Abstract

The light pollution model is employed to analyze spatial behavior of luminance at the night sky under cloudless and overcast conditions. Enhanced light excess is particularly identified at cloudy skies, because the clouds efficiently contribute to the downward luminous flux. It is evident that size of ground-based light sources can play an important role in the case of overcast sky conditions. Nevertheless, the realistically sized light sources are rarely embedded into light pollution modeling, and rather they are replaced by simple point sources. We discuss the discrepancies between sky luminance distributions when at first the planar light sources are considered and at second the point-source approximation is accepted. The found differences are noticeable if the size of the light source, distance to the observer, and altitude of a cloudy layer are comparable one to the other. Compared with point-source approximation, an inclusion of the size factor into modeling the light sources leads to partial elimination of the steep changes of sky luminance (typical for point sources of light). The narrow and sharp light pillars normally presented on the sky illuminated by point light sources can disappear or fuse together when two or more nearby light sources are considered with their real sizes. Sky elements situated close to the horizon will glow efficiently if luminous flux originates from two-dimensional ground-based entities (such as cities or villages).

© 2008 Optical Society of America

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  6. R. Dick and A. Weeks, “Fighting light pollution in the Ottawa area--technical elements,” J. R. Astron. Soc. Can. 91, 193-197 (1997).
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  14. P. Cinzano, F. Falchi, and C. D. Elvidge, “The first world atlas of the artificial night sky brightness,” Mon. Not. R. Astron. Soc. 328, 689-707 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  16. A. Penttilä and K. Lumme, “The effect of particle shape on scattering--a study with a collection of axisymmetric particles and sphere clusters,” J. Quant. Spectrosc. Radiat. Transfer 89, 291-301 (2004).
    [CrossRef]
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    [CrossRef]
  19. C. Chalkias, M. Petrakis, B. Psiloglou, and M. Lianou, “Modelling of light pollution in suburban areas using remotely sensed imagery and GIS,” J. Environ. Manage. 79, 57-63(2006).
    [CrossRef]
  20. R. Berry, “Light pollution in southern Ontario,” J. R. Astron. Soc. Can. 70, 97-115 (1976).
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    [CrossRef]
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  24. M. Kocifaj, H. Horvath, and J. Hrvol, “Optical properties of urban aerosols in the region Bratislava--Vienna--II: comparisons and results,” Atmos. Environ. 40, 1935-1948 (2006).
    [CrossRef]
  25. N. Lagrosas, Y. Yoshii, H. Kuze, N. Takeuchi, S. Naito, A. Sone, and H. Kan, “Observation of boundary layer aerosols using a continuously operated, portable lidar system,” Atmos. Environ. 38, 3885-3892 (2004).
    [CrossRef]
  26. N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
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    [CrossRef]
  30. C. J. Braak, J. F. de Haan, C. V. M. Van der Mee, J. W. Hovenier, and L. D. Travis, “Parameterized scattering matrices for small particles in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 69, 585-604 (2001).
    [CrossRef]
  31. O. V. Kalashnikova and I. N. Sokolik, “Modeling the radiative properties of nonspherical soil-derived mineral aerosols,” J. Quant. Spectrosc. Radiat. Transfer 87, 137-166(2004).
    [CrossRef]
  32. R. H. Garstang, “Night-sky brightness at observatories and sites,” Publ. Astron. Soc. Pac. 101, 306-329 (1989).
    [CrossRef]
  33. R. N. Green, B. A. Wielicki, J. A. Coakley, L. L. Stowe, P. O'R. Hinton, and Y. Hu, “Clouds and the Earth's Radiant Energy System (CERES) algorithm theoretical basis document,” CERES inversion to instantaneous top of the atmosphere fluxes, Release 2.2, 2 June (1997).
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2007 (1)

2006 (3)

D. X. Kerola, “Modelling artificial night-sky brightness with a polarized multiple scattering radiative transfer computer code,” Mon. Not. R. Astron. Soc. 365, 1295-1299 (2006).
[CrossRef]

C. Chalkias, M. Petrakis, B. Psiloglou, and M. Lianou, “Modelling of light pollution in suburban areas using remotely sensed imagery and GIS,” J. Environ. Manage. 79, 57-63(2006).
[CrossRef]

