Abstract

We use the discrete dipole approximation (DDA) method to calculate the intensity and the linear polarization degree of light scattered by agglomerated debris particles with hierarchical structure as functions of size parameter (varying from x = 2 to x = 14) and phase angle. Such structures are important, e.g., for cometary and interplanetary dust particles. Calculations for three combinations of refractive index were made, which correspond to regions of water ice, organic matter, and silicates. We examine the photometric and polarization properties of agglomerated particles with prefractal (Whitten–Sander model) and nonfractal porous structures of particle fragments formed by dipoles. We find that the aggregated particles can produce significant negative polarization at small phase angles. Increasing the packing density of dipoles and/or refractive index makes the negative polarization more prominent. The depth of the negative polarization branch depends on the type of internal structure: the negative polarization branch of particles having nonfractal structure is noticeably shallower in comparison with that of those having a prefractal structure. The negative polarization branch depth strongly depends on the imaginary part of the refractive index and increases with decreasing absorption. Polarization phase curves for agglomerated debris particles become smoother as the number of hierarchical levels increases.

© 2005 Optical Society of America

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  1. G. Videen, W. Sun, Q. Fu, D. R. Secker, R. Greenaway, P. H. Kaye, E. Hirst, D. Bartley, “Light scattering from deformed droplets and droplets with inclusions: II. Theoretical treatment,” Appl. Opt. 39, 5031–5039 (2000).
    [CrossRef]
  2. D. R. Secker, R. Greenaway, P. H. Kaye, E. Hirst, D. Bartley, G. Videen, “Light scattering from deformed droplets and droplets with inclusions: I. Experimental results,” Appl. Opt. 39, 5023–5030 (2000).
    [CrossRef]
  3. M. I. Mishchenko, J. J. Hovenier, W. J. Wiscombe, L. D. Travis, “Light scattering by nonspherical particles: theory, measurements, and applications,” M. I. Mishchenko, J. W. Hovenier, L. D. Travis eds. (Academic Press, 2000) pp. 29–60.
    [CrossRef]
  4. B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1498 (1994).
    [CrossRef]
  5. K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
    [CrossRef]
  6. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys, J. 186, 705–714 (1973).
    [CrossRef]
  7. E. S. Zubko, Y. G. Shkuratov, N. N. Kiselev, G. Videen, “DDA simulations of light scattering by small irregular particles with various structure,” J. Quant. Spectrosc. Radiat. Transfer (2005) (in press).
  8. G. Wurm, J. Blum, “Experiments on preplanetary dust aggregation,” Icarus 132, 125–136 (1998).
    [CrossRef]
  9. J. A. Nuth, F. J. Rietmeijer, H. G. Hill, “Condensation processes in astrophysical environments: the composition and structure of cometary grains,” Meteorit. Planet. Sci. 37, 1579–1590 (2002).
    [CrossRef]
  10. A. Li, J. Greenberg, “A unified model of interstellar dust,” Astron. Astrophys. 232, 566–584 (1997).
  11. J. Feder, “Fractals,” (Plenum, 1988).
    [CrossRef]
  12. B. M. Smirnov, “Fractal clusters,” Usp. Fiz. Nauk 149, 177–217 (1986).
    [CrossRef]
  13. T. Whitten, L. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
    [CrossRef]
  14. A. V. Belko, A. V. Nikitin, “The methods of fractal structure objects creation,” Vestr. Grodno State Univ. 2, 133–137 (2002).
  15. H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
    [CrossRef]
  16. K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.
  17. Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

2002 (2)

J. A. Nuth, F. J. Rietmeijer, H. G. Hill, “Condensation processes in astrophysical environments: the composition and structure of cometary grains,” Meteorit. Planet. Sci. 37, 1579–1590 (2002).
[CrossRef]

A. V. Belko, A. V. Nikitin, “The methods of fractal structure objects creation,” Vestr. Grodno State Univ. 2, 133–137 (2002).

2001 (1)

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

2000 (2)

1998 (1)

G. Wurm, J. Blum, “Experiments on preplanetary dust aggregation,” Icarus 132, 125–136 (1998).
[CrossRef]

1997 (1)

A. Li, J. Greenberg, “A unified model of interstellar dust,” Astron. Astrophys. 232, 566–584 (1997).

1994 (2)

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1498 (1994).
[CrossRef]

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

1986 (1)

B. M. Smirnov, “Fractal clusters,” Usp. Fiz. Nauk 149, 177–217 (1986).
[CrossRef]

1981 (1)

T. Whitten, L. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys, J. 186, 705–714 (1973).
[CrossRef]

Bartley, D.

Belko, A. V.

A. V. Belko, A. V. Nikitin, “The methods of fractal structure objects creation,” Vestr. Grodno State Univ. 2, 133–137 (2002).

Belskaya, I.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

Blum, J.

G. Wurm, J. Blum, “Experiments on preplanetary dust aggregation,” Icarus 132, 125–136 (1998).
[CrossRef]

Clark, B.

K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.

de Haan, J.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Draine, B. T.

