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OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Modeling light scattering by forsterite particles Evgenij Zubko Institute of Astronomy, V.N. Karazin Kharkiv National University, 35 Sumskaya St., Kharkov 61022, Ukraine ([email protected]) Received January 13, 2015; accepted February 17, 2015; posted February 18, 2015 (Doc. ID 232413); published March 17, 2015 Laboratory optical measurements of forsterite particles reveal remarkably similar light-scattering responses in two samples that were thought to obey different size distributions. These measurements are modeled with irregularly shaped agglomerated debris particles having a refractive index of m 1.6 0.0005i that is representative of forsterite in the visible. Modeling closely reproduces the measurements of both samples, making retrieval of their size distributions possible. © 2015 Optical Society of America OCIS codes: (280.4991) Passive remote sensing; (290.0290) Scattering; (290.5850) Scattering, particles; (290.5855) Scattering, polarization; (290.5890) Scattering, stimulated. http://dx.doi.org/10.1364/OL.40.001204
Laboratory optical measurements provide important clues for better understanding of the interrelation between the chemical and physical characteristics of small dust particles and their light-scattering response. In the case of naturally occurring dust particles, however, it often is difficult to characterize precisely the sample particles. An even more complicated problem is to control those characteristics. For instance, investigation of the impact of size distribution on light scattering implies an ability for controllable variation of the size distribution in sample particles. This is an extremely difficult technical problem to accomplish in practice. In this Letter, we analyze laboratory measurements of light scattering by forsterite particles that are thought to obey different size distributions [1]. Forsterite is a magnesium-rich olivine that is an abundant constituent of, for example, cometary dust (e.g., [2]). The forsterite sample investigated in [1] has a chemical composition of Mg1.9 Fe0.1 SiO4 . The sample was obtained by crushing a large piece of material to a finegrained powder. As a consequence, the particles reveal highly irregular shapes with sharp edges and corners. This can be seen in the SEM images presented in [1]. The forsterite powder was dry sieved first with a 50 μm sieve and, then, with a 20 μm sieve. The particles that did not pass through the second sieve were collected and designated as the initial sample. While the size of these particles is expected to be squeezed between 20 and 50 μm, inspection of the SEM images and measurement with a Fritsch laser particle sizer reveal a considerable amount of submicron and micron-sized particles. This is the result of electrostatic forces that cause particles to adhere to each other, resulting in agglomerates that cannot pass through a 20 μm sieve. To eliminate electrostatic forces, the authors moistened the forsterite powder with pure ethanol and repeated the sieving procedure with the wet powder. Particles that passed through a 20 μm sieve were collected, dried, and designated as a small sample, whereas the remaining particles collected on the 20 μm sieve are designated as the washed sample. Light scattering by all three samples was measured in the single-particle scattering regime. In this regime, an enormous number of sample particles is suspended in the air and irradiated with the incident light. The light scattered from such a cloud is representative simultaneously of a large number of particle shapes in random 0146-9592/15/071204-04$15.00/0
positions and orientations. However, the amount of ejected material is limited to keep the cloud optically thin and, thus, to ensure a negligible contribution from the multiple light scattering between particles. A beam of light can be fully characterized with a fourdimension Stokes vector [3], whereas the response from the target is expressed in terms of a 4×4 matrix that is referred to as the Mueller matrix [3] or the scattering matrix [1]. Within this formalism, the interaction of light with sample particles is given with a product of their Mueller matrix and the Stokes vector of the incident light. This yields another Stokes vector corresponding with the scattered light. In the case of irregular particles with random shape and orientation, the Mueller matrix takes on a simple, block-diagonal form with only six nonzero elements: F11 , F12 F21 , F22 , F33 , F34 −F43 , and F44 [3]. These elements are dependent on the geometry of illumination and observation of the target particles that are described with the scattering angle θ. It is significant that in [1], all nonzero elements of the Mueller matrix were measured in the forsterite particles over almost the entire range of scattering angle θ 5°–173°; the measurements were conducted at a single wavelength of λ 0.633 μm, though. The experimental results reported in [1] are publically available via the Amsterdam–Granada LightScattering Database [4]. Interestingly, all three samples reveal quite similar light-scattering responses. In this work, we analyze the initial sample and the small sample. The washed sample is not considered because its light-scattering response nearly coincides with what is in the initial sample, except for the Mueller matrix element F34 . Figure 1 shows all nonzero elements of the Mueller matrix as a function of the scattering angle θ at λ 0.633 μm. Because of technical difficulties accompanying the absolute radiometry, the element F11 is normalized at θ 90°, whereas other elements are given relative to F11 . We draw particular attention to element F11 and the ratio −F12 ∕F11 that are key parameters in passive remote sensing, exploiting the solar radiation as an incident light source. One distinctive feature of solar radiation is that it is highly unpolarized. However, scattered solar radiation acquires partial polarization that is quantified with the ratio and referred to as the degree of linear polarization P. Simultaneously, the element F11 corresponds with the total intensity of scattered sunlight. © 2015 Optical Society of America