M. Kocifaj, H. Horvath, and J. Hrvol, “Optical properties of urban aerosols in the region Bratislava--Vienna--II: comparisons and results,” Atmos. Environ. 40, 1935-1948 (2006).
[CrossRef]

2004 (6)

N. Lagrosas, Y. Yoshii, H. Kuze, N. Takeuchi, S. Naito, A. Sone, and H. Kan, “Observation of boundary layer aerosols using a continuously operated, portable lidar system,” Atmos. Environ. 38, 3885-3892 (2004).
[CrossRef]

A. Penttilä and K. Lumme, “The effect of particle shape on scattering--a study with a collection of axisymmetric particles and sphere clusters,” J. Quant. Spectrosc. Radiat. Transfer 89, 291-301 (2004).
[CrossRef]

P. Cinzano and C. D. Elvidge, “Night sky brightness at sites from DMSP-OLS satellite measurements,” Mon. Not. R. Astron. Soc. 353, 1107-1116 (2004).
[CrossRef]

M. Pedani, “Light pollution at the Roque de los Muchachos Observatory,” New Astron. Rev. 9, 641-650 (2004).
[CrossRef]

P. Cinzano, “A portable spectrophotometer for light pollution measurements,” Mem. S.A.It. Suppl. 5, 395-398 (2004).

O. V. Kalashnikova and I. N. Sokolik, “Modeling the radiative properties of nonspherical soil-derived mineral aerosols,” J. Quant. Spectrosc. Radiat. Transfer 87, 137-166(2004).
[CrossRef]

2003 (2)

K. M. Lata, K. V. S. Badarinath, T. V. R. Rao, R. R. Reddy, Y. N. Ahammed, K. R. Gopal, and P. A. Azeem, “Studies on aerosol optical properties over urban and semi-urban environments of Hyderabad and Anantapur,” J. Quant. Spectrosc. Radiat. Transfer 78, 257-268 (2003).
[CrossRef]

A. Barducci, P. Marcoionni, I. Pippi, and M. Poggesi, “Effects of light pollution revealed during a nocturnal aerial survey by two hyperspectral imagers,” Appl. Opt. 42, 4349-4361 (2003).
[CrossRef] [PubMed]

2001 (2)

C. J. Braak, J. F. de Haan, C. V. M. Van der Mee, J. W. Hovenier, and L. D. Travis, “Parameterized scattering matrices for small particles in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 69, 585-604 (2001).
[CrossRef]

P. Cinzano, F. Falchi, and C. D. Elvidge, “The first world atlas of the artificial night sky brightness,” Mon. Not. R. Astron. Soc. 328, 689-707 (2001).
[CrossRef]

2000 (4)

S. Isobe and S. Hamamura, “Light pollution and its energy loss,” Astrophys. Space Sci. 273, 289-294 (2000).
[CrossRef]

M. Di Sora, “The fight against light pollution in Italy,” Mem. Soc. Astron. Ital. 71, 271-279 (2000).

P. Cinzano, “Modelling light pollution from searchlights,” Mem. Soc. Astron. Ital. 71, 239-250 (2000).

A. Hännel, “The situation of light pollution in Germany,” Mem. Soc. Astron. Ital. 71, 153-158 (2000).

1999 (2)

F. D. Prugna, “Visual measurements and spectra survey of night sky brightness in Venezuela and Italy,” Astron. Astrophys. Suppl. Ser. 140, 345-349 (1999).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, E. G. Zanovitskij, and N. T. Yakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409-432 (1999).
[CrossRef]

1998 (2)

C. R. Benn and S. L. Ellison, “Brightness of the night sky over La Palma,” New Astron. Rev. 42, 503-507 (1998).
[CrossRef]

N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
[CrossRef]

1997 (1)

R. Dick and A. Weeks, “Fighting light pollution in the Ottawa area--technical elements,” J. R. Astron. Soc. Can. 91, 193-197 (1997).

1996 (1)

D. L. Crawford, “Light pollution: the problem and the potential solutions,” Baltic Astron. 5, 263-269 (1996).

1995 (1)

L. M. Celnikier, “Understanding the physics of meteoritic descent,” Am. J. Phys. 63, 524-535 (1995).
[CrossRef]

1993 (1)

1991 (1)

R. H. Garstang, “Dust and light pollution,” Publ. Astron. Soc. Pac. 103, 1109-1116 (1991).
[CrossRef]

1989 (1)

R. H. Garstang, “Night-sky brightness at observatories and sites,” Publ. Astron. Soc. Pac. 101, 306-329 (1989).
[CrossRef]

1986 (1)

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364-375 (1986).
[CrossRef]

1976 (1)

R. Berry, “Light pollution in southern Ontario,” J. R. Astron. Soc. Can. 70, 97-115 (1976).

Ahammed, Y. N.

