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Light scattering by large clusters of dipoles as an analog for cometary dust aggregates
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Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 155–164 www.elsevier.com/locate/jqsrt
Light scattering by large clusters of dipoles as an analog for cometary dust aggregates Hiroshi Kimura, Ingrid Mann Institut fu¨r Planetologie, Westfa¨lische Wilhelms-Universita¨t, Wilhelm-Klemm-StraX e 10, D-48149 Mu¨nster, Germany
Abstract This paper addresses the question as to whether the characteristics of solar radiation scattered by cometary dust can be intrinsically attributed to light scattering by a number of interacting electric dipoles. We calculate light scattering by an ensemble of dipoles having the polarizability of an isolated sphere using the discrete dipole approximation. Our results are consistent with the recent successful model that describes cometary dust as large aggregate particles consisting of optically dark submicrometer-size monomers. We show that by calculating electric dipole–dipole interactions the overall trend of the optical properties can be studied. Calculating higher scattering orders is currently limited by computer capabilities, but is required for a better quantitative description of light scattering by cometary dust. We finally discuss the different model parameters considered for investigating the optical properties of cometary dust. r 2004 Elsevier Ltd. All rights reserved. PACS: 42.25.Fx; 42.25.Ja; 47.53.+n; 95.75.z; 95.85.Kr; 96.50.Dj; 96.50.Gn Keywords: Cometary dust; Light scattering; Discrete dipole approximation; Geometric albedo; Linear polarization; Aggregate particles
1. Introduction A comparison between observations of solar radiation scattered by cometary dust and numerical calculations of light scattering by nonspherical particles helps to characterize the Corresponding author. Fax: +49-251-83-36-301.
E-mail addresses: [email protected] (H. Kimura), [email protected] (I. Mann). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.019
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physical and chemical properties of cometary dust. Recently, the optical characteristics of cometary dust are found to be qualitatively consistent with light-scattering properties of large aggregate particles consisting of optically dark submicrometer-size grains, irrespective of the configuration of the constituent grains [1–3]. These studies have utilized the superposition Tmatrix method that allows us to rigorously solve light-scattering problems for aggregate particles consisting of homogeneous spherical monomers. It is a powerful tool, in particular, to calculate the orientation-averaged scattering matrix, but the currently available computing memory limits the study to a small number of monomers [4–6]. Discrete dipole approximation (DDA) can be used to supplement light-scattering calculations for an aggregate of arbitrarily shaped particles [7–9]. Since the DDA approximates arbitrarily shaped particles by an ensemble of interacting electric dipoles, the number of dipoles mainly determines the size of memory required. For a cluster of spheres, the memory usage can be greatly reduced by the a1 -term method, in which the spherical monomers are replaced by single dipoles having the polarizability of a sphere [10–12]. Therefore, the DDA a1 -term method is of great advantage to calculate light-scattering properties of large aggregates consisting of spherical monomers. The DDA a1 -term method takes into account only the electric dipole term of the Mie solution, while the superposition T-matrix method contains higher scattering orders of expansion coefficients derived by Mie theory. It should be noted that DDA provides rigorous solutions for light scattering by an ensemble of interacting electric dipoles. Accordingly, the difference in the solutions between the DDA a1 -term method and the superposition T-matrix method becomes insignificant, when spherical monomers are small compared to wavelength [13]. It is most likely that constituent grains of aggregate particles in comets are smaller than visible wavelengths [1,14–16]. In fact, constructive interference between interacting dipoles has been proposed to account for the negative linear polarization at small phase angles observed for cometary dust [17,18]. This gives rise to the question as to whether the optical characteristics of cometary dust can be attributed to intrinsic properties of an ensemble of electric dipoles with a certain set of dipole parameters. The study of light scattering by an ensemble of dipoles is, therefore, useful to search for the fundamental mechanisms which cause the observed phenomena on light scattering by cometary dust. In this paper, we apply the DDA a1 -term method to study light-scattering properties of large clusters of electric dipoles, in particular, the dependence of geometric albedo and linear polarization on phase angle with different assumptions for configurations and overall sizes of dipole-clusters. We also discuss the importance of electric dipole–dipole interactions for describing the light-scattering properties of cometary dust by comparing the results between the DDA a1 -term method and the superposition T-matrix method.
