Abstract

We have calculated the scattering of light by media consisting of densely packed spherical particles by applying geometric optics and Monte Carlo simulation. We found that when the packing density is increased, dark surfaces are clearly brightened, especially near grazing emergence and incidence. Also, the transmission through a finite layer is reduced with opaque particles but not with transparent ones. The results indicate that previous models that use low-density approximations (Lommel–Seeliger, Hapke, Lumme–Bowell, etc.) are not accurate for typical regoliths.

© 1992 Optical Society of America

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References

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  1. H. Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wiss. 2 Kl. 16, 405–516 (1887).
  2. H. Seeliger, “Theorie der Beleuchtung staubförmiger kosmischen Masses Insbesonder des Saturnsringes,” Abh. Bayer. Akad. Wiss. 2 Kl. 18, 1–72 (1893).
  3. E. Schoenberg, “Theoretische Photometrie,” in Handbuch der Astrophysik (Springer-Verlag, Berlin, 1929), Band II/1, pp. 130–170.
  4. M. S. Bobrov, “Physical properties of Saturn’s rings,” in Surfaces and Interiors of Planets and Satellites, A. Dolfus, ed. (Academic, New York, 1970), pp. 376–461.
  5. W. M. Irvine, “Shadowing effect in diffuse reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
    [CrossRef]
  6. K. Lumme, “On photometric properties of Saturn’s rings,” Astrophys. Space Sci. 8, 90–101 (1970).
    [CrossRef]
  7. K. Lumme, “Interpretation of the light curves of some non-atmospheric bodies in the solar system,” Astrophys. Space Sci. 13, 219–230 (1971).
    [CrossRef]
  8. K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies I. Theory,” Astron. J. 82, 1694–1704 (1982).
  9. E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).
  10. L. W. Esposito, “Extension to the classical calculation of the effect of mutual shadowing in diffuse reflection,” Icarus 39, 69–80 (1979).
    [CrossRef]
  11. B. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68, 4571–4586 (1963).
    [CrossRef]
  12. B. Hapke, “Bidirectional reflection spectroscopy 1, theory,” J. Geophys. Res. 86, 3039–3054 (1981).
    [CrossRef]
  13. B. Hapke, “Bidirectional reflection spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
    [CrossRef]
  14. J. I. Peltoniemi, “Light scattering by non-plane-parallel stochastic clouds,” Proc. Finn. Astron. Soc. (1990), pp. 23–26.
  15. J. I. Peltoniemi, K. Lumme, “Radiative transfer in stochastically bounded, densely packed, particulate media,” presented at the International Commission for Optics Meeting on Atmospheric, Volume, and Surface Scattering and Propagation, Florence, Italy, 1991.
  16. C. K. Chan, C. L. Tien, “Radiative transfer in packed spheres,” J. Heat Transfer 96, 52–58 (1974).
    [CrossRef]
  17. K. O. Muinonen, “Light scattering by inhomogeneous media: backward enhancement and reversal of linear polarization,” Ph.D. dissertation (University of Helsinki, Helsinki, Finland, 1990).
  18. Yu. G. Shkuratov, “New mechanism of negative polarization of light scattered by atmospheric cosmic bodies,” Astron. Vestn. 23, 176–180 (1989) (in Russian).
  19. F. A. Franklin, A. F. Cook, “Optical properties of Saturn’s rings. II. Two-color phase curves of the two bright rings,” Astron. J. 70, 704–720 (1965).
    [CrossRef]
  20. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1982).
  21. J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
    [CrossRef] [PubMed]
  22. L. E. Reichl, A Modern Course in Statistical Physics (U. Texas Press, Austin, Tex., 1980), pp. 143–144.
  23. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 479–489.
  24. A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  25. B. Hapke, E. Wells, “Bidirectional reflection spectroscopy 2, experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
    [CrossRef]
  26. K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Diffuse reflection from a stochastically bounded, semi-infinite medium,” Trans. Theory Stat. Phys. 19, 317–332 (1990).
    [CrossRef]
  27. K. Muinonen, K. Lumme, “Light scattering by solar system dust: the opposition effect and the reversal of polarization,” in Proceedings of the 126th IAU Colloquium (Kluwer, Tokyo, 1991).

