Abstract

We investigate the magnification due to refraction of the apparent horizontal sizes of finite celestial bodies, such as the Sun or Moon. Two models are discussed and compared with the earlier works of Biot and Chauvenet. It is shown that the apparent horizontal size of the object varies with respect to its true horizontal size as a function of altitude or zenith distance, from a reduction of about 0.0276% at the zenith, to an amplification of about 0.0045% when the object appears just at the horizon (namely, when the true altitude γ is negative and related to the corresponding refraction R by γ=R). It is also shown that the apparent horizontal size is equal to the true size when the true altitude γ is related to the corresponding refraction R by γ=R/2. Thus, the total magnification (and reduction) range for differently sized objects is about 0.032%–0.033% and depends on the refraction.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. D. Cassini, Ephemerides Novissimæ Motuum Coelestium Marchionis Cornelii Malvasiæ (ex typographia Andreæ Cassiani, Mvtinæ impensis avthoris, 1662).
  2. B. Oriani, “De refractionibus astronomicis,” in Ephemerides astronomicae anni 1788: Appendix ad ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp. 164–277.
  3. H. Atkinson, “On astronomical and other refractions; with a connected Inquiry into the law of temperature in different latitudes and at different altitudes,” Mem. R. Astronom. Soc. 2, 137–260 (1826).
  4. H. Atkinson, “On the fluctuations of the atmosphere near the earth’s surface; and On the effect of such fluctuations upon the refraction at the horizon, and at very low altitudes, especially on the dip of the horizon at sea,” Mon. Not. R. Astronom. Soc. 1, 192–193 (1830).
  5. H. Atkinson, “Of the fluctuations of the atmosphere near the earth’s surface; and of their effects upon the refraction at very low altitudes. With an appendix on the dip of the horizon,” Mem. R. Astronom. Soc. 4, 517–530 (1831).
  6. J. B. Biot, “Sur les réfractions astronomiques,” Additions a la Connaissance des Tems 1839, 3–114 (1836).
  7. A. T. Young, “Sunset science. IV. Low-altitude refraction,” Astronom. J. 127, 3622–3637 (2004).
    [CrossRef]
  8. A. T. Young, “Understanding astronomical refraction,” Observatory 126, 82–115 (2006).
  9. J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome I (Bernard, 1810), p. 233.
  10. J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome II (Bernard, 1811), pp. 558–560.
  11. R. Main, Practical and Spherical Astronomy (Deighton, Bell, 1863), pp. 137–138.
  12. W. Chauvenet, Manual of Spherical and Practical Astronomy (J. B. Lippincott, 1863), Vol.  I, pp. 187–188.
  13. C. Y. Hohenkerk and A. T. Sinclair, “The computation of angular atmospheric refraction at large zenith angles,” Nautical Almanac Office Tech. Note 63 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1985).
  14. S. N. Kivalov, “Improved ray tracing air mass numbers model,” Appl. Opt. 46, 7091–7098 (2007).
    [CrossRef] [PubMed]
  15. L. Auer and E. M. Standish, Astronomical Refraction: Computational Method for all Zenith Angles (Yale University Astronomy Dept., 1979).
  16. L. Auer and E. M. Standish, “Astronomical refraction: computational method for all zenith angles,” Astronom. J. 119, 2472–2474 (2000).
    [CrossRef]
  17. A. T. Sinclair, “The effect of atmospheric refraction on laser ranging data,” Nautical Almanac Office Tech. Note 59 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1982).
  18. United States Committee on Extension to the Standard Atmosphere, “U.S. standard atmosphere” (U.S. Government Printing Office, 1976).