2. Discrete dipole approximation We only briefly describe the DDA a1 -term method below since the details of DDA computations used in this paper have been given elsewhere [10–12,19]. To solve scattering problems using DDA, one needs to specify the electric polarizability and location of dipoles [9,20,21].
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2.1. Polarizability of dipoles Each electric dipole is assumed to have the polarizability of an isolated sphere derived from Mie theory [10–12,22]: 3a1 ; ð1Þ 2k3 where k ¼ 2p=l is the wave number of incident solar radiation at a wavelength of l. The first scattering coefficient of Mie theory a1 is given as am ¼ i
a1 ¼
mc1 ðmxm Þc01 ðxm Þ c1 ðxm Þc01 ðmxm Þ ; mc1 ðmxm Þx01 ðxm Þ x1 ðxm Þc01 ðmxm Þ
ð2Þ
where c1 and x1 are the Riccati–Bessel functions and the prime indicates the derivative [12]. Note that the a1 -term is here given as a function of the size parameter, xm , and complex refractive index, m, of the monomer. Throughout this paper, we use the complex refractive index of m ¼ 1:98 þ 0:48i, which has been suggested for cometary dust at l ¼ 0:6 mm [1–3]. 2.2. Location of dipoles We first consider clusters of identical spherical monomers with radius am and complex refractive index m. Then we locate a single dipole in the center of each spherical monomer, the diameter of which gives the separation of neighboring two dipoles. The Cartesian coordinates of monomers are determined by three-dimensional numerical simulations of coagulation processes [23]. To investigate the influence of the dipole configuration on the optical properties, we consider two types of aggregation processes: particles grown under the ballistic cluster–cluster aggregation (BCCA) and particles grown under the ballistic particle-cluster aggregation (BPCA) [23,24]. Note that in this way we do not locate the dipoles on a periodic lattice while common DDA computations use an array of dipoles located on a cubic lattice. The projections of BCCA and BPCA particles consisting of identical 1024 spherical monomers are illustrated in Fig. 1. It is common for DDA computations to describe the size of a particle using the radius of a volume-equivalent sphere aV ¼ am N 1=3 m , where N m denotes the number of spherical monomers. Here we use am ¼ 0:1 mm and N m ¼ 256, 512, and 1024, corresponding to aV ¼ 0:63, 0:80, and 1:0 mm, respectively. As clearly seen in Fig. 1, the apparent sizes of the aggregates are larger than their volume-equivalent radii. The size parameter of monomers xm ¼ kam is xm ¼ 1:0 at l ¼ 0:6 mm and the size parameter of the aggregate xV ¼ kaV is xV ¼ 6:6 for N m ¼ 256, xV ¼ 8:4 for N m ¼ 512, and xV ¼ 11 for N m ¼ 1024.
3. Numerical results We calculate orientationally averaged properties of light scattering by a large number of dipoles using the DDA a1 -term method. Three angles, b, Y, and F specify the orientation of the particle in the ranges 0pbp360 , 0pYp180 , and 0pFp360 . Giving nb , nY , and nF angles uniformly
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Ballistic ClusterCluster Aggregate
Ballistic ParticleCluster Aggregate
1 m Fig. 1. The ballistic cluster–cluster and particle-cluster aggregations of 1024 identical spheres having radius of 0:1 mm. The volume of both aggregate particles, regardless of their apparent sizes, is equal to space that is enclosed by a sphere having radius of 1 mm. The bar on the right bottom indicates the scale of 1 mm.