K. M. Lata, K. V. S. Badarinath, T. V. R. Rao, R. R. Reddy, Y. N. Ahammed, K. R. Gopal, and P. A. Azeem, “Studies on aerosol optical properties over urban and semi-urban environments of Hyderabad and Anantapur,” J. Quant. Spectrosc. Radiat. Transfer 78, 257-268 (2003).
[CrossRef]

Azeem, P. A.

K. M. Lata, K. V. S. Badarinath, T. V. R. Rao, R. R. Reddy, Y. N. Ahammed, K. R. Gopal, and P. A. Azeem, “Studies on aerosol optical properties over urban and semi-urban environments of Hyderabad and Anantapur,” J. Quant. Spectrosc. Radiat. Transfer 78, 257-268 (2003).
[CrossRef]

Badarinath, K. V. S.

K. M. Lata, K. V. S. Badarinath, T. V. R. Rao, R. R. Reddy, Y. N. Ahammed, K. R. Gopal, and P. A. Azeem, “Studies on aerosol optical properties over urban and semi-urban environments of Hyderabad and Anantapur,” J. Quant. Spectrosc. Radiat. Transfer 78, 257-268 (2003).
[CrossRef]

Barducci, A.

Benn, C. R.

C. R. Benn and S. L. Ellison, “Brightness of the night sky over La Palma,” New Astron. Rev. 42, 503-507 (1998).
[CrossRef]

Berry, R.

R. Berry, “Light pollution in southern Ontario,” J. R. Astron. Soc. Can. 70, 97-115 (1976).

Bhartia, B. K.

N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Braak, C. J.

C. J. Braak, J. F. de Haan, C. V. M. Van der Mee, J. W. Hovenier, and L. D. Travis, “Parameterized scattering matrices for small particles in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 69, 585-604 (2001).
[CrossRef]

Celnikier, L. M.

L. M. Celnikier, “Understanding the physics of meteoritic descent,” Am. J. Phys. 63, 524-535 (1995).
[CrossRef]

Chalkias, C.

C. Chalkias, M. Petrakis, B. Psiloglou, and M. Lianou, “Modelling of light pollution in suburban areas using remotely sensed imagery and GIS,” J. Environ. Manage. 79, 57-63(2006).
[CrossRef]

Cinzano, P.

P. Cinzano, “A portable spectrophotometer for light pollution measurements,” Mem. S.A.It. Suppl. 5, 395-398 (2004).

P. Cinzano and C. D. Elvidge, “Night sky brightness at sites from DMSP-OLS satellite measurements,” Mon. Not. R. Astron. Soc. 353, 1107-1116 (2004).
[CrossRef]

P. Cinzano, F. Falchi, and C. D. Elvidge, “The first world atlas of the artificial night sky brightness,” Mon. Not. R. Astron. Soc. 328, 689-707 (2001).
[CrossRef]

P. Cinzano, “Modelling light pollution from searchlights,” Mem. Soc. Astron. Ital. 71, 239-250 (2000).

Coakley, J. A.

R. N. Green, B. A. Wielicki, J. A. Coakley, L. L. Stowe, P. O'R. Hinton, and Y. Hu, “Clouds and the Earth's Radiant Energy System (CERES) algorithm theoretical basis document,” CERES inversion to instantaneous top of the atmosphere fluxes, Release 2.2, 2 June (1997).

Crawford, D. L.

D. L. Crawford, “Light pollution: the problem and the potential solutions,” Baltic Astron. 5, 263-269 (1996).

de Haan, J. F.

C. J. Braak, J. F. de Haan, C. V. M. Van der Mee, J. W. Hovenier, and L. D. Travis, “Parameterized scattering matrices for small particles in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 69, 585-604 (2001).
[CrossRef]

Di Sora, M.

M. Di Sora, “The fight against light pollution in Italy,” Mem. Soc. Astron. Ital. 71, 271-279 (2000).

Dick, R.

