Abstract

Air mass numbers have traditionally been obtained by techniques that use height as the integration variable. This introduces an inherent singularity at the horizon, and ad hoc solutions have been invented to cope with it. A survey of the possible options including integration by height, zenith angle, and horizontal distance or path length is presented. Ray tracing by path length is shown to avoid singularities both at the horizon and in the zenith. A fourth-order Runge–Kutta numerical integration scheme is presented, which treats refraction and air mass as path integrals. The latter may optionally be split out into separate contributions of the atmosphere's constituents.

© 2008 Optical Society of America

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References

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    [CrossRef]
  2. F. Link and L. Neuzil, Tables of Light Trajectories in the Terrestrial Atmosphere (Hermann, 1969).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. C. Runge, "Ueber die numerische Auflösung von Differentialgleichungen," Math. Ann. 46, 167-178 (1895).
    [CrossRef]
  6. W. M. Smart, Textbook on Spherical Astronomy, 6th ed., revised by R. M. Green (Cambridge U. Press, 1977).
  7. L. H. Auer and E. M. Standish, "Astronomical refraction: computational method for all zenith angles," Astron. J. 119, 2472-2477 (2000).
    [CrossRef]
  8. C. Y. Hohenkerk and A. T. Sinclair, "The computation of angular atmospheric refraction at large zenith angles," NAO Technical Note No. 63 (HM Nautical Almanac Office, 1985).
  9. P. K. Seidelmann, ed., Explanatory Supplement to the Astronomical Almanac (University Science Books, 1992).
  10. A. T. Young, "Sunset science. IV. Low-altitude refraction," Astron. J. 127, 3622-3637 (2004).
    [CrossRef]
  11. W. H. Lehn, "A simple parabolic model for the optics of the atmospheric surface layer," Appl. Math. Model. 9, 447-453 (1985).
    [CrossRef]
  12. D. Bruton, "Optical determination of atmospheric temperature profiles," Ph.D. thesis (Texas A&M University, 1996).
  13. S. Y. van der Werf, "Ray tracing and refraction in the modified US1976 atmosphere," Appl. Opt. 42, 354-366 (2003).
    [CrossRef] [PubMed]
  14. S. Y. van der Werf, G. P. Können, and W. H. Lehn, "Novaya Zemlya effect and sunsets," Appl. Opt. 42, 367-378 (2003).
    [CrossRef] [PubMed]
  15. D. Gutierrez, F. J. Seron, A. Munoz, and O. Anson, "Simulation of atmospheric phenomena," Comput. Graph. 30, 994-1010 (2006).
    [CrossRef]

2007 (1)

2006 (1)

D. Gutierrez, F. J. Seron, A. Munoz, and O. Anson, "Simulation of atmospheric phenomena," Comput. Graph. 30, 994-1010 (2006).
[CrossRef]

2004 (1)

A. T. Young, "Sunset science. IV. Low-altitude refraction," Astron. J. 127, 3622-3637 (2004).
[CrossRef]

2003 (2)

2000 (1)

L. H. Auer and E. M. Standish, "Astronomical refraction: computational method for all zenith angles," Astron. J. 119, 2472-2477 (2000).
[CrossRef]

1989 (1)

1985 (1)

W. H. Lehn, "A simple parabolic model for the optics of the atmospheric surface layer," Appl. Math. Model. 9, 447-453 (1985).
[CrossRef]

1965 (1)

F. Kasten, "A new table and approximation formula for the relative optical air mass," Arch. Meteorol. Geophys. Bioklimatol. Ser. B 14, 206-223 (1965).
[CrossRef]

1895 (1)

C. Runge, "Ueber die numerische Auflösung von Differentialgleichungen," Math. Ann. 46, 167-178 (1895).
[CrossRef]

Appl. Math. Model. (1)

W. H. Lehn, "A simple parabolic model for the optics of the atmospheric surface layer," Appl. Math. Model. 9, 447-453 (1985).
[CrossRef]

Appl. Opt. (4)