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
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Fig. 1. Six nonzero elements of the Mueller matrix as a function of the scattering angle θ measured in the forsterite samples. Data are adapted from [1].
As one can see in Fig. 1, the light-scattering responses are remarkably similar in the initial sample and the small sample, yet the degree of linear polarization at side scattering θ ∼ 90° appears to be sensitive enough to discriminate one sample from another. However, the overall similarity in Fig. 1 indicates little difference in size distribution of the forsterite samples, at least in the range of sizes governing the light-scattering response. Interestingly, the projected-surface-area distribution reveals little difference in the initial sample and the small sample over the range of particle radii from r 0.1 to 3 μm [1]. The difference mainly arises at r > 3 μm. On the other hand, it is long known that submicron and micron-sized grains can be dramatically oversized with the Fritsch laser particle sizer [5 and therein]. Therefore, the measured size distribution likely is accompanied by significant uncertainties. It is of high practical interest to model the light-scattering responses in the forsterite samples, and clarify whether the resemblance in Fig. 1 is indeed caused by the minor differences in their size distributions over the range of submicron and micron-sized particles. Forsterite particles are modeled with the so-called agglomerated debris particles. This particle morphology has been developed to reproduce the phase functions of cometary dust. However, it also has been used to model laboratory optical measurements of feldspar particles having terrestrial origin [5]. The similarity of the forsterite samples with the feldspar samples is significant. Both were obtained by crushing large pieces of material to a fine-grained powder. Therefore, one can expect resemblance in their particle morphology. Agglomerated debris particles have highly irregular agglomerate morphologies, providing shapes similar to many classes of natural particles. For a detailed description of the algorithm for generation of agglomerated debris particles see, e.g., [5]. Six sample agglomerated debris particles are shown on the top of Fig. 2. We note
that the particles have fluffy morphology. On average, only 23.6% of the volume of their circumscribed spheres is occupied with material, corresponding with the filling factor or packing density. The average geometric cross section of the particles is equal to 0.61 of the projected area corresponding to the circumscribing sphere. Light scattering from agglomerated debris particles is computed with the discrete dipole approximation (DDA) (e.g., [6]), using the same programming implementation as in [5]. More details on this code, practical issues in the DDA simulation of light scattering from irregularly shaped particles, and control of the computational accuracy can be found in [5]. As noted in [1], there is a lack of information on the refractive index m in forsterite in visible. The authors have estimated it to be m ≈ 1.63 0.00001i at λ 0.633 μm. Imm seems to be well constrained, taking into account that the deposited forsterite samples are white in appearance, whereas Rem is inferred with uncertainty. For agglomerated debris particles in this work, the refractive index is preset to m 1.6 0.0005i, which is quite close to the estimation in [1]. Note also that for irregularly shaped particles comparable with wavelength, the Mueller matrix elements at Imm 0.001 are indistinguishable from those at Imm 0 [5]. Therefore, the difference in Imm suggested in [1] and in this work is not significant. The light-scattering response from micron-sized agglomerated debris particles is dependent on the size parameter x 2πr∕λ that is the ratio of the radius r of the circumscribing sphere to the wavelength of incident light λ. The considered range of size parameter x is 1–32; the upper limit of the range is limited by convergence of the DDA. At λ 0.633 μm, this range of x corresponds to particle radii from about 0.1 to 3.2 μm. The increment of x is 1 for the range x 1–16 and 2 for x > 16. For each x,
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OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Fig. 2. Six sample agglomerated debris particles (top) and results of modeling of the Mueller matrix elements in the initial sample (first and second rows) and the small sample (third and fourth rows). The best fit to the initial sample is obtained with a power law size distribution r −3.05 and to the small sample with r −3.2 .
