Abstract

An improved ray tracing air mass model to calculate the air mass numbers for the entire zenith angle range is developed. The improved model uses the approach when the trajectory element of light in the atmosphere is approximated by an arc of a circle. This way the angles at the beginning and at the end of the trajectory element can be counted simultaneously. This approach gives the second-order approximation for the real light trajectory with more accurate results than the results of the approaches of Link and Neuzil (Tables of Light Trajectories in the Terrestrial Atmosphere, Hermann, 1969) and the Kasten and Young models [Appl. Opt. 28, 4735 (1989)]. The developed model allows us to avoid the calculation problems of the Link and Neuzil and Kasten models when the zenith angle is close to or equal to 90°. As a result, we deliver the new air mass number table for the entire zenith angle range and provide the comparison of the developed model results with the results of the Link and Neuzil and the Kasten models.

© 2007 Optical Society of America

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References

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  1. A. T. Young, "Cassini's Model," http://mintaka.sdsu.edu/GF/explain/atmos_refr/models/Cassini.html.
  2. B. Oriani, De Refractionibus Astronomicis Ephemerides Astronomicae Anni 1788: Appendix ad Ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp. 164-277.
  3. J. B. Biot, Sur les Réfractions Astronomiques (Additions à la Conn. des Temps de 1839) (De l'Académie des Sciences, 1836), pp. 3-114.
  4. J. D. Forbes, "On the transparency of the atmosphere and the law of extinction of the solar rays in passing through it," Philos. Trans. R. Soc. London 132, 225-273 (1842).
    [CrossRef]
  5. P. Bouguer, Traité d'Optique sur la Gradation de la Lumière (Mem. de l'Académie Royale des Sciences. H. L. Guerin and L. F. Delatour, 1760).
  6. P. S. Laplace, Traité de Mecanique Celeste (Chez J. B. M. Duhrat, 1805), Vol. 4, Chap. 3.
  7. P. S. Laplace, Celestial Mechanics. Translated with a commentary, by Nathaniel Bowditch (Chelsea Publishing, 1966).
  8. A. T. Young, "Understanding astronomical refraction," Observatory 126, 82-115 (2006).
  9. A. Bemporad, "Sulla teoria d'estinzione di Bouguer," Mem. Soc. Astron. Ital. 30, 217-236 (1901).
  10. A. Bemporad, Zur Theorie der Extinktion des Lichtes (Mitteilungen der Grossh. Sternwarte, 1904).
  11. F. Link and L. Neuzil, Tables of Light Trajectories in the Terrestrial Atmosphere (Hermann, 1969).
  12. U.S. Standard Atmosphere Supplements, 1966 (U.S. Government Printing Office, 1966).
  13. F. Kasten, "A new table and approximation formula for the relative optical air mass," Arch. Meteorol. , Geophys. Bioklimatol., Ser. B 14, 206-223 (1965).
  14. International Organization for Standardization, Standard Atmosphere, International Standard ISO2533 (1972).
  15. F. Kasten and A. T. Young, "Revised optical air mass tables and approximations formula," Appl. Opt. 28, 4735-4738 (1989).
    [CrossRef] [PubMed]
  16. A. T. Young, "Laplace's extinction theorem" (2006), http://mintaka.sdsu.edu/GF/explain/extinction/Laplace.html.
  17. L. H. Auer and E. M. Standish, "Astronomical refraction: computational method for all zenith angles," Astron. J. 119, 2472-2477 (2000).
    [CrossRef]
  18. C. Y. Hohenkerk and A. T. Sinclair, The Computation of Angular Atmospheric Refraction at Large Zenith Angles (NAO technical note, Royal Greenwich Observatory, 1985).
  19. S. Y. van der Werf, "Ray tracing and refraction in the modified US1976 atmosphere," Appl. Opt. 42, 354-366 (2003).
    [CrossRef] [PubMed]
  20. S. Khromov and L. Mamontova, Meteorological Handbook (Gidrometeoizdat, 1963).
  21. U.S. Standard Atmosphere, 1976 (U.S. Government Printing Office, 1976).
  22. U.S. Standard Atmosphere, 1976, as published by NOAA, NASA, and USAF, http://scipp.ucsc.edu/outreach/balloon/atmos/1976%20Standard%20Atmosphere.htm.
  23. "Properties of the U.S. Standard Atmosphere 1976" (2005), http://www.pdas.com/atmos.htm.
  24. M. J. Mahoney, "A Brief Note on Standard/Reference Atmospheres" (2005), http://mtp.jpl.nasa.gov/notes/altitude/ReferenceAtmospheres.html.
  25. A. T. Young, Department of Astronomy, San Diego State University, 5500 Campanile Drive, San Diego, California 92182-1221, USA (personal communication, 2006).