1990

J. I. Peltoniemi, “Light scattering by non-plane-parallel stochastic clouds,” Proc. Finn. Astron. Soc. (1990), pp. 23–26.

K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Diffuse reflection from a stochastically bounded, semi-infinite medium,” Trans. Theory Stat. Phys. 19, 317–332 (1990).
[CrossRef]

1989

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

Yu. G. Shkuratov, “New mechanism of negative polarization of light scattered by atmospheric cosmic bodies,” Astron. Vestn. 23, 176–180 (1989) (in Russian).

1986

B. Hapke, “Bidirectional reflection spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

1982

A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies I. Theory,” Astron. J. 82, 1694–1704 (1982).

1981

B. Hapke, “Bidirectional reflection spectroscopy 1, theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

B. Hapke, E. Wells, “Bidirectional reflection spectroscopy 2, experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

1979

L. W. Esposito, “Extension to the classical calculation of the effect of mutual shadowing in diffuse reflection,” Icarus 39, 69–80 (1979).
[CrossRef]

1974

C. K. Chan, C. L. Tien, “Radiative transfer in packed spheres,” J. Heat Transfer 96, 52–58 (1974).
[CrossRef]

1971

K. Lumme, “Interpretation of the light curves of some non-atmospheric bodies in the solar system,” Astrophys. Space Sci. 13, 219–230 (1971).
[CrossRef]

1970

K. Lumme, “On photometric properties of Saturn’s rings,” Astrophys. Space Sci. 8, 90–101 (1970).
[CrossRef]

1966

W. M. Irvine, “Shadowing effect in diffuse reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
[CrossRef]

1965

F. A. Franklin, A. F. Cook, “Optical properties of Saturn’s rings. II. Two-color phase curves of the two bright rings,” Astron. J. 70, 704–720 (1965).
[CrossRef]

1963

B. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68, 4571–4586 (1963).
[CrossRef]

1893

H. Seeliger, “Theorie der Beleuchtung staubförmiger kosmischen Masses Insbesonder des Saturnsringes,” Abh. Bayer. Akad. Wiss. 2 Kl. 18, 1–72 (1893).

1887

H. Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wiss. 2 Kl. 16, 405–516 (1887).

Bobrov, M. S.

M. S. Bobrov, “Physical properties of Saturn’s rings,” in Surfaces and Interiors of Planets and Satellites, A. Dolfus, ed. (Academic, New York, 1970), pp. 376–461.

Bowell, E.

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies I. Theory,” Astron. J. 82, 1694–1704 (1982).

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

Chan, C. K.

C. K. Chan, C. L. Tien, “Radiative transfer in packed spheres,” J. Heat Transfer 96, 52–58 (1974).
[CrossRef]

Cook, A. F.

F. A. Franklin, A. F. Cook, “Optical properties of Saturn’s rings. II. Two-color phase curves of the two bright rings,” Astron. J. 70, 704–720 (1965).
[CrossRef]

Dominigue, D.

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

Esposito, L. W.

L. W. Esposito, “Extension to the classical calculation of the effect of mutual shadowing in diffuse reflection,” Icarus 39, 69–80 (1979).
[CrossRef]

Franklin, F. A.

F. A. Franklin, A. F. Cook, “Optical properties of Saturn’s rings. II. Two-color phase curves of the two bright rings,” Astron. J. 70, 704–720 (1965).
[CrossRef]

Hapke, B.

B. Hapke, “Bidirectional reflection spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

B. Hapke, “Bidirectional reflection spectroscopy 1, theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

B. Hapke, E. Wells, “Bidirectional reflection spectroscopy 2, experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

B. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68, 4571–4586 (1963).
[CrossRef]

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

Harris, A. W.

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

Irvine, W. M.

K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Diffuse reflection from a stochastically bounded, semi-infinite medium,” Trans. Theory Stat. Phys. 19, 317–332 (1990).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

W. M. Irvine, “Shadowing effect in diffuse reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
[CrossRef]

Ishimaru, A.