the light-scattering responses of agglomerated debris particles are averaged over a minimum of 500 sample particles and their orientations. An analysis of the number density of particles nr in the initial sample and the small sample (data adapted from [4]) reveals that over the range r 0.6–3.2 μm,
nr in both samples can be satisfactorily approximated with the power law r −3 , whereas, at r 0.1–0.6 μm, there are significant deviations from this trend. On the other hand, retrievals of size distribution in the submicron domain are highly uncertain because of limitations of the Fritsch laser particle sizer (e.g., [5]). Therefore, the true
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
size distribution of particles with r 0.1–0.6 μm is unknown. The simplest solution of this dilemma is an extrapolation of the size distribution r −3 toward the submicron sizes. As was demonstrated in [5], such an extrapolation may provide agreeable fits to the measured light-scattering response and, therefore, we adapt it here. The size distribution r −3 is treated only as an initial estimate, whereas the power index a is considered to be a tunable parameter. By varying the power index a, the modeling results are adjusted to fit the measurements. As one can see in Fig. 1, the degree of linear polarization at side scattering θ ∼ 90° appears to be sensitive enough to discriminate unambiguously between the initial sample and the small sample. Therefore, the maximum of the positive polarization branch P max has been chosen as a leading characteristic for fitting. A reasonable fit can be achieved with the power index a −3.05 in the initial sample (first and second rows in Fig. 2) and a −3.2 in the small sample (third and fourth rows in Fig. 2). Both values appear to be quite close to the initial estimate a −3. Note also that the greater absolute value of a implies a higher relative abundance of small particles. Therefore, the present retrieval result is consistent with the differences expected in the two samples. However, the present modeling also makes it possible to perform a quantitative assessment of the differences in distributions of the samples. When approximating their size distribution with a power-law function, the measured difference in light scattering by the initial sample and by the small sample arises from only a small difference in the power index jΔaj 0.15. It is important to emphasize that while the modeling results were adjusted to optical measurements using only P max , the agglomerated debris particles almost perfectly fit the whole angular profiles of the degree of linear polarization in both samples. In particular, the model quite satisfactorily reproduces the phenomenon of the negative polarization that appears near the backscattering regime (at θ > 150°) in various natural samples and model particles (e.g., [7–9]). In addition to the degree of linear polarization for which the fit was optimized, the model also reveals reasonably good fits in the other Mueller matrix elements, although some differences can be noted in the ratios M22 ∕M11 , M44 ∕M11 , and M11 ∕M11 90°. A different optimization scheme could be used to fit all these parameters simultaneously. It is important that even when the
1207
modeling mismatches the measurements, it still adequately reproduces the relative difference between the initial sample and the small sample that can be seen in Fig. 1. With regard to the quantitative differences between the model simulations and measurements, this can be explained not only by the differences in morphologies of the model and sample particles, but also by the simplification of the particle distribution used in the modeling. Though agglomerated debris particles are indeed a reasonable approximation for the forsterite particles, they obviously were not designed to reproduce all the morphological features in the target particles. Another reason for the differences could be the contribution of particles