opacity, planets and satellites: atmospheres, protoplanetary disks, radiative transfer, scattering

\published

2018 June 14

1 Introduction

Dust aggregates are ubiquitous at various astrophysical environments where coagulation of dust particles can take place, such as molecular clouds (Ormel et al., 2009, 2011; Steinacker et al., 2010), protoplanetary disks (Blum & Wurm, 2008; Okuzumi et al., 2012), and planetary atmospheres (Marley et al., 2013). Optical properties of dust aggregates are key ingredients of the radiative transfer in these environments, and these often govern observational appearance of the environments.

Optical properties of dust aggregates have been calculated by using numerical methods, such as the T-Matrix Method (TMM; Mishchenko et al., 1996) and the Discrete Dipole Approximation (DDA; Purcell & Pennypacker, 1973; Draine & Flatau, 1994). By using these techniques, a multitude of numerical studies have been performed (e.g., Kimura et al., 2016; Silsbee & Draine, 2016). Unfortunately, these numerical methods are still computationally demanding, in particular, when the size parameter of aggregate particles becomes large, and hence approximate methods are useful for many astrophysical purposes. Many approximate methods have been proposed so far (e.g., Bohren & Huffman, 1983; Ossenkopf, 1991; Mukai et al., 1992; Stognienko et al., 1995; Botet et al., 1997; Voshchinnikov et al., 2005; Voshchinnikov et al., 2007; Min et al., 2003, 2005, 2008, 2016); however, it still remains unclear which of them can reliably reproduce optical properties of aggregate particles.

In the series of this paper, we study optical properties of dust aggregates using TMM and investigate a fast and reliable approximate method which is able to reproduce the TMM results. A byproduct of finding a reliable approximation is that it leads to better, or sometimes intuitive, understanding of their optical properties. Tazaki et al. (2016, henceforth Paper I, ) studied the angular dependence of the scattering of fractal dust aggregates, such as the phase function and the degree of polarization and showed that the Rayleigh–Gans–Debye (RGD) theory is an useful method for calculating these properties. This study examines the angle-integrated optical properties of dust aggregates, such as the opacity and the asymmetry parameter.

This paper also attempts to relax a limitation of the RGD theory studied in Paper I . Because the RGD theory is a single scattering theory, it is applicable when multiple scattering is sub-dominant. This limitation is expected to be relaxed by considering the multiple scattering effect. For this purpose, we adopt the mean field assumption (Berry & Percival, 1986). With this assumption, multiple scattering can be solved self-consistently, while keeping most of the formulation of the RGD theory preserved (Botet et al., 1997). We refer to this approach as the mean field theory (MFT). In this paper, we test the validity of MFT and discuss whether it is a reliable approximation or not. If not, we propose an alternative method.

Furthermore, we test the validity of other widely used approximate methods, such as the effective medium theory (EMT; Mukai et al., 1992; Henning & Stognienko, 1996; Min et al., 2008) and the distribution of hollow spheres method (DHS; Min et al., 2003, 2005, 2016). For example, these methods have been commonly used in both observational and theoretical studies of circumstellar environments (e.g.. Ormel et al., 2011; Min et al., 2012; Kataoka et al., 2014; Cuzzi et al., 2014; Woitke et al., 2016).

In the EMT method, a dust aggregate is approximated by a single homogeneous sphere, and then the Mie theory (Bohren & Huffman, 1983) is applied to obtain the optical properties. In this method, information of the dust aggregate, such as the size, the porosity and the composition, is reduced to a quantity, so-called the effective dielectric function. Paper I showed that EMT cannot reproduce the phase function of fluffy dust aggregates. In this paper, we study the validity of EMT in the calculations of the opacity and the asymmetry parameter. Our primary focus is large relatively porous dust aggregates; ballistic cluster cluster agglomerates (BCCA) and ballistic particle cluster agglomerates (BPCA). Shen et al. (2008) studied the validity of EMT for small and compact ballistic dust aggregates, such as BPCA and more compact dust aggregates.

DHS is an approximate method to mimic optical properties of irregularly shaped particles (Min et al., 2005, 2016). In the DHS method, optical properties of dust aggregates are obtained by using those of hollow spheres. DHS seems to capture optical properties of irregularly shaped particles in the Rayleigh domain, such as an enhancement of the absorption opacity and the red shifted solid state feature (Min et al., 2003, 2006). The applicability of DHS to the larger particles has been mainly tested using compact dust aggregates (porosity of 25% in Min et al. (2016)); therefore, in this paper, we test the applicability of DHS to large fluffy dust aggregates (porosity $\gtrsim 85\%$).

It is known that coagulation of dust particles leads to an enhancement of the absorption opacity in the Rayleigh domain (Wright, 1987; Bazell & Dwek, 1990; Kozasa et al., 1992; Mackowski, 1995; Henning et al., 1995; Stognienko et al., 1995; Mackowski, 2006; Köhler et al., 2011). In the case of protoplanetary disks, a value of the absorption opacity at millimeter wavelength is important because it is often used to derive the disk dust mass (Beckwith et al., 1990). We investigate how an enhancement of the absorption opacity of the dust aggregate depends on its number of monomers, structure, and composition. In addition, we also discuss the methods that might be used to approximate this effect.

This paper is organized as follows. In Section 2, we summarize light scattering models of fractal dust aggregates based on the RGD theory and MFT. In Section 3, we show that MFT can reproduce the extinction opacity; however, MFT fails to reproduce the scattering and absorption opacities. We propose a modification to MFT, a modified mean field theory (MMF), whose validity is also tested in this section. In Section 4, we compare the optical properties obtained by other approximate methods (EMT and DHS) to those obtained by TMM and discuss their validity. In Section 5, we study an enhancement of the absorption opacity of dust aggregates in the Rayleigh domain. Section 6 summarizes the results of this study. Table 1 shows the list of abbreviations of various methods used in this paper.