2006

A. T. Young, "Understanding astronomical refraction," Observatory 126, 82-115 (2006).

2003

2000

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

1989

1965

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

1901

A. Bemporad, "Sulla teoria d'estinzione di Bouguer," Mem. Soc. Astron. Ital. 30, 217-236 (1901).

1842

J. D. Forbes, "On the transparency of the atmosphere and the law of extinction of the solar rays in passing through it," Philos. Trans. R. Soc. London 132, 225-273 (1842).
[CrossRef]

Appl. Opt.

Arch. Meteorol.

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

Astron. J.

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

Mem. Soc. Astron. Ital.

A. Bemporad, "Sulla teoria d'estinzione di Bouguer," Mem. Soc. Astron. Ital. 30, 217-236 (1901).

Observatory

A. T. Young, "Understanding astronomical refraction," Observatory 126, 82-115 (2006).

Philos. Trans. R. Soc. London

J. D. Forbes, "On the transparency of the atmosphere and the law of extinction of the solar rays in passing through it," Philos. Trans. R. Soc. London 132, 225-273 (1842).
[CrossRef]

Other

P. Bouguer, Traité d'Optique sur la Gradation de la Lumière (Mem. de l'Académie Royale des Sciences. H. L. Guerin and L. F. Delatour, 1760).

P. S. Laplace, Traité de Mecanique Celeste (Chez J. B. M. Duhrat, 1805), Vol. 4, Chap. 3.

P. S. Laplace, Celestial Mechanics. Translated with a commentary, by Nathaniel Bowditch (Chelsea Publishing, 1966).

A. Bemporad, Zur Theorie der Extinktion des Lichtes (Mitteilungen der Grossh. Sternwarte, 1904).

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

U.S. Standard Atmosphere Supplements, 1966 (U.S. Government Printing Office, 1966).

C. Y. Hohenkerk and A. T. Sinclair, The Computation of Angular Atmospheric Refraction at Large Zenith Angles (NAO technical note, Royal Greenwich Observatory, 1985).

A. T. Young, "Laplace's extinction theorem" (2006), http://mintaka.sdsu.edu/GF/explain/extinction/Laplace.html.

International Organization for Standardization, Standard Atmosphere, International Standard ISO2533 (1972).

A. T. Young, "Cassini's Model," http://mintaka.sdsu.edu/GF/explain/atmos_refr/models/Cassini.html.

B. Oriani, De Refractionibus Astronomicis Ephemerides Astronomicae Anni 1788: Appendix ad Ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp. 164-277.

J. B. Biot, Sur les Réfractions Astronomiques (Additions à la Conn. des Temps de 1839) (De l'Académie des Sciences, 1836), pp. 3-114.

S. Khromov and L. Mamontova, Meteorological Handbook (Gidrometeoizdat, 1963).

U.S. Standard Atmosphere, 1976 (U.S. Government Printing Office, 1976).

U.S. Standard Atmosphere, 1976, as published by NOAA, NASA, and USAF, http://scipp.ucsc.edu/outreach/balloon/atmos/1976%20Standard%20Atmosphere.htm.

"Properties of the U.S. Standard Atmosphere 1976" (2005), http://www.pdas.com/atmos.htm.

M. J. Mahoney, "A Brief Note on Standard/Reference Atmospheres" (2005), http://mtp.jpl.nasa.gov/notes/altitude/ReferenceAtmospheres.html.

A. T. Young, Department of Astronomy, San Diego State University, 5500 Campanile Drive, San Diego, California 92182-1221, USA (personal communication, 2006).

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Figures (5)

Fig. 1
Fig. 1

Scheme of Link and Neuzil's ray tracing model, where r 0 is the radius of the Earth; h is the height of the atmospheric layer above the ground; dh, n, and ρ are height, refraction index, and density of the layer, respectively; i and i′ are angles of light direction at the beginning and at the end of the trajectory element inside the layer; z is the zenith angle; and dφ is the angular element for the geocentric angle φ.

Fig. 2
Fig. 2

Scheme of the improved ray tracing model, where r 0 is the radius of the Earth; h is the height of the atmospheric layer above the ground; Δh is the finite height of the layer; i and i′ are angles of light direction at the beginning and at the end of the finite trajectory element (Δs) inside the layer, respectively; L is the chord corresponding to Δs; z is the zenith angle; Δφ is the finite angular element for the geocentric angle φ; and Δη is the central angle of the arc of the circle that approximates Δs.