with radii in excess of 3.2 μm, which are not taken into account in the DDA modeling. Nevertheless, despite these shortcomings, the differences between the model and measured results appear small. What is most significant is that the differences between the two samples can be demonstrated to be caused by the differences in their size distributions. The author thanks Dr. Gorden Videen and Dr. Olga Muñoz for valuable comments on this work and two anonymous referees for their constructive reviews. References 1. H. Volten, O. Muñoz, J. R. Brucato, J. W. Hovenier, L. Colangeli, L. B. F. M. Waters, and W. J. van der Zande, J. Quant. Spectrosc. Radiat. Transfer 100, 429 (2006). 2. D. H. Wooden, D. E. Harker, C. E. Woodward, H. M. Butner, C. Koike, F. C. Witteborn, and C. W. McMurtry, Astrophys. J. 517, 1034 (1999). 3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 4. O. Muñoz, F. Moreno, D. Guirado, D. D. Dabrowska, H. Volten, and J. W. Hovenier, J. Quant. Spectrosc. Radiat. Transfer 113, 565 (2012). 5. E. Zubko, K. Muinonen, O. Muñoz, T. Nousiainen, Yu. Shkuratov, W. Sun, and G. Videen, J. Quant. Spectrosc. Radiat. Transfer 131, 175 (2013). 6. B. T. Draine and P. J. Flatau, J. Opt. Soc. Am. A 11, 1491 (1994). 7. H. Volten, O. Muñoz, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, J. Geophys. Res. 106, 17375 (2001). 8. M. I. Mishchenko and L. Liu, J. Quant. Spectrosc. Radiat. Transfer 106, 616 (2007). 9. E. S. Zubko, G. Videen, and Yu. G. Shkuratov, J. Quant. Spectrosc. Radiat. Transfer 151, 38 (2015).
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Modeling light scattering by forsterite particles Evgenij Zubko Institute of Astronomy, V.N. Karazin Kharkiv National University, 35 Sumskaya St., Kharkov 61022, Ukraine ([email protected]) Received January 13, 2015; accepted February 17, 2015; posted February 18, 2015 (Doc. ID 232413); published March 17, 2015 Laboratory optical measurements of forsterite particles reveal remarkably similar light-scattering responses in two samples that were thought to obey different size distributions. These measurements are modeled with irregularly shaped agglomerated debris particles having a refractive index of m 1.6 0.0005i that is representative of forsterite in the visible. Modeling closely reproduces the measurements of both samples, making retrieval of their size distributions possible. © 2015 Optical Society of America OCIS codes: (280.4991) Passive remote sensing; (290.0290) Scattering; (290.5850) Scattering, particles; (290.5855) Scattering, polarization; (290.5890) Scattering, stimulated. http://dx.doi.org/10.1364/OL.40.001204
Laboratory optical measurements provide important clues for better understanding of the interrelation between the chemical and physical characteristics of small dust particles and their light-scattering response. In the case of naturally occurring dust particles, however, it often is difficult to characterize precisely the sample particles. An even more complicated problem is to control those characteristics. For instance, investigation of the impact of size distribution on light scattering implies an ability for controllable variation of the size distribution in sample particles. This is an extremely difficult technical problem to accomplish in practice. In this Letter, we analyze laboratory measurements of light scattering by forsterite particles that are thought to obey different size distributions [1]. Forsterite is a magnesium-rich olivine that is an abundant constituent of, for example, cometary dust (e.g., [2]). The forsterite sample investigated in [1] has a chemical composition of Mg1.9 Fe0.1 SiO4 . The sample was obtained by crushing a large piece of material to a finegrained powder. As a consequence, the particles reveal highly irregular shapes with sharp edges and corners. This can be seen in the SEM images presented in [1]. The forsterite powder was dry sieved first with a 50 μm sieve and, then, with a 20 μm sieve. The