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 479–489.

Kuga, Y.

Lumme, K.

K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Diffuse reflection from a stochastically bounded, semi-infinite medium,” Trans. Theory Stat. Phys. 19, 317–332 (1990).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies I. Theory,” Astron. J. 82, 1694–1704 (1982).

K. Lumme, “Interpretation of the light curves of some non-atmospheric bodies in the solar system,” Astrophys. Space Sci. 13, 219–230 (1971).
[CrossRef]

K. Lumme, “On photometric properties of Saturn’s rings,” Astrophys. Space Sci. 8, 90–101 (1970).
[CrossRef]

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

J. I. Peltoniemi, K. Lumme, “Radiative transfer in stochastically bounded, densely packed, particulate media,” presented at the International Commission for Optics Meeting on Atmospheric, Volume, and Surface Scattering and Propagation, Florence, Italy, 1991.

K. Muinonen, K. Lumme, “Light scattering by solar system dust: the opposition effect and the reversal of polarization,” in Proceedings of the 126th IAU Colloquium (Kluwer, Tokyo, 1991).

Muinonen, K.

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

K. Muinonen, K. Lumme, “Light scattering by solar system dust: the opposition effect and the reversal of polarization,” in Proceedings of the 126th IAU Colloquium (Kluwer, Tokyo, 1991).

Muinonen, K. O.

K. O. Muinonen, “Light scattering by inhomogeneous media: backward enhancement and reversal of linear polarization,” Ph.D. dissertation (University of Helsinki, Helsinki, Finland, 1990).

Peltoniemi, J. I.

J. I. Peltoniemi, “Light scattering by non-plane-parallel stochastic clouds,” Proc. Finn. Astron. Soc. (1990), pp. 23–26.

K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Diffuse reflection from a stochastically bounded, semi-infinite medium,” Trans. Theory Stat. Phys. 19, 317–332 (1990).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
[CrossRef] [PubMed]

J. I. Peltoniemi, K. Lumme, “Radiative transfer in stochastically bounded, densely packed, particulate media,” presented at the International Commission for Optics Meeting on Atmospheric, Volume, and Surface Scattering and Propagation, Florence, Italy, 1991.

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

Reichl, L. E.

L. E. Reichl, A Modern Course in Statistical Physics (U. Texas Press, Austin, Tex., 1980), pp. 143–144.

Schoenberg, E.

E. Schoenberg, “Theoretische Photometrie,” in Handbuch der Astrophysik (Springer-Verlag, Berlin, 1929), Band II/1, pp. 130–170.

Seeliger, H.

H. Seeliger, “Theorie der Beleuchtung staubförmiger kosmischen Masses Insbesonder des Saturnsringes,” Abh. Bayer. Akad. Wiss. 2 Kl. 18, 1–72 (1893).

H. Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wiss. 2 Kl. 16, 405–516 (1887).

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 479–489.

Shkuratov, Yu. G.

Yu. G. Shkuratov, “New mechanism of negative polarization of light scattered by atmospheric cosmic bodies,” Astron. Vestn. 23, 176–180 (1989) (in Russian).

Tien, C. L.

C. K. Chan, C. L. Tien, “Radiative transfer in packed spheres,” J. Heat Transfer 96, 52–58 (1974).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 479–489.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1982).

Wells, E.

B. Hapke, E. Wells, “Bidirectional reflection spectroscopy 2, experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

Abh. Bayer. Akad. Wiss. 2 Kl.

H. Seeliger, “Zur Theorie der Beleuchtung der grossen Planeten, Insbesondere des Saturn,” Abh. Bayer. Akad. Wiss. 2 Kl. 16, 405–516 (1887).

H. Seeliger, “Theorie der Beleuchtung staubförmiger kosmischen Masses Insbesonder des Saturnsringes,” Abh. Bayer. Akad. Wiss. 2 Kl. 18, 1–72 (1893).

Appl. Opt.

Astron. J.