Fig. 3
Fig. 3

Graph of the curvature ratio of the real light trajectory and the approximating circle of the improved model, where Δh is the finite height of the layer and κ light , κ circle , and κ light / κ circle are the curvature of light, the curvature of the approximating circle, and the ratio of these curvatures related to the finite point of the finite height interval, respectively.

Fig. 4
Fig. 4

Scheme of the relative positioning of the real light trajectory, the standard model approximation, and the improved model approximation, where d h Δ h is the height of the layer, ds is the trajectory element of the standard model, Δs is the finite trajectory element of the improved model, and L is the chord correspondent to Δs.

Fig. 5
Fig. 5

Graphs of Δφ, Δi, and Δ η = Δ φ + Δ i and their square root approximations, where Δh is the finite height of the layer, Δφ is the diamond marked curve, Δi is the triangle marked curve, Δ η = Δ φ + Δ i is the square marked curve, an approximation for Δφ is the diamond shapes, and an approximation for Δi is the circle shapes.

Tables (2)

Tables Icon

Table 1 Approximation for the U.S. Standard Atmosphere, 1976 Density Profile

Tables Icon

Table 2 Comparison of the Relative Air Mass Numbers for the Different Models

Equations (27)

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sin   i = r n r n   sin   i = r 0 n 0 r h n   sin   z .
d s = sec   i d h ,
d M = ρ d s = ρ   sec   i d h .
M a R 58.36   sin   θ z .
M abs = h 0 h ρ ( h ) sec i d h = h 0 h ρ ( h ) 1 sin 2 i  d h = h 0 h ρ ( h ) [ 1 ( r 0 r ) 2 ( n 0 n ) 2 sin 2 z ] 1 / 2 d h .
M a 1 ρ 0 H 0 ρ d h ( 1 { 1 + 2 δ 0 [ 1 ρ ρ 0 ] } { ( r 0 r 0 + h ) cos   γ } 2 ) 1 / 2 .
n 2 1 n 2 + 2 1 ρ = const .
M a 1 sin   γ + a ( γ + b ) c .
φ = z i + R .
d φ = 1 r h   tan   i d h ,
d φ = d i + d R ,
d R = 1 n   tan   i d n .
Δ η = Δ φ + Δ i = Δ R .
Δ s = r circle Δ η = L 2   sin ( Δ η 2 ) Δ η L .
tan ( Δ φ 2 ) = 1 r h + r h tan ( i ¯ ) Δ h ,
Δ φ 1 r ¯ h tan ( i ¯ ) Δ h .
| L | 2 = r h 2 + r h 2 2 r h r h   cos ( Δ φ ) = r h 2 + r h 2 2 r h r h 4 r ¯ h 2 cos 2 ( i ¯ ) Δ h 2 sin 2 ( i ¯ ) 4 r ¯ h 2 cos 2 ( i ¯ ) + Δ h 2 sin 2 ( i ¯ ) .
Δ s L = ( r h 2 + r h 2 2 r h r h 4 r ¯ h 2 cos 2 ( i ¯ ) Δ h 2 sin 2 ( i ¯ ) 4 r ¯ h 2 cos 2 ( i ¯ ) + Δ h 2 sin 2 ( i ¯ ) ) 1 / 2 ,
Δ M = ρ ( h ¯ ) Δ s ρ ( h ¯ ) ( r h 2 + r h 2 2 r h r h 4 r ¯ h 2 cos 2 ( i ¯ ) Δ h 2 sin 2 ( i ¯ ) 4 r ¯ h 2 cos 2 ( i ¯ ) + Δ h 2 sin 2 ( i ¯ ) ) 1 / 2 .
κ circle = 1 r circle = 2   sin ( Δ η 2 ) L .
κ light = 1 r = sin ( i ) n ( h ) d n d h .
Δ φ + Δ i < 2.8 ° .
Δ φ = 1.113072567 Δ h ,
Δ i = 0.930149184 Δ h .
H g p = r 0 H g m r 0 + H g m ,
s [ h 0 , h 1 ] = Δ h Δ s Δ h ( r h 2 + r h 2 2 r h r h × 4 r ¯ h 2 cos 2 ( i ¯ ) Δ h 2 sin 2 ( i ¯ ) 4 r ¯ h 2 cos 2 ( i ¯ ) + Δ h 2 sin 2 ( i ¯ ) ) 1 / 2 ,
M [ h 0 , h 1 ] = Δ h Δ M Δ h ρ ( h ¯ ) ( r h 2 + r h 2 2 r h r h × 4 r ¯ h 2 cos 2 ( i ¯ ) Δ h 2 sin 2 ( i ¯ ) 4 r ¯ h 2 cos 2 ( i ¯ ) + Δ h 2 sin 2 ( i ¯ ) ) 1 / 2 .

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