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

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Light Scattering by Fractal Dust Aggregates. II. Opacity and Asymmetry Parameter The Astrophysical Journal, 860:79 (17pp), 2018 June 10 https://doi.org/10.3847/1538-4357/aac32d © 2018. The American Astronomical Society. All rights reserved. Light Scattering by Fractal Dust Aggregates. II. Opacity and Asymmetry Parameter Ryo Tazaki and Hidekazu Tanaka Astronomical Institute, Graduate School of Science Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai 980-8578, Japan; [email protected] Received 2018 March 9; revised 2018 April 26; accepted 2018 May 6; published 2018 June 14 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, 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 dimensions 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. Key words: opacity – planets and satellites: atmospheres – protoplanetary disks – radiative transfer – scattering 1. Introduction Rayleigh–Gans–Debye (RGD) theory is a useful method for calculating these properties. This study examines the angle- Dust aggregates are ubiquitous at various astrophysical integrated optical properties of dust aggregates, such as the environments where coagulation of dust particles can take opacity and the asymmetry parameter. place, such as molecular clouds (Ormel et al. 2009, 2011; This paper also attempts to relax a limitation of the RGD Steinacker et al. 2010), protoplanetary disks (Blum & Wurm theory studied in Paper I. Because the RGD theory is a single- 2008; Okuzumi et al. 2012), and planetary atmospheres ( ) scattering theory, it is applicable when multiple scattering is Marley et al. 2013 . Optical properties of dust aggregates are subdominant. This limitation is expected to be relaxed by key ingredients of the radiative transfer in these environments, considering the multiple-scattering effect. For this purpose, we and these often govern the observational appearance of the adopt the mean field assumption (Berry & Percival 1986). With environments. this assumption, multiple scattering can be solved self- Optical properties of dust aggregates have been calculated by ( consistently, while keeping most of the formulation of the using numerical methods, such as the T-Matrix Method TMM; RGD theory preserved (Botet et al. 1997). We refer to this ) Mishchenko et al. 1996 and the Discrete Dipole Approx- approach as the mean field theory (MFT). In this paper, we test ( imation DDA; Purcell & Pennypacker 1973; Draine & Flatau the validity of MFT and discuss whether it is a reliable ) 1994 . By using these techniques, a multitude of numerical approximation or not. If not, we propose an alternative method. ( studies have been performed e.g., Kimura et al. 2016; Silsbee Furthermore, we test the validity of other widely used ) & Draine 2016 . Unfortunately, these numerical methods are approximate methods, such as the effective medium theory still computationally demanding, in particular, when the size (EMT; Mukai et al. 1992; Henning & Stognienko 1996; Min parameter of aggregate particles becomes large, and hence et al. 2008) and the distribution of hollow spheres method approximate methods are useful for many astrophysical (DHS; Min et al. 2003, 2005, 2016). For example, these purposes. Many approximate methods have been proposed so methods have been commonly used in both observational and far (e.g., Bohren & Huffman 1983; Ossenkopf 1991; Mukai theoretical studies of circumstellar environments (e.g.. Ormel et al. 1992; Stognienko et al. 1995; Botet et al. 1997; Min et al. et al. 2011; Min et al. 2012; Cuzzi et al. 2014; Kataoka et al. 2003, 2005, 2008, 2016; Voshchinnikov et al. 2005, 2007); 2014; Woitke et al. 2016). however, it still remains unclear which of them can reliably In the EMT method, a dust aggregate is approximated by a reproduce optical properties of aggregate particles. single homogeneous sphere, and then the Mie theory (Bohren We study optical properties of dust aggregates using TMM & Huffman 1983) is applied to obtain the optical properties. In and investigate a fast and reliable approximate method that is this method, information of the dust aggregate, such as the size, able to reproduce the TMM results. A by-product of finding a the porosity, and the composition, is reduced to a quantity, the reliable approximation is that it leads to better, or sometimes so-called effective dielectric function. Paper I showed that intuitive, understanding of their optical properties. Tazaki EMT cannot reproduce the phase function of fluffy dust et al. (2016, hereafter Paper I) studied the angular dependence aggregates. In this paper, we study the validity of EMT in the of the scattering of fractal dust aggregates, such as the phase calculations of the opacity and the asymmetry parameter. Our function and the degree of polarization, and showed that the primary focus is large, relatively porous dust aggregates, 1 The Astrophysical Journal, 860:79 (17pp), 2018 June 10 Tazaki & Tanaka 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) ballistic cluster–cluster agglomerates (BCCAs), and ballistic aggregates are important. Hence, we consider optical properties particle cluster agglomerates (BPCAs). Shen et al. (2008) of the dust aggregates averaged over random orientations. studied the validity of EMT for small and compact ballistic dust Light scattering models described below are based on the aggregates, such as BPCAs and more compact dust aggregates. statistical distribution of monomers in the aggregate. Once the DHS is an approximate method to mimic optical properties optical properties are averaged over random orientations, these of irregularly shaped particles (Min et al. 2005, 2016). In the properties are well described by the aggregate model with a DHS method, optical properties of dust aggregates are obtained statistically isotropic distribution of monomers. As a conse- by using those of hollow spheres. DHS seems to capture optical quence, the random orientation assumption makes light properties of irregularly shaped particles in the Rayleigh scattering models rather simple, e.g., scattering properties do domain, such as an enhancement of the absorption opacity and not depend on the zenith angle (see, e.g., Paper I). A caveat of the redshifted solid-state feature (Min et al. 2003, 2006). The the statistically isotropic assumption is that applications to applicability of DHS to the larger particles has been mainly dichroic extinction or polarized emission are precluded. tested using compact dust aggregates (porosity of 25% in Min The calculation of MFT has three steps. The first step is to et al. 2016); therefore, in this paper we test the applicability of calculate optical properties of a monomer particle (Section 2.1). DHS to large fluffy dust aggregates (porosity 85%). In the second step, interference of scattered waves from each It is known that coagulation of dust particles leads to an monomer is calculated under the single-scattering assumption ( ) enhancement of the absorption opacity in the Rayleigh domain Section 2.2 , where the statistical arrangement of monomers is ( ) fi (Wright 1987; Bazell & Dwek 1990; Kozasa et al. 1992; assumed see Section 2.4 . The nal step is to calculate fi Henning et al. 1995; Mackowski 1995, 2006; Stognienko multiple scattering using the mean eld approximation ( ) fi et al. 1995; Köhler et al. 2011). In the case of protoplanetary Section 2.3 . Note that the rst and second steps correspond ( ) disks, a value of the absorption opacity at millimeter to the RGD theory Paper I . wavelength is important because it is often used to derive the disk dust mass (Beckwith et al. 1990). We investigate how an 2.1. Optical Properties of a Spherical Monomer enhancement of the absorption opacity of the dust aggregate Light scattering models presented below are applicable to a depends on its number of monomers, structure, and composi- single monomer size and single isotropic composition. In this tion. In addition, we also discuss the methods that might be paper, all monomers are assumed to be spherical, and hence the used to approximate this effect. Mie theory (Bohren & Huffman 1983) can be used to obtain This paper is organized as follows. In Section 2,we optical properties of the monomer. summarize light scattering models of fractal dust aggregates In what follows, we adopt the same notation as Bohren & based on the RGD theory and MFT.
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