Feder, J.

J. Feder, “Fractals,” (Plenum, 1988).
[CrossRef]

Flatau, P. J.

Fu, Q.

Greenaway, R.

Greenberg, J.

A. Li, J. Greenberg, “A unified model of interstellar dust,” Astron. Astrophys. 232, 566–584 (1997).

Hill, H. G.

J. A. Nuth, F. J. Rietmeijer, H. G. Hill, “Condensation processes in astrophysical environments: the composition and structure of cometary grains,” Meteorit. Planet. Sci. 37, 1579–1590 (2002).
[CrossRef]

Hirst, E.

Hovenier, J.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Hovenier, J. J.

M. I. Mishchenko, J. J. Hovenier, W. J. Wiscombe, L. D. Travis, “Light scattering by nonspherical particles: theory, measurements, and applications,” M. I. Mishchenko, J. W. Hovenier, L. D. Travis eds. (Academic Press, 2000) pp. 29–60.
[CrossRef]

Kaydash, V.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

Kaye, P. H.

Kiselev, N. N.

E. S. Zubko, Y. G. Shkuratov, N. N. Kiselev, G. Videen, “DDA simulations of light scattering by small irregular particles with various structure,” J. Quant. Spectrosc. Radiat. Transfer (2005) (in press).

Kreslavsky, M.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

Li, A.

A. Li, J. Greenberg, “A unified model of interstellar dust,” Astron. Astrophys. 232, 566–584 (1997).

Lumme, K.

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, J. J. Hovenier, W. J. Wiscombe, L. D. Travis, “Light scattering by nonspherical particles: theory, measurements, and applications,” M. I. Mishchenko, J. W. Hovenier, L. D. Travis eds. (Academic Press, 2000) pp. 29–60.
[CrossRef]

Muinonen, K.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.

Muñoz, O.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Nikitin, A. V.

A. V. Belko, A. V. Nikitin, “The methods of fractal structure objects creation,” Vestr. Grodno State Univ. 2, 133–137 (2002).

Nousianen, T.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Nuth, J. A.

J. A. Nuth, F. J. Rietmeijer, H. G. Hill, “Condensation processes in astrophysical environments: the composition and structure of cometary grains,” Meteorit. Planet. Sci. 37, 1579–1590 (2002).
[CrossRef]

Opanasenko, N.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

Ovcharenko, A.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys, J. 186, 705–714 (1973).
[CrossRef]

Piironen, J.

K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys, J. 186, 705–714 (1973).
[CrossRef]

Rahola, J.

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

Rietmeijer, F. J.

J. A. Nuth, F. J. Rietmeijer, H. G. Hill, “Condensation processes in astrophysical environments: the composition and structure of cometary grains,” Meteorit. Planet. Sci. 37, 1579–1590 (2002).
[CrossRef]

Rol, E.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Sander, L.

T. Whitten, L. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Secker, D. R.

Shkuratov, Y. G.

E. S. Zubko, Y. G. Shkuratov, N. N. Kiselev, G. Videen, “DDA simulations of light scattering by small irregular particles with various structure,” J. Quant. Spectrosc. Radiat. Transfer (2005) (in press).

Shkuratov, Yu

K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.

Shkuratov, Yu.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

Smirnov, B. M.

B. M. Smirnov, “Fractal clusters,” Usp. Fiz. Nauk 149, 177–217 (1986).
[CrossRef]

Sun, W.

Travis, L. D.

M. I. Mishchenko, J. J. Hovenier, W. J. Wiscombe, L. D. Travis, “Light scattering by nonspherical particles: theory, measurements, and applications,” M. I. Mishchenko, J. W. Hovenier, L. D. Travis eds. (Academic Press, 2000) pp. 29–60.
[CrossRef]

Vassen, W.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Videen, G.

G. Videen, W. Sun, Q. Fu, D. R. Secker, R. Greenaway, P. H. Kaye, E. Hirst, D. Bartley, “Light scattering from deformed droplets and droplets with inclusions: II. Theoretical treatment,” Appl. Opt. 39, 5031–5039 (2000).
[CrossRef]

D. R. Secker, R. Greenaway, P. H. Kaye, E. Hirst, D. Bartley, G. Videen, “Light scattering from deformed droplets and droplets with inclusions: I. Experimental results,” Appl. Opt. 39, 5023–5030 (2000).
[CrossRef]

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

E. S. Zubko, Y. G. Shkuratov, N. N. Kiselev, G. Videen, “DDA simulations of light scattering by small irregular particles with various structure,” J. Quant. Spectrosc. Radiat. Transfer (2005) (in press).

Volten, H.

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

Whitten, T.

T. Whitten, L. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Wiscombe, W. J.

M. I. Mishchenko, J. J. Hovenier, W. J. Wiscombe, L. D. Travis, “Light scattering by nonspherical particles: theory, measurements, and applications,” M. I. Mishchenko, J. W. Hovenier, L. D. Travis eds. (Academic Press, 2000) pp. 29–60.
[CrossRef]

Wurm, G.