in b, cos Y, and F, respectively, we totally obtain nb nY nF orientations for our consideration. The scattering matrix averaged over the orientations is further calculated for two orthogonal scattering planes, the results of which should coincide for randomly-orientated particles. We choose ðnb ; nY ; nF Þ ¼ ð120; 60; 120Þ for BCCA particles and ðnb ; nY ; nF Þ ¼ ð90; 45; 90Þ for BPCA particles so that the results at the two planes converge. We present the geometric albedo Ap and the degree of linear polarization P as a function of phase angle a. The geometric albedo is defined as Ap ¼ ðS 11 =k2 Þðp=GÞ, where S11 is the orientation-averaged ð1; 1Þ component of the scattering matrix and G is the geometric cross section [25]. It is obvious from Fig. 1 that the geometric cross section of a BCCA particle is larger than that of a BPCA particle with equal volume. We calculate the geometric cross section of a cluster of spheres based on a Monte-Carlo method [23,26]. The degree of linear polarization is calculated by P ¼ S12 =S11 , where S 12 is the orientation-averaged ð1; 2Þ component of the scattering matrix [27]. Fig. 2 shows numerical results of geometric albedo (upper panel) and linear polarization (lower panel) for 256, 512, and 1024 interacting electric dipoles whose configuration is determined by the BCCA. The upper left corner of each lower panel depicts enlarged phase-angle dependences of linear polarization in the phase-angle range of 0pap15 . The geometric albedo and the linear polarization vary smoothly with phase angle and show no oscillations. The geometric albedo has a wide and strong enhancement toward large phase angles and a weak rise toward small phase angles. The degree of linear polarization shows a broad positive maximum at an intermediate phase angle with negative values at small phase angles. All the results obtained here are entirely consistent with light-scattering properties of cometary dust derived from optical observations that are currently discussed [28,29].
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Fig. 2. The geometric albedo Ap (upper panel) and the degree of linear polarization P (lower panel) as a function of phase angle a for ballistic cluster–cluster aggregations (BCCA) of 256, 512, and 1024 interacting dipoles with the minimum separation of 0:2 mm. For comparison, the results from the superposition T-matrix method are also plotted as dotted curves for N ¼ 256.
For comparison, numerical results based on the superposition T-matrix method are plotted for the same aggregate of 256 monomers in the left figure as dotted lines. The comparison indicates that taking into account the multipole terms results in a lower geometric albedo at small phase angles as well as a lower linear polarization near the maximum. Fig. 3 shows numerical results for 256, 512, and 1024 interacting dipoles with the dipole configurations of BPCAs. The geometric albedo and linear polarization for the BPCA particles are almost identical to those for the BCCA particles. Although the geometric albedo and linear polarization are nearly independent of the overall size and structure of aggregates, there are certain tendencies that could be noticed. The negative polarization branch at small phase angles becomes deeper as the sizes of aggregate particles increase. It should also be noted that the maximum polarization at a 90 seems to decrease with increasing size of aggregates. In addition, larger aggregates tend to have larger values of the inversion angle, namely, the phase angle at which the polarization turns from negative values to positive values. In Figs. 2 and 3, we also plot the corresponding values of S11 =k2 , which is a measure of intensity, along the right axis of each upper panel. It is clear that, in contrast to geometric albedo, intensity increases with the size and geometric cross section of aggregate particles.
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Fig. 3. The same as Fig. 2, but for ballistic particle-cluster aggregations (BPCA).
4. Discussion 4.1. Multipole terms We have shown that dipole–dipole interactions alone produce the major optical characteristics of cometary dust. This justifies the importance of dipole–dipole interactions in describing the light-scattering properties of aggregate particles consisting of small grains. Therefore, our study of dipole–dipole interactions allows us to estimate the trend of the properties with increasing size of particles. However, this does not mean that multipole terms can be neglected, since multipole terms improve the results as inferred from the calculations with the superposition T-matrix method. Namely, the contribution of multipole terms reduces the albedo and the maximum polarization and, therefore, provides a better agreement with observational data. The importance of multipole terms increases with the size of monomers, but larger monomers generate oscillations in the albedo and polarization curves as already demonstrated numerically and experimentally [30,31]. A significant contribution of multipole terms may be ruled out by a lack of oscillations in the phase-angle dependences of albedo and polarization observed for cometary dust. Therefore, the size of monomers (am ¼ 0:1 mm) that we have chosen here should be a reasonable estimate for cometary dust in order for the dipole–dipole interactions to determine the major optical features.