particles that did not pass through the second sieve were collected and designated as the initial sample. While the size of these particles is expected to be squeezed between 20 and 50 μm, inspection of the SEM images and measurement with a Fritsch laser particle sizer reveal a considerable amount of submicron and micron-sized particles. This is the result of electrostatic forces that cause particles to adhere to each other, resulting in agglomerates that cannot pass through a 20 μm sieve. To eliminate electrostatic forces, the authors moistened the forsterite powder with pure ethanol and repeated the sieving procedure with the wet powder. Particles that passed through a 20 μm sieve were collected, dried, and designated as a small sample, whereas the remaining particles collected on the 20 μm sieve are designated as the washed sample. Light scattering by all three samples was measured in the single-particle scattering regime. In this regime, an enormous number of sample particles is suspended in the air and irradiated with the incident light. The light scattered from such a cloud is representative simultaneously of a large number of particle shapes in random 0146-9592/15/071204-04$15.00/0
positions and orientations. However, the amount of ejected material is limited to keep the cloud optically thin and, thus, to ensure a negligible contribution from the multiple light scattering between particles. A beam of light can be fully characterized with a fourdimension Stokes vector [3], whereas the response from the target is expressed in terms of a 4×4 matrix that is referred to as the Mueller matrix [3] or the scattering matrix [1]. Within this formalism, the interaction of light with sample particles is given with a product of their Mueller matrix and the Stokes vector of the incident light. This yields another Stokes vector corresponding with the scattered light. In the case of irregular particles with random shape and orientation, the Mueller matrix takes on a simple, block-diagonal form with only six nonzero elements: F11 , F12 F21 , F22 , F33 , F34 −F43 , and F44 [3]. These elements are dependent on the geometry of illumination and observation of the target particles that are described with the scattering angle θ. It is significant that in [1], all nonzero elements of the Mueller matrix were measured in the forsterite particles over almost the entire range of scattering angle θ 5°–173°; the measurements were conducted at a single wavelength of λ 0.633 μm, though. The experimental results reported in [1] are publically available via the Amsterdam–Granada LightScattering Database [4]. Interestingly, all three samples reveal quite similar light-scattering responses. In this work, we analyze the initial sample and the small sample. The washed sample is not considered because its light-scattering response nearly coincides with what is in the initial sample, except for the Mueller matrix element F34 . Figure 1 shows all nonzero elements of the Mueller matrix as a function of the scattering angle θ at λ 0.633 μm. Because of technical difficulties accompanying the absolute radiometry, the element F11 is normalized at θ 90°, whereas other elements are given relative to F11 . We draw particular attention to element F11 and the ratio −F12 ∕F11 that are key parameters in passive remote sensing, exploiting the solar radiation as an incident light source. One distinctive feature of solar radiation is that it is highly unpolarized. However, scattered solar radiation acquires partial polarization that is quantified with the ratio and referred to as the degree of linear polarization P. Simultaneously, the element F11 corresponds with the total intensity of scattered sunlight. © 2015 Optical Society of America
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
1205
Fig. 1. Six nonzero elements of the Mueller matrix as a function of the scattering angle θ measured in the forsterite samples. Data are adapted from [1].