R. Dick and A. Weeks, “Fighting light pollution in the Ottawa area--technical elements,” J. R. Astron. Soc. Can. 91, 193-197 (1997).

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Zanovitskij, and N. T. Yakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409-432 (1999).
[CrossRef]

Draine, B. T.

B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT.6.1,” Freeware, http://arxiv.org/abs/astro-ph/0409262 (2004).

Ellison, S. L.

C. R. Benn and S. L. Ellison, “Brightness of the night sky over La Palma,” New Astron. Rev. 42, 503-507 (1998).
[CrossRef]

Elvidge, C. D.

P. Cinzano and C. D. Elvidge, “Night sky brightness at sites from DMSP-OLS satellite measurements,” Mon. Not. R. Astron. Soc. 353, 1107-1116 (2004).
[CrossRef]

P. Cinzano, F. Falchi, and C. D. Elvidge, “The first world atlas of the artificial night sky brightness,” Mon. Not. R. Astron. Soc. 328, 689-707 (2001).
[CrossRef]

Falchi, F.

P. Cinzano, F. Falchi, and C. D. Elvidge, “The first world atlas of the artificial night sky brightness,” Mon. Not. R. Astron. Soc. 328, 689-707 (2001).
[CrossRef]

Fioletov, V.

N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
[CrossRef]

Flatau, P. J.

B. T. Draine and P. J. Flatau, “User guide for the discrete dipole approximation code DDSCAT.6.1,” Freeware, http://arxiv.org/abs/astro-ph/0409262 (2004).

Garstang, R. H.

R. H. Garstang, “Dust and light pollution,” Publ. Astron. Soc. Pac. 103, 1109-1116 (1991).
[CrossRef]

R. H. Garstang, “Night-sky brightness at observatories and sites,” Publ. Astron. Soc. Pac. 101, 306-329 (1989).
[CrossRef]

R. H. Garstang, “Model for artificial night-sky illumination,” Publ. Astron. Soc. Pac. 98, 364-375 (1986).
[CrossRef]

Gopal, K. R.

K. M. Lata, K. V. S. Badarinath, T. V. R. Rao, R. R. Reddy, Y. N. Ahammed, K. R. Gopal, and P. A. Azeem, “Studies on aerosol optical properties over urban and semi-urban environments of Hyderabad and Anantapur,” J. Quant. Spectrosc. Radiat. Transfer 78, 257-268 (2003).
[CrossRef]

Green, R. N.

R. N. Green, B. A. Wielicki, J. A. Coakley, L. L. Stowe, P. O'R. Hinton, and Y. Hu, “Clouds and the Earth's Radiant Energy System (CERES) algorithm theoretical basis document,” CERES inversion to instantaneous top of the atmosphere fluxes, Release 2.2, 2 June (1997).

Hamamura, S.

S. Isobe and S. Hamamura, “Light pollution and its energy loss,” Astrophys. Space Sci. 273, 289-294 (2000).
[CrossRef]

Hännel, A.

A. Hännel, “The situation of light pollution in Germany,” Mem. Soc. Astron. Ital. 71, 153-158 (2000).

Herman, J. R.

N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
[CrossRef]

Hinton, P. O'R.

R. N. Green, B. A. Wielicki, J. A. Coakley, L. L. Stowe, P. O'R. Hinton, and Y. Hu, “Clouds and the Earth's Radiant Energy System (CERES) algorithm theoretical basis document,” CERES inversion to instantaneous top of the atmosphere fluxes, Release 2.2, 2 June (1997).

Horvath, H.

M. Kocifaj, H. Horvath, and J. Hrvol, “Optical properties of urban aerosols in the region Bratislava--Vienna--II: comparisons and results,” Atmos. Environ. 40, 1935-1948 (2006).
[CrossRef]

Hovenier, J. W.

C. J. Braak, J. F. de Haan, C. V. M. Van der Mee, J. W. Hovenier, and L. D. Travis, “Parameterized scattering matrices for small particles in planetary atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 69, 585-604 (2001).
[CrossRef]

Hrvol, J.

M. Kocifaj, H. Horvath, and J. Hrvol, “Optical properties of urban aerosols in the region Bratislava--Vienna--II: comparisons and results,” Atmos. Environ. 40, 1935-1948 (2006).
[CrossRef]

Hu, Y.