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4.2. Refractive index Another important parameter to influence light-scattering properties of aggregates is the refractive index of the monomers. Previous studies with DDA have often assumed aggregate particles consisting of optically transparent materials to be representative of cometary dust [32–34]. This originates possibly from the fact that spherical particles consisting of optically dark materials do not show negative polarization at small phase angles, which is inconsistent with the polarization data for cometary dust [35]. However, a shallow branch of negative polarization could appear at small phase angles for aggregates of optically dark materials [13,36,37]. Moreover, we have clearly demonstrated with the DDA a1 -term method that the negative branch becomes deeper and wider for larger dipole clusters with high refractive index. The most recent calculations with the superposition T-matrix method have also shown large clusters of optically dark submicrometersize spheres to have all the characteristics for cometary dust [1–3]. Consequently, we conclude that the assumption of optically transparent materials forming aggregate particles is a long-standing misconception about the optical properties of cometary dust. 4.3. Dipole configuration In contrast to the refractive index, the morphology of aggregates does not show significant influence on the albedo and polarization curves for aggregates of NX256. This is the case for the size parameter of monomers xm X1 as proved by rigorous solutions for smaller aggregates of Np256 [1–3,13]. On the other hand, it has been shown that light scattering by aggregate particles consisting of tiny monomers with xm o1 depends on the configuration of the Rayleigh scatterers [13,30,38]. This could be understood if the interactions are mainly limited to the monomers in‘‘neighborhood’’, which may be marked out by scales of k1 . The neighbors for xm X1 are exclusively touching monomers, whereas those for xm o1 contain distant monomers, the number of which depends on the configuration. The uniqueness of the optical characteristics observed for cometary dust may indicate the monomer’s size to be xm X1. 4.4. Overall cluster size The optical properties of clusters of dipoles are insensitive not only to the morphologies, but also to the overall sizes of the clusters. Nevertheless, slight changes in the polarization indicate that the polarization curves of larger aggregates have a lower maximum, a deeper negative branch, and a larger inversion angle than the results we have shown here. This sheds light on the problem that the polarization of clusters of spheres with Np256 has a higher maximum, a shallower negative branch, and smaller inversion angle than observed. Therefore, the currently existing quantitative mismatch between the polarization observed for cometary dust and that calculated for clusters of spheres may be solved once we could deal with much larger aggregates. 4.5. Random orientation For larger aggregates (i.e., N4103 ), it is inevitable for us to consider a larger number of orientations for random-orientation averaging. We have here calculated the scattering matrices
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for randomly oriented BCCAs and BPCAs of Np1024 with 864,000 and 364,500 orientations, respectively. These numbers are already extremely high compared to the numbers of orientations used in previous studies, which typically considered tens or hundreds of orientations. Although the calculations for each orientation are fast, orientational averaging is highly time consuming for large aggregates. It is, therefore, worthwhile applying an analytical averaging method rather than a numerical quadrature method for larger aggregates [39]. Alternatively, one could calculate the T-matrix for the large aggregates using DDA and then apply the analytical orientational averaging to the results taking advantage of the T-matrix method [40]. It is also likely that larger computing memory would be available in the future, enabling us to utilize the superposition Tmatrix method for larger aggregates.
Acknowledgements We are grateful to Ludmilla Kolokolova for valuable discussion. We would like to thank Bruce T. Draine and Piotr J. Flatau for providing the original DDA code, and Hajime Okamoto for implanting the a1 -term method into the DDA code. This work is supported by the German Aerospace Center DLR (Deutschen Zentrum fu¨r Luft- und Raumfahrt) under the projects ‘‘Kosmischer Staub: Der Kreislauf interstellarer und interplanetarer Materie’’ (RD-RX-50 OO 0101-ZA) and ‘‘Mikro-Impakte’’ (RD-RX-50 OO 0203).