F. A. Franklin, A. F. Cook, “Optical properties of Saturn’s rings. II. Two-color phase curves of the two bright rings,” Astron. J. 70, 704–720 (1965).
[CrossRef]

K. Lumme, E. Bowell, “Radiative transfer in the surfaces of atmosphereless bodies I. Theory,” Astron. J. 82, 1694–1704 (1982).

Astron. Vestn.

Yu. G. Shkuratov, “New mechanism of negative polarization of light scattered by atmospheric cosmic bodies,” Astron. Vestn. 23, 176–180 (1989) (in Russian).

Astrophys. Space Sci.

K. Lumme, “On photometric properties of Saturn’s rings,” Astrophys. Space Sci. 8, 90–101 (1970).
[CrossRef]

K. Lumme, “Interpretation of the light curves of some non-atmospheric bodies in the solar system,” Astrophys. Space Sci. 13, 219–230 (1971).
[CrossRef]

Icarus

L. W. Esposito, “Extension to the classical calculation of the effect of mutual shadowing in diffuse reflection,” Icarus 39, 69–80 (1979).
[CrossRef]

B. Hapke, “Bidirectional reflection spectroscopy 4. The extinction coefficient and the opposition effect,” Icarus 67, 264–280 (1986).
[CrossRef]

J. Geophys. Res.

B. Hapke, “A theoretical photometric function for the lunar surface,” J. Geophys. Res. 68, 4571–4586 (1963).
[CrossRef]

B. Hapke, “Bidirectional reflection spectroscopy 1, theory,” J. Geophys. Res. 86, 3039–3054 (1981).
[CrossRef]

W. M. Irvine, “Shadowing effect in diffuse reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
[CrossRef]

B. Hapke, E. Wells, “Bidirectional reflection spectroscopy 2, experiments and observations,” J. Geophys. Res. 86, 3055–3060 (1981).
[CrossRef]

J. Heat Transfer

C. K. Chan, C. L. Tien, “Radiative transfer in packed spheres,” J. Heat Transfer 96, 52–58 (1974).
[CrossRef]

J. Opt. Soc. Am.

Proc. Finn. Astron. Soc.

J. I. Peltoniemi, “Light scattering by non-plane-parallel stochastic clouds,” Proc. Finn. Astron. Soc. (1990), pp. 23–26.

Trans. Theory Stat. Phys.

K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Diffuse reflection from a stochastically bounded, semi-infinite medium,” Trans. Theory Stat. Phys. 19, 317–332 (1990).
[CrossRef]

Other

K. Muinonen, K. Lumme, “Light scattering by solar system dust: the opposition effect and the reversal of polarization,” in Proceedings of the 126th IAU Colloquium (Kluwer, Tokyo, 1991).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1982).

L. E. Reichl, A Modern Course in Statistical Physics (U. Texas Press, Austin, Tex., 1980), pp. 143–144.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), pp. 479–489.

J. I. Peltoniemi, K. Lumme, “Radiative transfer in stochastically bounded, densely packed, particulate media,” presented at the International Commission for Optics Meeting on Atmospheric, Volume, and Surface Scattering and Propagation, Florence, Italy, 1991.

K. O. Muinonen, “Light scattering by inhomogeneous media: backward enhancement and reversal of linear polarization,” Ph.D. dissertation (University of Helsinki, Helsinki, Finland, 1990).

E. Bowell, B. Hapke, D. Dominigue, K. Lumme, J. I. Peltoniemi, A. W. Harris, “Application of photometric models to asteroids,” in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, eds. (U. Arizona Press, Tucson, Ariz., 1989).

E. Schoenberg, “Theoretische Photometrie,” in Handbuch der Astrophysik (Springer-Verlag, Berlin, 1929), Band II/1, pp. 130–170.

M. S. Bobrov, “Physical properties of Saturn’s rings,” in Surfaces and Interiors of Planets and Satellites, A. Dolfus, ed. (Academic, New York, 1970), pp. 376–461.

