Abstract

The approximate bulk-scattering phase function of a polydisperse system of dust particles is derived in an analytical form. In the theoretical solution, the particle size distribution is modeled by a modified gamma function that can satisfy various media differing in modal radii. Unlike the frequently applied power law, the modified gamma distribution shows no singularity when the particle radius approaches zero. The approximate scattering phase function is related to the parameters of the size distribution function. This is an important advantage compared to the empirical Henyey–Greenstein (HG) approximation, which is a simple function of the average cosine. However, any optimized value of average cosine of the HG function cannot provide the information on particle microphysical characteristics, such as the size distribution function. In this paper, the mapping between average cosine and the parameters of size distribution function is given by a semianalytical expression that is applicable in rapid numerical simulations on various dust populations. In particular, the modal radius and half-width can be quickly estimated using the presented formulas.

© 2011 Optical Society of America

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  3. M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
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    [CrossRef]
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    [CrossRef]
  24. C. M. Sorensen and D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
    [CrossRef]
  25. R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
    [CrossRef]
  26. B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT7.1,” http://arXiv.org/abs/1002.1505v1 (2007).
  27. H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
    [CrossRef]
  28. M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

2010 (4)

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

M. Vincendon and Y. Langevin, “A spherical Monte Carlo model of aerosols: validation and first applications to Mars and Titan,” Icarus 207, 923–931 (2010).
[CrossRef]

M. Ádámkovics, J. W. Barnes, M. Hartung, and I. de Pater, “Observations of a stationary mid-latitude cloud system on Titan,” Icarus 208, 868–877 (2010).
[CrossRef]

S. L. Gratiy, D. A. Levin, and A. C. Walker, “Rassvet: backward Monte Carlo radiative transfer in spherical-shell planetary atmospheres,” Icarus 206, 366–379 (2010).
[CrossRef]

2009 (1)

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

2008 (3)

A. Goobar, “Low Rv from circumstellar dust around supernovae,” Astrophys. J. 686, L103–L106 (2008).
[CrossRef]

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

2007 (1)

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

2006 (2)

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
[CrossRef]

2005 (1)

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

2004 (1)

J. Warell, “Properties of the Herman regolith: IV. Photometric parameters of Mercury and the Moon contrasted with Hapke modelling,” Icarus 167, 271–286 (2004).
[CrossRef]

2001 (2)

M. Caldas and V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

Z. M. Dlugach and A. V. Morozhenko, “Parameters of dust particles in the Martian atmosphere,” Solar System Research 35, 421–430 (2001).
[CrossRef]

2000 (3)

S. K. Sharma and A. K. Roy, “New approximate phase functions: test for nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 64, 327–337 (2000).
[CrossRef]

C. M. Sorensen and D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

1998 (1)

B. Hartman and D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

1996 (1)

N. C. Wickramasinghe and F. Hoyle, “Very small dust particles (Vsdps) in comet C/1996 B2 (Hyakutake),” Astrophys. Space Sci. 239, 121–123 (1996).
[CrossRef]

1994 (1)

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

1979 (1)

A’Hearn, M. F.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Ádámkovics, M.

M. Ádámkovics, J. W. Barnes, M. Hartung, and I. de Pater, “Observations of a stationary mid-latitude cloud system on Titan,” Icarus 208, 868–877 (2010).
[CrossRef]

Baines, K. H.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Baratoux, D.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Barnes, J. W.

M. Ádámkovics, J. W. Barnes, M. Hartung, and I. de Pater, “Observations of a stationary mid-latitude cloud system on Titan,” Icarus 208, 868–877 (2010).
[CrossRef]

Belton, J. S.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Bianchi, L.

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

Bibring, J. P.

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

Bohren, K.

K. Bohren and D. Hufmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Box, M. A.

Brown, R. H.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Buratti, B. J.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Caldas, M.

M. Caldas and V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

Cantor, B. A.

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

Chevrel, S.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Clancy, R. T.

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

Clark, R. N.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Coquilla, R.

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

Cord, A.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Crockett, C. J.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Daydou, Y.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

de Pater, I.

M. Ádámkovics, J. W. Barnes, M. Hartung, and I. de Pater, “Observations of a stationary mid-latitude cloud system on Titan,” Icarus 208, 868–877 (2010).
[CrossRef]

Deepak, A.

Dlugach, Z. M.