As one can see in Fig. 1, the light-scattering responses are remarkably similar in the initial sample and the small sample, yet the degree of linear polarization at side scattering θ ∼ 90° appears to be sensitive enough to discriminate one sample from another. However, the overall similarity in Fig. 1 indicates little difference in size distribution of the forsterite samples, at least in the range of sizes governing the light-scattering response. Interestingly, the projected-surface-area distribution reveals little difference in the initial sample and the small sample over the range of particle radii from r 0.1 to 3 μm [1]. The difference mainly arises at r > 3 μm. On the other hand, it is long known that submicron and micron-sized grains can be dramatically oversized with the Fritsch laser particle sizer [5 and therein]. Therefore, the measured size distribution likely is accompanied by significant uncertainties. It is of high practical interest to model the light-scattering responses in the forsterite samples, and clarify whether the resemblance in Fig. 1 is indeed caused by the minor differences in their size distributions over the range of submicron and micron-sized particles. Forsterite particles are modeled with the so-called agglomerated debris particles. This particle morphology has been developed to reproduce the phase functions of cometary dust. However, it also has been used to model laboratory optical measurements of feldspar particles having terrestrial origin [5]. The similarity of the forsterite samples with the feldspar samples is significant. Both were obtained by crushing large pieces of material to a fine-grained powder. Therefore, one can expect resemblance in their particle morphology. Agglomerated debris particles have highly irregular agglomerate morphologies, providing shapes similar to many classes of natural particles. For a detailed description of the algorithm for generation of agglomerated debris particles see, e.g., [5]. Six sample agglomerated debris particles are shown on the top of Fig. 2. We note
that the particles have fluffy morphology. On average, only 23.6% of the volume of their circumscribed spheres is occupied with material, corresponding with the filling factor or packing density. The average geometric cross section of the particles is equal to 0.61 of the projected area corresponding to the circumscribing sphere. Light scattering from agglomerated debris particles is computed with the discrete dipole approximation (DDA) (e.g., [6]), using the same programming implementation as in [5]. More details on this code, practical issues in the DDA simulation of light scattering from irregularly shaped particles, and control of the computational accuracy can be found in [5]. As noted in [1], there is a lack of information on the refractive index m in forsterite in visible. The authors have estimated it to be m ≈ 1.63 0.00001i at λ 0.633 μm. Imm seems to be well constrained, taking into account that the deposited forsterite samples are white in appearance, whereas Rem is inferred with uncertainty. For agglomerated debris particles in this work, the refractive index is preset to m 1.6 0.0005i, which is quite close to the estimation in [1]. Note also that for irregularly shaped particles comparable with wavelength, the Mueller matrix elements at Imm 0.001 are indistinguishable from those at Imm 0 [5]. Therefore, the difference in Imm suggested in [1] and in this work is not significant. The light-scattering response from micron-sized agglomerated debris particles is dependent on the size parameter x 2πr∕λ that is the ratio of the radius r of the circumscribing sphere to the wavelength of incident light λ. The considered range of size parameter x is 1–32; the upper limit of the range is limited by convergence of the DDA. At λ 0.633 μm, this range of x corresponds to particle radii from about 0.1 to 3.2 μm. The increment of x is 1 for the range x 1–16 and 2 for x > 16. For each x,
1206
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
Fig. 2. Six sample agglomerated debris particles (top) and results of modeling of the Mueller matrix elements in the initial sample (first and second rows) and the small sample (third and fourth rows). The best fit to the initial sample is obtained with a power law size distribution r −3.05 and to the small sample with r −3.2 .