Arch. Meteorol. Geophys. Bioklimatol. Ser. B (1)

F. Kasten, "A new table and approximation formula for the relative optical air mass," Arch. Meteorol. Geophys. Bioklimatol. Ser. B 14, 206-223 (1965).
[CrossRef]

Astron. J. (2)

L. H. Auer and E. M. Standish, "Astronomical refraction: computational method for all zenith angles," Astron. J. 119, 2472-2477 (2000).
[CrossRef]

A. T. Young, "Sunset science. IV. Low-altitude refraction," Astron. J. 127, 3622-3637 (2004).
[CrossRef]

Comput. Graph. (1)

D. Gutierrez, F. J. Seron, A. Munoz, and O. Anson, "Simulation of atmospheric phenomena," Comput. Graph. 30, 994-1010 (2006).
[CrossRef]

Math. Ann. (1)

C. Runge, "Ueber die numerische Auflösung von Differentialgleichungen," Math. Ann. 46, 167-178 (1895).
[CrossRef]

Other (5)

W. M. Smart, Textbook on Spherical Astronomy, 6th ed., revised by R. M. Green (Cambridge U. Press, 1977).

C. Y. Hohenkerk and A. T. Sinclair, "The computation of angular atmospheric refraction at large zenith angles," NAO Technical Note No. 63 (HM Nautical Almanac Office, 1985).

P. K. Seidelmann, ed., Explanatory Supplement to the Astronomical Almanac (University Science Books, 1992).

F. Link and L. Neuzil, Tables of Light Trajectories in the Terrestrial Atmosphere (Hermann, 1969).

D. Bruton, "Optical determination of atmospheric temperature profiles," Ph.D. thesis (Texas A&M University, 1996).

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Figures (1)

Fig. 1
Fig. 1

Ray segment and explanation of its parameters.

Tables (1)

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Table 1 Ray Tracing Integration Schemes

Equations (16)

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d h d ϕ = ( R 0 + h ) tan ( β ) ,
d β d ϕ = 1 + ( R 0 + h ) r cos ( β ) ,
d s d ϕ = ( R 0 + h ) cos ( β ) .
1 r = cos ( β ) 1 n d n d h .
1 r = 1 n [ cos ( β ) n h sin ( β ) ( R 0 + h ) n ϕ ] .
d ξ = ( R 0 + h ) d n / d h n + ( R 0 + h ) d n / d h d β ,
s 2 = s 1 + Δ s ,
h 2 = h 1 + ( k h , 1 + 2 k h ,2 + 2 k h ,3 + k h ,4 ) Δ s / 6 ,
β 2 = β 1 + ( k β , 1 + 2 k β ,2 + 2 k β ,3 + k β ,4 ) Δ s / 6 ,
ϕ 2 = ϕ 1 + ( k ϕ , 1 + 2 k ϕ ,2 + 2 k ϕ ,3 + k ϕ ,4 ) Δ s / 6 ,
ξ 2 = ξ 1 + ( k ξ , 1 + 2 k ξ ,2 + 2 k ξ ,3 + k ξ ,4 ) Δ s / 6 ,
M 2 = M 1 + ( k M , 1 + 2 k M ,2 + 2 k M ,3 + k M ,4 ) Δ s / 6 ,
k X , 1 = | d X d s | s 1 , h 1 , β 1 , ϕ 1 ,
k X , 2 = | d X d s | s 1 + 1 2 Δ s , h 1 + 1 2 k h , 1 Δ s , β 1 + 1 2 k β , 1 Δ s , ϕ 1 + 1 2 k ϕ , 1 Δ s ,
k X ,3 = | d X d s | s 1 + 1 2 Δ s , h 1 + 1 2 k h , 2 Δ s , β 1 + 1 2 k β , 2 Δ s , ϕ 1 + 1 2 k ϕ , 2 Δ s ,
k X ,4 = | d X d s | s 1 + Δ s , h 1 + k h , 3 Δ s , β 1 + k β , 3 Δ s , ϕ 1 + k ϕ , 3 Δ s .

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