Z. M. Dlugach and A. V. Morozhenko, “Parameters of dust particles in the Martian atmosphere,” Solar System Research 35, 421–430 (2001).
[CrossRef]

Domingue, D.

B. Hartman and D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

Draine, B. T.

B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT7.1,” http://arXiv.org/abs/1002.1505v1 (2007).

Fadeev, V. Ya.

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

Farnham, T. L.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Fischbach, D. J.

C. M. Sorensen and D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

Flatau, P. J.

B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT7.1,” http://arXiv.org/abs/1002.1505v1 (2007).

Goguen, J. D.

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

Gondet, B.

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

Goobar, A.

A. Goobar, “Low Rv from circumstellar dust around supernovae,” Astrophys. J. 686, L103–L106 (2008).
[CrossRef]

Gratiy, S. L.

S. L. Gratiy, D. A. Levin, and A. C. Walker, “Rassvet: backward Monte Carlo radiative transfer in spherical-shell planetary atmospheres,” Icarus 206, 366–379 (2010).
[CrossRef]

Greeley, R.

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

Gwinner, K.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Haberle, R.

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

Hartman, B.

B. Hartman and D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

Hartung, M.

M. Ádámkovics, J. W. Barnes, M. Hartung, and I. de Pater, “Observations of a stationary mid-latitude cloud system on Titan,” Icarus 208, 868–877 (2010).
[CrossRef]

Henry, R. C.

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

Heuripeau, F.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Hicks, M. D.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Hoffman, H.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Horvath, H.

H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
[CrossRef]

Hoyle, F.

N. C. Wickramasinghe and F. Hoyle, “Very small dust particles (Vsdps) in comet C/1996 B2 (Hyakutake),” Astrophys. Space Sci. 239, 121–123 (1996).
[CrossRef]

Hufmann, D.

K. Bohren and D. Hufmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jaumann, R.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Jehl, A.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Jouglet, D.

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

Kabanov, M. V.

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

Karnataki, A.

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

Kasahara, M.

H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
[CrossRef]

Kauhanen, J.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Kocifaj, M.

H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
[CrossRef]

Kulmala, M.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

A. A. Lacis and M. I. Mishchenko, “Climate forcing, climate sensitivity, and climate response: a radiative modeling perspective on atmospheric aerosols,” in Aerosol Forcing of Climate, R.L.Charlson and J.Heintzenberg, eds. (Wiley, 1994).

Langevin, Y.

M. Vincendon and Y. Langevin, “A spherical Monte Carlo model of aerosols: validation and first applications to Mars and Titan,” Icarus 207, 923–931 (2010).
[CrossRef]

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

Lauri, A.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Levin, D. A.

S. L. Gratiy, D. A. Levin, and A. C. Walker, “Rassvet: backward Monte Carlo radiative transfer in spherical-shell planetary atmospheres,” Icarus 206, 366–379 (2010).
[CrossRef]

Li, J. Y.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Lisse, C. M.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Määttänen, A.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Malin, M. C.

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

Manaud, N.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

McCord, T. B.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

McFadden, L. A.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Meech, K. J.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Merikallio, S.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

A. A. Lacis and M. I. Mishchenko, “Climate forcing, climate sensitivity, and climate response: a radiative modeling perspective on atmospheric aerosols,” in Aerosol Forcing of Climate, R.L.Charlson and J.Heintzenberg, eds. (Wiley, 1994).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

Momary, T.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Morozhenko, A. V.

Z. M. Dlugach and A. V. Morozhenko, “Parameters of dust particles in the Martian atmosphere,” Solar System Research 35, 421–430 (2001).
[CrossRef]

Mosher, J. A.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Murthy, J.

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

Neukum, G.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Nicholson, P. D.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Panchenko, M. V.

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

Pinet, P.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Pkhalagov, Yu. A.

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

Poulet, F.

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

Roatsch, T.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Rosemberg, C.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Roy, A. K.

S. K. Sharma and A. K. Roy, “New approximate phase functions: test for nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 64, 327–337 (2000).
[CrossRef]

Savijärvi, H.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Scholten, F.

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

Semião, V.

M. Caldas and V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

Sharma, S. K.

S. K. Sharma and A. K. Roy, “New approximate phase functions: test for nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 64, 327–337 (2000).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, “The study of the properties of matter from single scattering,” in Theoretical and Applied Problems of Light Scattering (Nauka i Tekhnika, 1971), pp. 228–244.