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Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 155–164 www.elsevier.com/locate/jqsrt
Light scattering by large clusters of dipoles as an analog for cometary dust aggregates Hiroshi Kimura, Ingrid Mann Institut fu¨r Planetologie, Westfa¨lische Wilhelms-Universita¨t, Wilhelm-Klemm-StraX e 10, D-48149 Mu¨nster, Germany
Abstract This paper addresses the question as to whether the characteristics of solar radiation scattered by cometary dust can be intrinsically attributed to light scattering by a number of interacting electric dipoles. We calculate light scattering by an ensemble of dipoles having the polarizability of an isolated sphere using the discrete dipole approximation. Our results are consistent with the recent successful model that describes cometary dust as large aggregate particles consisting of optically dark submicrometer-size monomers. We show that by calculating electric dipole–dipole interactions the overall trend of the optical properties can be studied. Calculating higher scattering orders is currently limited by computer capabilities, but is required for a better quantitative description of light scattering by cometary dust. We finally discuss the different model parameters considered for investigating the optical properties of cometary dust. r 2004 Elsevier Ltd. All rights reserved. PACS: 42.25.Fx; 42.25.Ja; 47.53.+n; 95.75.z; 95.85.Kr; 96.50.Dj; 96.50.Gn Keywords: Cometary dust; Light scattering; Discrete dipole approximation; Geometric albedo; Linear polarization; Aggregate particles
1. Introduction A comparison between observations of solar radiation scattered by cometary dust and numerical calculations of light scattering by nonspherical particles helps to characterize the Corresponding author. Fax: +49-251-83-36-301.
E-mail addresses: [email protected] (H. Kimura), [email protected] (I. Mann). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.019
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physical and chemical properties of cometary dust. Recently, the optical characteristics of cometary dust are found to be qualitatively consistent with light-scattering properties of large aggregate particles consisting of optically dark submicrometer-size grains, irrespective of the configuration of the constituent grains [1–3]. These studies have utilized the superposition Tmatrix method that allows us to rigorously solve light-scattering problems for aggregate particles consisting of homogeneous spherical monomers. It is a powerful tool, in particular, to calculate the orientation-averaged scattering matrix, but the currently available computing memory limits the study to a small number of monomers [4–6]. Discrete dipole approximation (DDA) can be used to supplement light-scattering calculations for an aggregate of arbitrarily shaped particles [7–9]. Since the DDA approximates arbitrarily shaped particles by an ensemble of interacting electric dipoles, the number of dipoles mainly determines the size of memory required. For a cluster of spheres, the memory usage can be greatly reduced by the a1 -term method, in which the spherical monomers are replaced by single dipoles having the polarizability of a sphere [10–12]. Therefore, the DDA a1 -term method is of great advantage to calculate light-scattering properties of large aggregates consisting of spherical monomers. The DDA a1 -term method takes into account only the electric dipole term of the Mie solution, while the superposition T-matrix method contains higher scattering orders of expansion coefficients derived by Mie theory. It should be noted that DDA provides rigorous solutions for light scattering by an ensemble of interacting electric dipoles. Accordingly, the difference in the solutions between the DDA a1 -term method and the superposition T-matrix method becomes insignificant, when spherical monomers are small compared to wavelength [13]. It is most likely that constituent grains of aggregate particles in comets are smaller than visible wavelengths [1,14–16]. In fact, constructive interference between interacting dipoles has been proposed to account for the negative linear polarization at small phase angles observed for cometary dust [17,18]. This gives rise to the question as to whether the optical characteristics of cometary dust can be attributed to intrinsic properties of an ensemble of electric dipoles with a certain set of dipole parameters. The study of light scattering by an ensemble of dipoles is, therefore, useful to search for the fundamental mechanisms which cause the observed phenomena on light scattering by cometary dust. In this paper, we apply the DDA a1 -term method to study light-scattering properties of large clusters of electric dipoles, in particular, the dependence of geometric albedo and linear polarization on phase angle with different assumptions for configurations and overall sizes of dipole-clusters. We also discuss the importance of electric dipole–dipole interactions for describing the light-scattering properties of cometary dust by comparing the results between the DDA a1 -term method and the superposition T-matrix method.