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Figures (4)

Fig. 1
Fig. 1

Brightness of the surface or scattered intensity with unit incident flux versus angle of emergence. First-order scattering only, normal incidence, n = 1.55 + i: (a) D = 0.1; (b) D = 0.4. The solid curves represent present calculations, the short-dashed curves represent cumulant expansion [Eq. (5)], the dotted curves represent Hapke’s function, and the long-dashed curves represent the Lumme–Bowell function [Eq. (11)]. The small oscillations in our simulations (especially near the ends) result from the statistical nature of the Monte Carlo simulation and are thus spurious.

Fig. 2
Fig. 2

Disk-integrated brightness, all orders, n = 1.55 + i, D = 0.05, 0.2, and 0.4.

Fig. 3
Fig. 3

Brightness for n = 1.55 + i, D = 0.1 and 0.4, rough spheres: (a) normal incidence, (b) disk integrated.

Fig. 4
Fig. 4

Scattering by perfectly reflecting spheres (n = ∞ + ∞i), normal incidence: (a) 20 scatterings, (b) extrapolation to all orders. The solid curves represent the limit D → 0, i.e., the classical radiative transfer result [(a) from Eq. (10), (b) from van de Hulst20]; the dashed curves represent D = 0.1, and the dotted curves represent D = 0.4.

Tables (3)

Tables Icon

Table 1 Plane A(μ0) and Spherical A* Albedos of Densely Packed Mediaa

Tables Icon

Table 2 Scattering by a Finite Layer with Normal Incidencea

Tables Icon

Table 3 Direct Transmissiona as a Function of Volume Density D and Classical Sparse-Medium Optical Thickness τ0

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

I ( r , Ω ) = lim K d 3 K r f K ( r 1 , , r K ) i = 1 I i ( r , Ω ) ,
I i ( r , Ω ) = d 2 i a δ ( r - a i l i - Ω ) V ( r , a i ) l i 2 S ( a i ) × V ( a i , a i - 1 ) l i - 1 2 S ( a 1 ) V 0 F 0 ,
I i ( r , Ω ) = S ( r - Ω l , Ω , Ω i ) × S ( r - Ω l - Ω i l i , Ω i , Ω i - 1 ) F 0 ,
f K ( r 1 , , r k ) = C exp ( i j F i j ) = C g i j Θ ( r i - r j - a i - a j ) ,
I 1 = 1 μ d z 0 f 1 ( r 0 ) d 2 a S ( a ) F 0 × exp [ - i U i + 1 2 i j ( U i U j - U i U j ) - 1 3 ! i j k ( U i U j U k - 3 U i U j U k + 2 U i U j U k ) - ] ,
U i U j d r i d r j U i U j f k ( r i , , r j r 0 ) ,
f k K ( r 1 , r k ) = d r k + 1 d r K f K ( r 1 , , r K ) ,
f k ( r 1 , , r k r 0 ) = f k + 1 ( r 0 , r 1 , , r k ) / f ( r 0 ) .
f k K ( r 1 , , r k ) = f 1 ( r 1 ) f 1 ( r k ) × Θ ( r 1 - r 2 - 2 a ) Θ ( r k - 1 - r k - 2 a ) ,
I i = d r 1 d r i δ ( r - r i l i - Ω ) exp ( - β l i ) β ω ˜ 0 P ( ϑ i ) l i 2 4 π × exp ( - β l i - 1 ) exp ( - β l 1 ) β ω ˜ 0 P ( ϑ 1 ) l 1 2 4 π exp ( - β l 0 ) F 0 ,
I 1 ( μ , μ 0 , α ; D ) = ω ˜ 0 P ( α ) 4 π μ 0 μ + μ 0 Φ ( μ , μ 0 , α ) F ,
Φ ( y ) = y e y 0 1 t 2 y - 1 e - y t d t y + 3 / 4 y + 3 / 2 , y = D 2.4 μ + μ 0 ( μ 2 + μ 0 2 - 2 μ μ 0 cos α ) 1 / 2 .
A ( μ 0 ) = d μ d φ μ I ( μ , μ 0 , φ ) μ 0 F 0 .
L ( α ) = 4 π d cos γ d λ μ 0 μ I ( μ , μ 0 , φ ) μ 0 F 0 ,
A * = d μ d μ 0 d φ μ μ 0 I ( μ , μ 0 , φ ) μ 0 F 0 .

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