2007 (1)

2006 (1)

A. T. Young, “Understanding astronomical refraction,” Observatory 126, 82–115 (2006).

2004 (1)

A. T. Young, “Sunset science. IV. Low-altitude refraction,” Astronom. J. 127, 3622–3637 (2004).
[CrossRef]

2000 (1)

L. Auer and E. M. Standish, “Astronomical refraction: computational method for all zenith angles,” Astronom. J. 119, 2472–2474 (2000).
[CrossRef]

1985 (1)

C. Y. Hohenkerk and A. T. Sinclair, “The computation of angular atmospheric refraction at large zenith angles,” Nautical Almanac Office Tech. Note 63 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1985).

1982 (1)

A. T. Sinclair, “The effect of atmospheric refraction on laser ranging data,” Nautical Almanac Office Tech. Note 59 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1982).

1979 (1)

L. Auer and E. M. Standish, Astronomical Refraction: Computational Method for all Zenith Angles (Yale University Astronomy Dept., 1979).

1976 (1)

United States Committee on Extension to the Standard Atmosphere, “U.S. standard atmosphere” (U.S. Government Printing Office, 1976).

1863 (2)

R. Main, Practical and Spherical Astronomy (Deighton, Bell, 1863), pp. 137–138.

W. Chauvenet, Manual of Spherical and Practical Astronomy (J. B. Lippincott, 1863), Vol.  I, pp. 187–188.

1836 (1)

J. B. Biot, “Sur les réfractions astronomiques,” Additions a la Connaissance des Tems 1839, 3–114 (1836).

1831 (1)

H. Atkinson, “Of the fluctuations of the atmosphere near the earth’s surface; and of their effects upon the refraction at very low altitudes. With an appendix on the dip of the horizon,” Mem. R. Astronom. Soc. 4, 517–530 (1831).

1830 (1)

H. Atkinson, “On the fluctuations of the atmosphere near the earth’s surface; and On the effect of such fluctuations upon the refraction at the horizon, and at very low altitudes, especially on the dip of the horizon at sea,” Mon. Not. R. Astronom. Soc. 1, 192–193 (1830).

1826 (1)

H. Atkinson, “On astronomical and other refractions; with a connected Inquiry into the law of temperature in different latitudes and at different altitudes,” Mem. R. Astronom. Soc. 2, 137–260 (1826).

1811 (1)

J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome II (Bernard, 1811), pp. 558–560.

1810 (1)

J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome I (Bernard, 1810), p. 233.

1787 (1)

B. Oriani, “De refractionibus astronomicis,” in Ephemerides astronomicae anni 1788: Appendix ad ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp. 164–277.

1662 (1)

G. D. Cassini, Ephemerides Novissimæ Motuum Coelestium Marchionis Cornelii Malvasiæ (ex typographia Andreæ Cassiani, Mvtinæ impensis avthoris, 1662).

Atkinson, H.

H. Atkinson, “Of the fluctuations of the atmosphere near the earth’s surface; and of their effects upon the refraction at very low altitudes. With an appendix on the dip of the horizon,” Mem. R. Astronom. Soc. 4, 517–530 (1831).

H. Atkinson, “On the fluctuations of the atmosphere near the earth’s surface; and On the effect of such fluctuations upon the refraction at the horizon, and at very low altitudes, especially on the dip of the horizon at sea,” Mon. Not. R. Astronom. Soc. 1, 192–193 (1830).

H. Atkinson, “On astronomical and other refractions; with a connected Inquiry into the law of temperature in different latitudes and at different altitudes,” Mem. R. Astronom. Soc. 2, 137–260 (1826).

Auer, L.

L. Auer and E. M. Standish, “Astronomical refraction: computational method for all zenith angles,” Astronom. J. 119, 2472–2474 (2000).
[CrossRef]

L. Auer and E. M. Standish, Astronomical Refraction: Computational Method for all Zenith Angles (Yale University Astronomy Dept., 1979).

Biot, J. B.

J. B. Biot, “Sur les réfractions astronomiques,” Additions a la Connaissance des Tems 1839, 3–114 (1836).

J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome II (Bernard, 1811), pp. 558–560.

J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome I (Bernard, 1810), p. 233.

Cassini, G. D.