2. Discrete dipole approximation We only briefly describe the DDA a1 -term method below since the details of DDA computations used in this paper have been given elsewhere [10–12,19]. To solve scattering problems using DDA, one needs to specify the electric polarizability and location of dipoles [9,20,21].
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2.1. Polarizability of dipoles Each electric dipole is assumed to have the polarizability of an isolated sphere derived from Mie theory [10–12,22]: 3a1 ; ð1Þ 2k3 where k ¼ 2p=l is the wave number of incident solar radiation at a wavelength of l. The first scattering coefficient of Mie theory a1 is given as am ¼ i
a1 ¼
mc1 ðmxm Þc01 ðxm Þ c1 ðxm Þc01 ðmxm Þ ; mc1 ðmxm Þx01 ðxm Þ x1 ðxm Þc01 ðmxm Þ
ð2Þ
where c1 and x1 are the Riccati–Bessel functions and the prime indicates the derivative [12]. Note that the a1 -term is here given as a function of the size parameter, xm , and complex refractive index, m, of the monomer. Throughout this paper, we use the complex refractive index of m ¼ 1:98 þ 0:48i, which has been suggested for cometary dust at l ¼ 0:6 mm [1–3]. 2.2. Location of dipoles We first consider clusters of identical spherical monomers with radius am and complex refractive index m. Then we locate a single dipole in the center of each spherical monomer, the diameter of which gives the separation of neighboring two dipoles. The Cartesian coordinates of monomers are determined by three-dimensional numerical simulations of coagulation processes [23]. To investigate the influence of the dipole configuration on the optical properties, we consider two types of aggregation processes: particles grown under the ballistic cluster–cluster aggregation (BCCA) and particles grown under the ballistic particle-cluster aggregation (BPCA) [23,24]. Note that in this way we do not locate the dipoles on a periodic lattice while common DDA computations use an array of dipoles located on a cubic lattice. The projections of BCCA and BPCA particles consisting of identical 1024 spherical monomers are illustrated in Fig. 1. It is common for DDA computations to describe the size of a particle using the radius of a volume-equivalent sphere aV ¼ am N 1=3 m , where N m denotes the number of spherical monomers. Here we use am ¼ 0:1 mm and N m ¼ 256, 512, and 1024, corresponding to aV ¼ 0:63, 0:80, and 1:0 mm, respectively. As clearly seen in Fig. 1, the apparent sizes of the aggregates are larger than their volume-equivalent radii. The size parameter of monomers xm ¼ kam is xm ¼ 1:0 at l ¼ 0:6 mm and the size parameter of the aggregate xV ¼ kaV is xV ¼ 6:6 for N m ¼ 256, xV ¼ 8:4 for N m ¼ 512, and xV ¼ 11 for N m ¼ 1024.
3. Numerical results We calculate orientationally averaged properties of light scattering by a large number of dipoles using the DDA a1 -term method. Three angles, b, Y, and F specify the orientation of the particle in the ranges 0pbp360 , 0pYp180 , and 0pFp360 . Giving nb , nY , and nF angles uniformly
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Ballistic ClusterCluster Aggregate
Ballistic ParticleCluster Aggregate
1 m Fig. 1. The ballistic cluster–cluster and particle-cluster aggregations of 1024 identical spheres having radius of 0:1 mm. The volume of both aggregate particles, regardless of their apparent sizes, is equal to space that is enclosed by a sphere having radius of 1 mm. The bar on the right bottom indicates the scale of 1 mm.