Sicardy, B.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Simonelli, D. P.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Sorensen, C. M.

C. M. Sorensen and D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

Sotin, C.

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Sujatha, N. V.

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

Sunshine, J. M.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Thomas, P. C.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Tohno, S.

H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

Uzhegov, V. N.

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

Vehkamäki, H.

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

Veretennikov, V. V.

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

Veverka, J.

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

Vincendon, M.

M. Vincendon and Y. Langevin, “A spherical Monte Carlo model of aerosols: validation and first applications to Mars and Titan,” Icarus 207, 923–931 (2010).
[CrossRef]

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

Walker, A. C.

S. L. Gratiy, D. A. Levin, and A. C. Walker, “Rassvet: backward Monte Carlo radiative transfer in spherical-shell planetary atmospheres,” Icarus 206, 366–379 (2010).
[CrossRef]

Warell, J.

J. Warell, “Properties of the Herman regolith: IV. Photometric parameters of Mercury and the Moon contrasted with Hapke modelling,” Icarus 167, 271–286 (2004).
[CrossRef]

White, B.

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

Wickramasinghe, N. C.

N. C. Wickramasinghe and F. Hoyle, “Very small dust particles (Vsdps) in comet C/1996 B2 (Hyakutake),” Astrophys. Space Sci. 239, 121–123 (1996).
[CrossRef]

Wilson, G.

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

Wolff, M. J.

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (2)

N. V. Sujatha, J. Murthy, A. Karnataki, R. C. Henry, and L. Bianchi, “GALEX observations of diffuse UV radiation at high spatial resolution from the Sandage nebulosity,” Astrophys. J. 692, 1333–1338 (2009).
[CrossRef]

A. Goobar, “Low Rv from circumstellar dust around supernovae,” Astrophys. J. 686, L103–L106 (2008).
[CrossRef]

Astrophys. Space Sci. (1)

N. C. Wickramasinghe and F. Hoyle, “Very small dust particles (Vsdps) in comet C/1996 B2 (Hyakutake),” Astrophys. Space Sci. 239, 121–123 (1996).
[CrossRef]

Icarus (9)

J. Warell, “Properties of the Herman regolith: IV. Photometric parameters of Mercury and the Moon contrasted with Hapke modelling,” Icarus 167, 271–286 (2004).
[CrossRef]

J. Y. Li, M. F. A’Hearn, J. S. Belton, C. J. Crockett, T. L. Farnham, C. M. Lisse, L. A. McFadden, K. J. Meech, J. M. Sunshine, P. C. Thomas, and J. Veverka, “Deep impact photometry of comet 9P/Temel 1,” Icarus 187, 41–55 (2007).
[CrossRef]

M. Ádámkovics, J. W. Barnes, M. Hartung, and I. de Pater, “Observations of a stationary mid-latitude cloud system on Titan,” Icarus 208, 868–877 (2010).
[CrossRef]

S. L. Gratiy, D. A. Levin, and A. C. Walker, “Rassvet: backward Monte Carlo radiative transfer in spherical-shell planetary atmospheres,” Icarus 206, 366–379 (2010).
[CrossRef]

M. J. Wolff, R. T. Clancy, J. D. Goguen, M. C. Malin, and B. A. Cantor, “Ultraviolet dust aerosol properties as observed by MARCI,” Icarus 208, 143–155 (2010).
[CrossRef]

M. Vincendon, Y. Langevin, F. Poulet, J. P. Bibring, B. Gondet, D. Jouglet, and OMEGA Team, “Dust aerosols above the south polar cap of Mars as seen by OMEGA,” Icarus 196, 488–505 (2008).
[CrossRef]

A. Jehl, P. Pinet, D. Baratoux, Y. Daydou, S. Chevrel, F. Heuripeau, N. Manaud, A. Cord, C. Rosemberg, G. Neukum, K. Gwinner, F. Scholten, H. Hoffman, T. Roatsch, and HRSC Team, “Gusev photometric variability as seen from orbit by HRSC/Mars-express,” Icarus 197, 403–428 (2008).
[CrossRef]

M. Vincendon and Y. Langevin, “A spherical Monte Carlo model of aerosols: validation and first applications to Mars and Titan,” Icarus 207, 923–931 (2010).
[CrossRef]