R. N. Green, B. A. Wielicki, J. A. Coakley, L. L. Stowe, P. O'R. Hinton, and Y. Hu, “Clouds and the Earth's Radiant Energy System (CERES) algorithm theoretical basis document,” CERES inversion to instantaneous top of the atmosphere fluxes, Release 2.2, 2 June (1997).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Isobe, S.

S. Isobe and S. Hamamura, “Light pollution and its energy loss,” Astrophys. Space Sci. 273, 289-294 (2000).
[CrossRef]

Kalashnikova, O. V.

O. V. Kalashnikova and I. N. Sokolik, “Modeling the radiative properties of nonspherical soil-derived mineral aerosols,” J. Quant. Spectrosc. Radiat. Transfer 87, 137-166(2004).
[CrossRef]

Kan, H.

N. Lagrosas, Y. Yoshii, H. Kuze, N. Takeuchi, S. Naito, A. Sone, and H. Kan, “Observation of boundary layer aerosols using a continuously operated, portable lidar system,” Atmos. Environ. 38, 3885-3892 (2004).
[CrossRef]

Kerola, D. X.

D. X. Kerola, “Modelling artificial night-sky brightness with a polarized multiple scattering radiative transfer computer code,” Mon. Not. R. Astron. Soc. 365, 1295-1299 (2006).
[CrossRef]

Kerr, J.

N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
[CrossRef]

Kocifaj, M.

M. Kocifaj, “Light pollution model for cloudy and cloudless night skies with ground-based light sources,” Appl. Opt. 46, 3013-3022 (2007).
[CrossRef] [PubMed]

M. Kocifaj, H. Horvath, and J. Hrvol, “Optical properties of urban aerosols in the region Bratislava--Vienna--II: comparisons and results,” Atmos. Environ. 40, 1935-1948 (2006).
[CrossRef]

Krotkov, N. A.

N. A. Krotkov, B. K. Bhartia, J. R. Herman, V. Fioletov, and J. Kerr, “Satellite estimation of spectral surface UV irradiance in the presence of tropospheric aerosols: 1. Cloud-free case,” J. Geophys. Res. 103, 8779-8793 (1998).
[CrossRef]

Kuze, H.

N. Lagrosas, Y. Yoshii, H. Kuze, N. Takeuchi, S. Naito, A. Sone, and H. Kan, “Observation of boundary layer aerosols using a continuously operated, portable lidar system,” Atmos. Environ. 38, 3885-3892 (2004).
[CrossRef]

Lagrosas, N.

N. Lagrosas, Y. Yoshii, H. Kuze, N. Takeuchi, S. Naito, A. Sone, and H. Kan, “Observation of boundary layer aerosols using a continuously operated, portable lidar system,” Atmos. Environ. 38, 3885-3892 (2004).
[CrossRef]

Lata, K. M.

K. M. Lata, K. V. S. Badarinath, T. V. R. Rao, R. R. Reddy, Y. N. Ahammed, K. R. Gopal, and P. A. Azeem, “Studies on aerosol optical properties over urban and semi-urban environments of Hyderabad and Anantapur,” J. Quant. Spectrosc. Radiat. Transfer 78, 257-268 (2003).
[CrossRef]

Lianou, M.

C. Chalkias, M. Petrakis, B. Psiloglou, and M. Lianou, “Modelling of light pollution in suburban areas using remotely sensed imagery and GIS,” J. Environ. Manage. 79, 57-63(2006).
[CrossRef]

Lumme, K.

A. Penttilä and K. Lumme, “The effect of particle shape on scattering--a study with a collection of axisymmetric particles and sphere clusters,” J. Quant. Spectrosc. Radiat. Transfer 89, 291-301 (2004).
[CrossRef]

Marcoionni, P.

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Zanovitskij, and N. T. Yakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409-432 (1999).
[CrossRef]

M. I. Mishchenko, “Light scattering by size-shape distributions of randomly oriented axially symmetric particles of size comparable to a wavelength,” Appl. Opt. 32, 4652-4666(1993).
[CrossRef] [PubMed]

Naito, S.

N. Lagrosas, Y. Yoshii, H. Kuze, N. Takeuchi, S. Naito, A. Sone, and H. Kan, “Observation of boundary layer aerosols using a continuously operated, portable lidar system,” Atmos. Environ. 38, 3885-3892 (2004).
[CrossRef]

Pedani, M.