G. D. Cassini, Ephemerides Novissimæ Motuum Coelestium Marchionis Cornelii Malvasiæ (ex typographia Andreæ Cassiani, Mvtinæ impensis avthoris, 1662).

Chauvenet, W.

W. Chauvenet, Manual of Spherical and Practical Astronomy (J. B. Lippincott, 1863), Vol.  I, pp. 187–188.

Hohenkerk, C. Y.

C. Y. Hohenkerk and A. T. Sinclair, “The computation of angular atmospheric refraction at large zenith angles,” Nautical Almanac Office Tech. Note 63 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1985).

Kivalov, S. N.

Main, R.

R. Main, Practical and Spherical Astronomy (Deighton, Bell, 1863), pp. 137–138.

Oriani, B.

B. Oriani, “De refractionibus astronomicis,” in Ephemerides astronomicae anni 1788: Appendix ad ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp. 164–277.

Sinclair, A. T.

C. Y. Hohenkerk and A. T. Sinclair, “The computation of angular atmospheric refraction at large zenith angles,” Nautical Almanac Office Tech. Note 63 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1985).

A. T. Sinclair, “The effect of atmospheric refraction on laser ranging data,” Nautical Almanac Office Tech. Note 59 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1982).

Standish, E. M.

L. Auer and E. M. Standish, “Astronomical refraction: computational method for all zenith angles,” Astronom. J. 119, 2472–2474 (2000).
[CrossRef]

L. Auer and E. M. Standish, Astronomical Refraction: Computational Method for all Zenith Angles (Yale University Astronomy Dept., 1979).

Young, A. T.

A. T. Young, “Understanding astronomical refraction,” Observatory 126, 82–115 (2006).

A. T. Young, “Sunset science. IV. Low-altitude refraction,” Astronom. J. 127, 3622–3637 (2004).
[CrossRef]

Additions a la Connaissance des Tems (1)

J. B. Biot, “Sur les réfractions astronomiques,” Additions a la Connaissance des Tems 1839, 3–114 (1836).

Appl. Opt. (1)

Astronom. J. (2)

L. Auer and E. M. Standish, “Astronomical refraction: computational method for all zenith angles,” Astronom. J. 119, 2472–2474 (2000).
[CrossRef]

A. T. Young, “Sunset science. IV. Low-altitude refraction,” Astronom. J. 127, 3622–3637 (2004).
[CrossRef]

Mem. R. Astronom. Soc. (2)

H. Atkinson, “On astronomical and other refractions; with a connected Inquiry into the law of temperature in different latitudes and at different altitudes,” Mem. R. Astronom. Soc. 2, 137–260 (1826).

H. Atkinson, “Of the fluctuations of the atmosphere near the earth’s surface; and of their effects upon the refraction at very low altitudes. With an appendix on the dip of the horizon,” Mem. R. Astronom. Soc. 4, 517–530 (1831).

Mon. Not. R. Astronom. Soc. (1)

H. Atkinson, “On the fluctuations of the atmosphere near the earth’s surface; and On the effect of such fluctuations upon the refraction at the horizon, and at very low altitudes, especially on the dip of the horizon at sea,” Mon. Not. R. Astronom. Soc. 1, 192–193 (1830).

Observatory (1)

A. T. Young, “Understanding astronomical refraction,” Observatory 126, 82–115 (2006).

Other (10)

J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome I (Bernard, 1810), p. 233.

J. B. Biot, Traité Elementaire d’Astronomie Physique, Tome II (Bernard, 1811), pp. 558–560.

R. Main, Practical and Spherical Astronomy (Deighton, Bell, 1863), pp. 137–138.

W. Chauvenet, Manual of Spherical and Practical Astronomy (J. B. Lippincott, 1863), Vol.  I, pp. 187–188.

C. Y. Hohenkerk and A. T. Sinclair, “The computation of angular atmospheric refraction at large zenith angles,” Nautical Almanac Office Tech. Note 63 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1985).