in b, cos Y, and F, respectively, we totally obtain nb nY nF orientations for our consideration. The scattering matrix averaged over the orientations is further calculated for two orthogonal scattering planes, the results of which should coincide for randomly-orientated particles. We choose ðnb ; nY ; nF Þ ¼ ð120; 60; 120Þ for BCCA particles and ðnb ; nY ; nF Þ ¼ ð90; 45; 90Þ for BPCA particles so that the results at the two planes converge. We present the geometric albedo Ap and the degree of linear polarization P as a function of phase angle a. The geometric albedo is defined as Ap ¼ ðS 11 =k2 Þðp=GÞ, where S11 is the orientation-averaged ð1; 1Þ component of the scattering matrix and G is the geometric cross section [25]. It is obvious from Fig. 1 that the geometric cross section of a BCCA particle is larger than that of a BPCA particle with equal volume. We calculate the geometric cross section of a cluster of spheres based on a Monte-Carlo method [23,26]. The degree of linear polarization is calculated by P ¼ S12 =S11 , where S 12 is the orientation-averaged ð1; 2Þ component of the scattering matrix [27]. Fig. 2 shows numerical results of geometric albedo (upper panel) and linear polarization (lower panel) for 256, 512, and 1024 interacting electric dipoles whose configuration is determined by the BCCA. The upper left corner of each lower panel depicts enlarged phase-angle dependences of linear polarization in the phase-angle range of 0pap15 . The geometric albedo and the linear polarization vary smoothly with phase angle and show no oscillations. The geometric albedo has a wide and strong enhancement toward large phase angles and a weak rise toward small phase angles. The degree of linear polarization shows a broad positive maximum at an intermediate phase angle with negative values at small phase angles. All the results obtained here are entirely consistent with light-scattering properties of cometary dust derived from optical observations that are currently discussed [28,29].
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Fig. 2. The geometric albedo Ap (upper panel) and the degree of linear polarization P (lower panel) as a function of phase angle a for ballistic cluster–cluster aggregations (BCCA) of 256, 512, and 1024 interacting dipoles with the minimum separation of 0:2 mm. For comparison, the results from the superposition T-matrix method are also plotted as dotted curves for N ¼ 256.
For comparison, numerical results based on the superposition T-matrix method are plotted for the same aggregate of 256 monomers in the left figure as dotted lines. The comparison indicates that taking into account the multipole terms results in a lower geometric albedo at small phase angles as well as a lower linear polarization near the maximum. Fig. 3 shows numerical results for 256, 512, and 1024 interacting dipoles with the dipole configurations of BPCAs. The geometric albedo and linear polarization for the BPCA particles are almost identical to those for the BCCA particles. Although the geometric albedo and linear polarization are nearly independent of the overall size and structure of aggregates, there are certain tendencies that could be noticed. The negative polarization branch at small phase angles becomes deeper as the sizes of aggregate particles increase. It should also be noted that the maximum polarization at a 90 seems to decrease with increasing size of aggregates. In addition, larger aggregates tend to have larger values of the inversion angle, namely, the phase angle at which the polarization turns from negative values to positive values. In Figs. 2 and 3, we also plot the corresponding values of S11 =k2 , which is a measure of intensity, along the right axis of each upper panel. It is clear that, in contrast to geometric albedo, intensity increases with the size and geometric cross section of aggregate particles.
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4. Discussion 4.1. Multipole terms We have shown that dipole–dipole interactions alone produce the major optical characteristics of cometary dust. This justifies the importance of dipole–dipole interactions in describing the light-scattering properties of aggregate particles consisting of small grains. Therefore, our study of dipole–dipole interactions allows us to estimate the trend of the properties with increasing size of particles. However, this does not mean that multipole terms can be neglected, since multipole terms improve the results as inferred from the calculations with the superposition T-matrix method. Namely, the contribution of multipole terms reduces the albedo and the maximum polarization and, therefore, provides a better agreement with observational data. The importance of multipole terms increases with the size of monomers, but larger monomers generate oscillations in the albedo and polarization curves as already demonstrated numerically and experimentally [30,31]. A significant contribution of multipole terms may be ruled out by a lack of oscillations in the phase-angle dependences of albedo and polarization observed for cometary dust. Therefore, the size of monomers (am ¼ 0:1 mm) that we have chosen here should be a reasonable estimate for cometary dust in order for the dipole–dipole interactions to determine the major optical features.