B. Hartman and D. Domingue, “Scattering of light by individual particles and the implications for models of planetary surfaces,” Icarus 131, 421–448 (1998).
[CrossRef]

J. Aerosol Sci. (1)

H. Horvath, M. Kasahara, S. Tohno, and M. Kocifaj, “Angular scattering of the Gobi Desert aerosol and its influence on radiative forcing,” J. Aerosol Sci. 37, 1287–1302 (2006).
[CrossRef]

J. Geophys. Res. (1)

A. Määttänen, H. Vehkamäki, A. Lauri, S. Merikallio, J. Kauhanen, H. Savijärvi, and M. Kulmala, “Nucleation studies in the Martian atmosphere,” J. Geophys. Res. 110, E02002 (2005).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (3)

M. I. Mishchenko, “Asymmetry parameters of the phase function for densely packed scattering grains,” J. Quant. Spectrosc. Radiat. Transfer 52, 95–110 (1994).
[CrossRef]

M. Caldas and V. Semião, “A new approximate phase function for isolated particles and polydispersions,” J. Quant. Spectrosc. Radiat. Transfer 68, 521–542 (2001).
[CrossRef]

S. K. Sharma and A. K. Roy, “New approximate phase functions: test for nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 64, 327–337 (2000).
[CrossRef]

Opt. Commun. (1)

C. M. Sorensen and D. J. Fischbach, “Patterns in Mie scattering,” Opt. Commun. 173, 145–153 (2000).
[CrossRef]

Planet. Space Sci. (2)

R. Greeley, G. Wilson, R. Coquilla, B. White, and R. Haberle, “Windblown dust on Mars: laboratory simulations on flux as a function of surface roughness,” Planet. Space Sci. 48, 1349–1355 (2000).
[CrossRef]

B. J. Buratti, C. Sotin, R. H. Brown, M. D. Hicks, R. N. Clark, J. A. Mosher, T. B. McCord, R. Jaumann, K. H. Baines, P. D. Nicholson, T. Momary, D. P. Simonelli, and B. Sicardy, “Titan: preliminary results on surface properties and photometry from VIMS observations of the early flybys,” Planet. Space Sci. 54, 1498–1509 (2006).
[CrossRef]

Solar System Research (1)

Z. M. Dlugach and A. V. Morozhenko, “Parameters of dust particles in the Martian atmosphere,” Solar System Research 35, 421–430 (2001).
[CrossRef]

Other (6)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University, 2002).

K. Bohren and D. Hufmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

A. A. Lacis and M. I. Mishchenko, “Climate forcing, climate sensitivity, and climate response: a radiative modeling perspective on atmospheric aerosols,” in Aerosol Forcing of Climate, R.L.Charlson and J.Heintzenberg, eds. (Wiley, 1994).

K. S. Shifrin, “The study of the properties of matter from single scattering,” in Theoretical and Applied Problems of Light Scattering (Nauka i Tekhnika, 1971), pp. 228–244.

B. T. Draine and P. J. Flatau, “User Guide to the Discrete Dipole Approximation Code DDSCAT7.1,” http://arXiv.org/abs/1002.1505v1 (2007).

M. V. Kabanov, M. V. Panchenko, Yu. A. Pkhalagov, V. V. Veretennikov, V. N. Uzhegov, and V. Ya. Fadeev, Optical Properties of the Maritime Smokes (Nauka, 1988) (In Russian).

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Figures (7)

Fig. 1
Fig. 1

Asymmetry parameter g = cos obtained by numerical integration (solid curves) and calculated using the approximate Eq. (11) (dashed curves).

Fig. 2
Fig. 2

Normalization factor P λ ( a , b ) obtained by numerical integration (solid curves) and calculated using the approximate formula in Eq. (15) (dashed curves).

Fig. 3
Fig. 3

(a) Real morphology and (b) digitized model of the nonspherical particle used for modeling the scattering phase function by DDSCAT tool [26].

Fig. 4
Fig. 4

Normalized scattering phase function p N , λ ( a , b , θ ) for sandlike particles calculated using different methods. The polydisperse system of dust particles is characterized by the modified size-distribution function with parameters a = 0 and b = 20 μm 1 and refractive index m = 1.61 + 0.0 i . The wavelength of an incident radiation is 400 nm .