the light-scattering responses of agglomerated debris particles are averaged over a minimum of 500 sample particles and their orientations. An analysis of the number density of particles nr in the initial sample and the small sample (data adapted from [4]) reveals that over the range r 0.6–3.2 μm,
nr in both samples can be satisfactorily approximated with the power law r −3 , whereas, at r 0.1–0.6 μm, there are significant deviations from this trend. On the other hand, retrievals of size distribution in the submicron domain are highly uncertain because of limitations of the Fritsch laser particle sizer (e.g., [5]). Therefore, the true
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
size distribution of particles with r 0.1–0.6 μm is unknown. The simplest solution of this dilemma is an extrapolation of the size distribution r −3 toward the submicron sizes. As was demonstrated in [5], such an extrapolation may provide agreeable fits to the measured light-scattering response and, therefore, we adapt it here. The size distribution r −3 is treated only as an initial estimate, whereas the power index a is considered to be a tunable parameter. By varying the power index a, the modeling results are adjusted to fit the measurements. As one can see in Fig. 1, the degree of linear polarization at side scattering θ ∼ 90° appears to be sensitive enough to discriminate unambiguously between the initial sample and the small sample. Therefore, the maximum of the positive polarization branch P max has been chosen as a leading characteristic for fitting. A reasonable fit can be achieved with the power index a −3.05 in the initial sample (first and second rows in Fig. 2) and a −3.2 in the small sample (third and fourth rows in Fig. 2). Both values appear to be quite close to the initial estimate a −3. Note also that the greater absolute value of a implies a higher relative abundance of small particles. Therefore, the present retrieval result is consistent with the differences expected in the two samples. However, the present modeling also makes it possible to perform a quantitative assessment of the differences in distributions of the samples. When approximating their size distribution with a power-law function, the measured difference in light scattering by the initial sample and by the small sample arises from only a small difference in the power index jΔaj 0.15. It is important to emphasize that while the modeling results were adjusted to optical measurements using only P max , the agglomerated debris particles almost perfectly fit the whole angular profiles of the degree of linear polarization in both samples. In particular, the model quite satisfactorily reproduces the phenomenon of the negative polarization that appears near the backscattering regime (at θ > 150°) in various natural samples and model particles (e.g., [7–9]). In addition to the degree of linear polarization for which the fit was optimized, the model also reveals reasonably good fits in the other Mueller matrix elements, although some differences can be noted in the ratios M22 ∕M11 , M44 ∕M11 , and M11 ∕M11 90°. A different optimization scheme could be used to fit all these parameters simultaneously. It is important that even when the
1207
modeling mismatches the measurements, it still adequately reproduces the relative difference between the initial sample and the small sample that can be seen in Fig. 1. With regard to the quantitative differences between the model simulations and measurements, this can be explained not only by the differences in morphologies of the model and sample particles, but also by the simplification of the particle distribution used in the modeling. Though agglomerated debris particles are indeed a reasonable approximation for the forsterite particles, they obviously were not designed to reproduce all the morphological features in the target particles. Another reason for the differences could be the contribution of particles with radii in excess of 3.2 μm, which are not taken into account in the DDA modeling. Nevertheless, despite these shortcomings, the differences between the model and measured results appear small. What is most significant is that the differences between the two samples can be demonstrated to be caused by the differences in their size distributions. The author thanks Dr. Gorden Videen and Dr. Olga Muñoz for valuable comments on this work and two anonymous referees for their constructive reviews. References 1. H. Volten, O. Muñoz, J. R. Brucato, J. W. Hovenier, L. Colangeli, L. B. F. M. Waters, and W. J. van der Zande, J. Quant. Spectrosc. Radiat. Transfer 100, 429 (2006). 2. D. H. Wooden, D. E. Harker, C. E. Woodward, H. M. Butner, C. Koike, F. C. Witteborn, and C. W. McMurtry, Astrophys. J. 517, 1034 (1999). 3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 4. O. Muñoz, F. Moreno, D. Guirado, D. D. Dabrowska, H. Volten, and J. W. Hovenier, J. Quant. Spectrosc. Radiat. Transfer 113, 565 (2012). 5. E. Zubko, K. Muinonen, O. Muñoz, T. Nousiainen, Yu. Shkuratov, W. Sun, and G. Videen, J. Quant. Spectrosc. Radiat. Transfer 131, 175 (2013). 6. B. T. Draine and P. J. Flatau, J. Opt. Soc. Am. A 11, 1491 (1994). 7. H. Volten, O. Muñoz, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, J. Geophys. Res. 106, 17375 (2001). 8. M. I. Mishchenko and L. Liu, J. Quant. Spectrosc. Radiat. Transfer 106, 616 (2007). 9. E. S. Zubko, G. Videen, and Yu. G. Shkuratov, J. Quant. Spectrosc. Radiat. Transfer 151, 38 (2015).