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4.2. Refractive index Another important parameter to influence light-scattering properties of aggregates is the refractive index of the monomers. Previous studies with DDA have often assumed aggregate particles consisting of optically transparent materials to be representative of cometary dust [32–34]. This originates possibly from the fact that spherical particles consisting of optically dark materials do not show negative polarization at small phase angles, which is inconsistent with the polarization data for cometary dust [35]. However, a shallow branch of negative polarization could appear at small phase angles for aggregates of optically dark materials [13,36,37]. Moreover, we have clearly demonstrated with the DDA a1 -term method that the negative branch becomes deeper and wider for larger dipole clusters with high refractive index. The most recent calculations with the superposition T-matrix method have also shown large clusters of optically dark submicrometersize spheres to have all the characteristics for cometary dust [1–3]. Consequently, we conclude that the assumption of optically transparent materials forming aggregate particles is a long-standing misconception about the optical properties of cometary dust. 4.3. Dipole configuration In contrast to the refractive index, the morphology of aggregates does not show significant influence on the albedo and polarization curves for aggregates of NX256. This is the case for the size parameter of monomers xm X1 as proved by rigorous solutions for smaller aggregates of Np256 [1–3,13]. On the other hand, it has been shown that light scattering by aggregate particles consisting of tiny monomers with xm o1 depends on the configuration of the Rayleigh scatterers [13,30,38]. This could be understood if the interactions are mainly limited to the monomers in‘‘neighborhood’’, which may be marked out by scales of k1 . The neighbors for xm X1 are exclusively touching monomers, whereas those for xm o1 contain distant monomers, the number of which depends on the configuration. The uniqueness of the optical characteristics observed for cometary dust may indicate the monomer’s size to be xm X1. 4.4. Overall cluster size The optical properties of clusters of dipoles are insensitive not only to the morphologies, but also to the overall sizes of the clusters. Nevertheless, slight changes in the polarization indicate that the polarization curves of larger aggregates have a lower maximum, a deeper negative branch, and a larger inversion angle than the results we have shown here. This sheds light on the problem that the polarization of clusters of spheres with Np256 has a higher maximum, a shallower negative branch, and smaller inversion angle than observed. Therefore, the currently existing quantitative mismatch between the polarization observed for cometary dust and that calculated for clusters of spheres may be solved once we could deal with much larger aggregates. 4.5. Random orientation For larger aggregates (i.e., N4103 ), it is inevitable for us to consider a larger number of orientations for random-orientation averaging. We have here calculated the scattering matrices
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for randomly oriented BCCAs and BPCAs of Np1024 with 864,000 and 364,500 orientations, respectively. These numbers are already extremely high compared to the numbers of orientations used in previous studies, which typically considered tens or hundreds of orientations. Although the calculations for each orientation are fast, orientational averaging is highly time consuming for large aggregates. It is, therefore, worthwhile applying an analytical averaging method rather than a numerical quadrature method for larger aggregates [39]. Alternatively, one could calculate the T-matrix for the large aggregates using DDA and then apply the analytical orientational averaging to the results taking advantage of the T-matrix method [40]. It is also likely that larger computing memory would be available in the future, enabling us to utilize the superposition Tmatrix method for larger aggregates.
Acknowledgements We are grateful to Ludmilla Kolokolova for valuable discussion. We would like to thank Bruce T. Draine and Piotr J. Flatau for providing the original DDA code, and Hajime Okamoto for implanting the a1 -term method into the DDA code. This work is supported by the German Aerospace Center DLR (Deutschen Zentrum fu¨r Luft- und Raumfahrt) under the projects ‘‘Kosmischer Staub: Der Kreislauf interstellarer und interplanetarer Materie’’ (RD-RX-50 OO 0101-ZA) and ‘‘Mikro-Impakte’’ (RD-RX-50 OO 0203).
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