Abstract

Monte Carlo radiative transfer simulation of light scattering in planetary atmospheres is not a simple problem, especially the study of angular distribution of light intensity. Approximate phase functions such as Henyey–Greenstein, modified Henyey–Greenstein, or Legendre polynomial decomposition are often used to simulate the Mie phase function. An alternative solution using an exact calculation alleviates these approximations.

© 1996 Optical Society of America

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References

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  1. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  2. W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 4, 1563–1575 (1965).
    [CrossRef]
  3. W. M. Cornette, J. G. Shanks, “Physically reasonable analytic expression for the single-scattering phase function,” Appl. Opt. 31, 3152–3160 (1992).
    [CrossRef] [PubMed]
  4. C. M. Chu, S. W. Churchill, “Representation of the angular distribution of radiation scattered by a spherical particle,” J. Opt. Soc. Am. 45, 958–962 (1955).
    [CrossRef]
  5. J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
    [CrossRef]
  6. W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976).
    [CrossRef] [PubMed]
  7. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  9. G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
    [CrossRef]
  10. C. P. McKay, J. B. Pollack, R. Courtin, “The thermal structure of Titan's atmosphere,” Icarus 80, 23–53 (1989).
    [CrossRef] [PubMed]
  11. D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
    [CrossRef] [PubMed]

1995 (1)

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

1992 (1)

1989 (1)

C. P. McKay, J. B. Pollack, R. Courtin, “The thermal structure of Titan's atmosphere,” Icarus 80, 23–53 (1989).
[CrossRef] [PubMed]

1980 (1)

1976 (1)

1975 (1)

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

1974 (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[CrossRef]

1965 (1)

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 4, 1563–1575 (1965).
[CrossRef]

1955 (1)

1941 (1)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Brillet, J.

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

Chu, C. M.

Churchill, S. W.

Cornette, W. M.

Courtin, R.

C. P. McKay, J. B. Pollack, R. Courtin, “The thermal structure of Titan's atmosphere,” Icarus 80, 23–53 (1989).
[CrossRef] [PubMed]

Gautier, D.

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[CrossRef]

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Irvine, W. M.

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 4, 1563–1575 (1965).
[CrossRef]

Kattawar, G. W.

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Lentz, W. J.

McKay, C. P.

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

C. P. McKay, J. B. Pollack, R. Courtin, “The thermal structure of Titan's atmosphere,” Icarus 80, 23–53 (1989).
[CrossRef] [PubMed]

Parisot, J. P.

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

Pollack, J. B.

C. P. McKay, J. B. Pollack, R. Courtin, “The thermal structure of Titan's atmosphere,” Icarus 80, 23–53 (1989).
[CrossRef] [PubMed]

Raulin, F.

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

Shanks, J. G.

Toublanc, D.

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wiscombe, W. J.

Appl. Opt. (3)

Astrophys. J. (2)

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

W. M. Irvine, “Multiple scattering by large particles,” Astrophys. J. 4, 1563–1575 (1965).
[CrossRef]

Icarus (2)

C. P. McKay, J. B. Pollack, R. Courtin, “The thermal structure of Titan's atmosphere,” Icarus 80, 23–53 (1989).
[CrossRef] [PubMed]

D. Toublanc, J. P. Parisot, J. Brillet, D. Gautier, F. Raulin, C. P. McKay, “Photochemical modeling of Titan's atmosphere,” Icarus 113, 2–26 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

G. W. Kattawar, “A three-parameter analytic phase function for multiple scattering calculations,” J. Quant. Spectrosc. Radiat. Transfer 15, 839–849 (1975).
[CrossRef]

Space Science Rev. (1)

J. E. Hansen, L. D. Travis, “Light scattering in planetary atmospheres,” Space Science Rev. 16, 527–610 (1974).
[CrossRef]

Other (1)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (4)

Fig. 1
Fig. 1

Mie phase function compared with Henyey–Greenstein (H-G) and modified Henyey–Greenstein (H-G*) phase functions for r = 01 μm, λ = 0.1 μm, n = 1.656−0.595i.

Fig. 2
Fig. 2

Mie phase function compared with Henyey–Greenstein (H-G) and modified Henyey–Greenstein (H-G*) phase functions for r = 1 μm, λ = 0.1 μm, n = 1.656–0.595i.

Fig. 3
Fig. 3

Relative difference between real and simulated Mie phase functions for r = 0.01 μm, λ = 0.1 μm, n = 1.656−0.595i.

Fig. 4
Fig. 4

Relative difference between real and simulated Mie phase functions for r = 1 μm, λ = 0.1 μm, n = 1.656−0.595i.

Tables (1)

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Table 1 Radiative Transfer Comparisons

Equations (12)

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P HG ( θ , g ) = 1 g 2 [ 1 + g 2 2 g cos ( θ ) ] 3 / 2 ,
g = cos ( θ ) = p ( u ) cos ( u ) d u p ( u ) d u
P DHG ( θ , a , g 1 , g 2 ) = a P HG ( θ , g 1 ) + ( 1 a ) P HG ( θ , g 2 ) ,
P HG * ( θ , g ) = 3 2 1 g 2 2 + g 2 1 + cos ( θ ) 2 [ 1 + g 2 2 g cos ( θ ) ] 3 / 2 .
P ( x ) d x = { d x 0 x < 1 0 otherwise .
+ p ( x ) d x = 1 .
| p ( y ) d y | = | p ( x ) d x |
P ( y ) = p ( x ) | d x d y | .
d x d y = p ( y ) ,
y ( x ) = P 1 ( x ) ,
y ( x ) = 1 2 g [ 1 + g 2 ( 1 g 2 1 + g 2 g x ) 2 ] ,
i = 1 υ 1 p i < x i = 1 υ p i ,

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