Fig. 5
Fig. 5

Same as in Fig. 4, but for slightly different parameters of b, so that depicted scattering phase functions satisfy condition cos 0.671 (except for Mie scattering pattern).

Fig. 6
Fig. 6

(a) Minimized deviation D = i = 1 Nr [ M i EC i ] 2 / ( Nr 1 ) of model (M) and exactly calculated (EC) scattering phase functions as a function of modal radius r M (x axis) and half-width σ (y axis) of size distribution s ( r ) . Nr represents the number of discrete scattering angles, and the optimum values of r M and σ correspond to the global minimum of D. The model of slightly absorbing sandlike particles with refractive index 1.52 + 0.034 i is assumed. Wavelength of an incident radiation is 0.55 μm . (b) Same as Fig. 6a, but for wavelength 1.2 μm . (c) Same as Fig. 6a, but for absorbing soot particles with refractive index 1.75 + 0.3 i . The wavelength is 0.8 μm .

Fig. 7
Fig. 7

(a) DDA (solid lines) and reproduced (dashed lines) scattering phase functions obtained for optimized r M and σ [consult Figs. 6a, 6b]. (b) Same as in Fig. 7a, but assuming r M and σ optimized based on Fig. 6c.

Equations (17)

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p HG ( θ ) = 1 g 2 ( 1 + g 2 2 g cos θ ) 3 / 2 ,
f ( r ) = A r a e b r ; a 0 , b > 0 ,
I λ ( θ , r ) = I 0 , λ | 3 4 π ( m 2 1 m 2 + 2 ) | 2 1 + cos 2 θ ( 1 cos θ ) 2 ( sin q q cos q q ) 2 ,
I λ ( θ ) = 0 I λ ( θ , r ) f ( r ) d r .
I λ ( a , b , θ ) = 1 + cos 2 θ ( 1 cos θ ) 2 { L λ ( ( a + 1 ) ( a + 2 ) , b , θ ) + [ L λ ( 0 , b , θ ) L λ ( 4 , b , θ ) ] a + 3 2 R λ ( a , b , θ ) } ,
R λ ( a , b , θ ) = i = 1 3 2 f i [ f i sin G λ , i ( a , b , θ ) + f i + 1 cos G λ , i ( a , b , θ ) ] L i 3 i 2 ( 4 , b , θ ) j = 0 i 1 ( 1 ) i + j ( a + j ) a , G λ , i ( a , b , θ ) = ( a + i ) arctan ( 2 L λ ( 0 , b , θ ) ) , L λ ( ξ , b , θ ) = ξ + ( λ b 4 π sin ( θ / 2 ) ) 2 , f i = ( 1 ) i + 1 2 .
R λ ( a , b , θ ) = ( a + 1 ) ( a + 2 ) cos [ G λ , 3 ( a , b , θ ) ] 2 ( a + 1 ) L λ ( 4 , a , θ ) sin [ G λ , 2 ( a , b , θ ) ] L λ ( 4 , a , θ ) cos [ G λ , 1 ( a , b , θ ) ] .
L λ mod ( ξ , b , θ ) = ( 2 | m | 1 m λ b 4 π sin ( θ / 2 ) ) 2 + ξ ,
lim θ 0 I λ ( a , b , θ ) = 8 L λ 2 ( 0 , b , π ) { ( a + 1 ) ( a + 2 ) 3 [ ( a + 3 ) 4 8 ( a + 2 ) 4 10 + ( a + 2 ) 2 ] + ( a + 1 ) 4 6 [ ( a + 1 ) 2 30 1 ] } .
I λ ( 0 , b , 0 ) 32 π L λ 2 ( 0 , b , π ) .
cos cos 2 G ( a , λ b ) 1 + G 2 ( a , λ b ) ,
G ( a , λ b ) = 10 γ 2 + λ b 8 π 8 10 + 5 a / ( γ 2 π 2 ) ,
1 2 0 π p N , λ ( a , b , θ ) sin θ d θ = 1 .
p N , λ ( a , b , θ ) = 2 I λ ( a , b , θ ) P λ ( a , b ) ,
P λ ( a , b ) π 2 ( 1 + γ a ) 4 ( 8 π λ b ) 4 γ ( 1 a 180 ) ,
s ( r ) = π r 2 f ( r ) N ,
N = 0 f ( r ) d r .

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