G. Wurm, J. Blum, “Experiments on preplanetary dust aggregation,” Icarus 132, 125–136 (1998).
[CrossRef]

Zubko, E.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

Zubko, E. S.

E. S. Zubko, Y. G. Shkuratov, N. N. Kiselev, G. Videen, “DDA simulations of light scattering by small irregular particles with various structure,” J. Quant. Spectrosc. Radiat. Transfer (2005) (in press).

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Li, J. Greenberg, “A unified model of interstellar dust,” Astron. Astrophys. 232, 566–584 (1997).

Astrophys, J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys, J. 186, 705–714 (1973).
[CrossRef]

Astrophys. J. (1)

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

Icarus (1)

G. Wurm, J. Blum, “Experiments on preplanetary dust aggregation,” Icarus 132, 125–136 (1998).
[CrossRef]

J. Geophys. Res. (1)

H. Volten, O. Muñoz, E. Rol, J. de Haan, W. Vassen, J. Hovenier, K. Muinonen, T. Nousianen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106, 17375–17401 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

Meteorit. Planet. Sci. (1)

J. A. Nuth, F. J. Rietmeijer, H. G. Hill, “Condensation processes in astrophysical environments: the composition and structure of cometary grains,” Meteorit. Planet. Sci. 37, 1579–1590 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

T. Whitten, L. Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon,” Phys. Rev. Lett. 47, 1400–1403 (1981).
[CrossRef]

Usp. Fiz. Nauk (1)

B. M. Smirnov, “Fractal clusters,” Usp. Fiz. Nauk 149, 177–217 (1986).
[CrossRef]

Vestr. Grodno State Univ. (1)

A. V. Belko, A. V. Nikitin, “The methods of fractal structure objects creation,” Vestr. Grodno State Univ. 2, 133–137 (2002).

Other (5)

K. Muinonen, J. Piironen, Yu Shkuratov, A. Ovcharenko, B. Clark, “Asteroid photometric and polarimetric phase effect,” in Asteroids III, W. Bottke, R. Binzel, A. Cellino, P. Paolicchi, eds. (University of Arizona Press, 2002), pp. 123–138.

Yu. Shkuratov, G. Videen, M. Kreslavsky, I. Belskaya, A. Ovcharenko, V. Kaydash, N. Opanasenko, E. Zubko, “Scattering properties of planetary regoliths near opposition,” in Photopolarimetry in Remote Sensing, G. Videen, Y. Yatskiv, M. Mishchenko, eds. NATO Science Series, (Kluwer Academic2004), pp. 191–208.

M. I. Mishchenko, J. J. Hovenier, W. J. Wiscombe, L. D. Travis, “Light scattering by nonspherical particles: theory, measurements, and applications,” M. I. Mishchenko, J. W. Hovenier, L. D. Travis eds. (Academic Press, 2000) pp. 29–60.
[CrossRef]

J. Feder, “Fractals,” (Plenum, 1988).
[CrossRef]

E. S. Zubko, Y. G. Shkuratov, N. N. Kiselev, G. Videen, “DDA simulations of light scattering by small irregular particles with various structure,” J. Quant. Spectrosc. Radiat. Transfer (2005) (in press).

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Figures (6)

Fig. 1
Fig. 1

Examples of hierarchical debris particles: (a) first-level, (b) second-level, (c) third-level.

Fig. 2
Fig. 2

Example of a 3–D Whitten–Sander’s fractal structure.

Fig. 3
Fig. 3

Examples of an agglomerated debris particle (top), and the internal structure of the subparticle shown by the arrow (bottom) constructed with (a) a densely packed structure, (b) a prefractal structure generated with Whitten–Sander’s algorithm, (c) a porous debris particle in which the dipoles are inserted randomly on the matrix.

Fig. 4
Fig. 4

Logarithm of intensity of scattered light (first row) and degree of linear polarization (second row) as functions of phase angle α and size parameter x for first-level hierarchical debris particles consisting of densely packed dipoles at (a) refractive index m = 1.31 (icy particles), (b) m = 1.5 + 0.1i (organic particles), (c) m = 1.6 + 5 × 10−4i (silicate particles).

Fig. 5
Fig. 5

Same as in Fig. 4 for second-level hierarchical debris particles (first row and second row) and for third-level hierarchical debris particles (third row and fourth row).

Fig. 6
Fig. 6

Logarithm of intensity of scattered light (first row) and degree of linear polarization (second row) as functions of phase angle α and size parameter x for first-level hierarchical debris particles, whose interiors are constructed using the Whitten–Sander’s model, and the same for first-level hierarchical debris particles whose interiors are constructed of randomly inserted dipoles (third row and fourth row) at (a) refractive index m = 1.31 (icy particles), (b) m = 1.5 + 0.1i (organic particles), (c) m = 1.6 + 5 × 10−4i (silicate particles).

Equations (1)

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E ( r ) = E inc ( r ) + ( graddiv + k 2 ) 1 4 π V ( ɛ ( r ) - 1 ) E ( r ) exp ( i k r - r ) r - r d r ,

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