M. Pedani, “Light pollution at the Roque de los Muchachos Observatory,” New Astron. Rev. 9, 641-650 (2004).
[CrossRef]

Penttilä, A.

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Figures (4)

Fig. 1
Fig. 1

Distribution of luminance under cloudless conditions when one ground-based light source is situated in the vicinity of the observer. Left pane: point-source approximation (isolines are given in relative units), and right pane: circular-source model (the values associated with isolines are in cd / m 2 ). The angle along the circle represents azimuth of the sky element. The zenith angle of the element is measured from the center to the margin of the polar graph. The model parameters: L 1 = 2 km , R 1 = 2 km , and φ C , 1 = 250 ° .

Fig. 2
Fig. 2

Distribution of luminance under overcast conditions ( H = 1 km , ρ ¯ = 0.46 ) when one ground-based light source is situated in the vicinity of the observer. The model parameters: L 1 = 2 km , R 1 = 2 km , and φ C , 1 = 250 ° . The rest of the description of the figure is the same as in Fig. 1.

Fig. 3
Fig. 3

Distribution of luminance under cloudless conditions when three ground-based light sources are situated in the vicinity of the observer. The model parameters: L 1 = 2 km , R 1 = 2 km , and φ C , 1 = 250 ° ; L 2 = 4 km , R 2 = 3 km , and φ C , 2 = 90 ; L 3 = 8 km , R 3 = 2 km , and φ C , 3 = 20 . The radiative properties of all light sources are equal (i.e., I 0 , 1 = I 0 , 2 = I 0 , 3 ). The rest of the description of the figure is the same as in Fig. 1.

Fig. 4
Fig. 4

Distribution of luminance under overcast conditions ( H = 1 km , ρ ¯ = 0.46 ) when three ground-based light sources are situated in the vicinity of the observer. The model parameters: L 1 = 2 km , R 1 = 2 km , and φ C , 1 = 250 ° ; L 2 = 4 km , R 2 = 3 km , and φ C , 2 = 90 ; L 3 = 8 km , R 3 = 2 km , and φ C , 3 = 20 . The radiative properties of all light-sources are equal (i.e., I 0 , 1 = I 0 , 2 = I 0 , 3 ). The rest of the description of the figure is the same as in Fig. 1.

Equations (7)

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T λ ( h , z , φ ) = exp { [ 1 cos z 0 , h + 1 cos z ] [ τ λ , 0 ( M ) ( e h / h 0 1 ) + τ λ , 0 ( A ) ( e γ h 1 ) ] } ,
τ λ , 0 ( A ) = 0 [ 0 C ext ( r , λ , m ) f ( r , h ) d r ] d h ,
C ext ( r , λ , m ) = 1 8 π 2 0 2 π d β 1 1 d cos Θ 0 2 π d Φ C ext ( r , λ , m , β , Θ , Φ ) ,
p λ ( A ) ( ϑ , h ) = ( λ 2 π ) 2 0 S 11 ( ϑ , λ , r ) f ( r , h ) d r ,
1 2 0 π p λ ( A ) ( ϑ , h ) sin ϑ d ϑ = 1 ,
Γ λ ( h , z , φ ) = 1 4 π { Ω λ ( M ) τ λ , 0 ( M ) h 0 P λ ( M ) ( ϑ ) e h / h 0 + Ω λ ( A ) P λ ( A ) ( ϑ ) [ 0 C ext ( r , λ , m ) f ( r , h ) d r ] } ,
J V ( z , φ ) = 1 π H 2 i = 1 N R = 0 R i ( φ 0 ) φ 0 = 0 2 π cos 4 z 0 , H , i B ( Q i , q i , z 0 , H , i ) × { λ 1 λ 2 ρ λ ( z 0 , H , i , z , ϑ H , i ) I λ , 0 , i V λ T λ ( H , z , φ ) d λ } R d R d φ 0 + 1 cos z R = 0 R i ( φ 0 ) φ 0 = 0 2 π 0 H B ( Q i , q i , z 0 , h , i ) cos 2 z 0 , h , i h 2 × { λ 1 λ 2 I λ , 0 , i V λ T λ ( h , z , φ ) Γ λ ( h , z , φ ) d λ } d h R d R d φ 0 ,

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