G. D. Cassini, Ephemerides Novissimæ Motuum Coelestium Marchionis Cornelii Malvasiæ (ex typographia Andreæ Cassiani, Mvtinæ impensis avthoris, 1662).

B. Oriani, “De refractionibus astronomicis,” in Ephemerides astronomicae anni 1788: Appendix ad ephemerides Anni 1788 (Appresso Giuseppe Galeazzi, 1787), pp. 164–277.

A. T. Sinclair, “The effect of atmospheric refraction on laser ranging data,” Nautical Almanac Office Tech. Note 59 (H. M. Nautical Almanac Office, Royal Greenwich Observatory, 1982).

United States Committee on Extension to the Standard Atmosphere, “U.S. standard atmosphere” (U.S. Government Printing Office, 1976).

L. Auer and E. M. Standish, Astronomical Refraction: Computational Method for all Zenith Angles (Yale University Astronomy Dept., 1979).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Ray-tracing geometry for a finite-sized object.

Fig. 2
Fig. 2

Angular differences between the true and apparent directions toward two points on the object.

Fig. 3
Fig. 3

Two approaches to the horizontal magnification.

Fig. 4
Fig. 4

Zenith cases.

Tables (2)

Tables Icon

Table 1 Minimal and Maximal Visible Horizontal Angles α for the Hohenkerk and Sinclair Method

Tables Icon

Table 2 Reductions, Amplifications and the Total Magnification Ranges for Different Wavelengths

Equations (26)

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

Δ = Δ sin ( Z ) sin ( Z ) ,
Δ S q = Δ S 1 cos 2 ( q ) ,
α = α δ R ,
cos α 2 = cos γ · cos ψ · cos β 2 + sin γ · sin ψ , sin ψ = cos α 2 · sin γ ,
cos γ · tan β 2 = tan α 2 , cos ψ · sin β 2 = sin α 2 , cos ψ = sin 2 α 2 + cos 2 α 2 · cos 2 γ ,
sin α 2 = sin α 2 · cos ( ψ + R ) cos ψ = sin α 2 ( cos R sin R sin γ · cos α 2 sin 2 α 2 + cos 2 α 2 · cos 2 γ ) .
cos R sin R · sin γ · cos α 2 sin 2 α 2 + cos 2 α 2 · cos 2 γ = 1 .
cos R sin R · sin γ cos γ = 1 ,
Δ = L alm = r · cos γ · β = sin ( Z ) · β .
L gc = r · α = α .
sin ( L gc 2 ) = sin α 2 = sin ( Z ) sin β 2
cos α 2 = cos 2 γ · cos β 2 + sin 2 γ , cos α 2 = cos 2 ( γ + R ) · cos β 2 + sin 2 ( γ + R ) ,
cos α 2 = cos α 2 · cos 2 ( γ + R ) cos 2 γ + [ sin 2 ( γ + R ) sin 2 γ · cos 2 ( γ + R ) cos 2 γ ] .
cos α = ( A , B ) = cos γ · cos ψ · cos β + sin γ · sin ψ .
sin ψ = cos α · sin γ .
cos ψ = 1 sin 2 ψ = 1 cos 2 α ( 1 cos 2 γ ) = sin 2 α + cos 2 α · cos 2 γ ,
cos α = cos γ · cos ψ · cos β + sin γ · sin ψ = cos β · cos γ · sin 2 α + cos 2 α · cos 2 γ + sin γ · cos α · sin γ .
cos α ( 1 sin 2 γ ) = cos β · cos γ · sin 2 α + cos 2 α · cos 2 γ ,
cos α = cos β cos γ · sin 2 α + cos 2 α · cos 2 γ .
cos 2 α = cos 2 β cos 2 γ · sin 2 α + cos 2 β · cos 2 α ,
cos 2 α ( 1 cos 2 β ) = cos 2 β cos 2 γ · sin 2 α .
cos α · sin β = cos β cos γ · sin α .
cos γ · tan β = tan α .
sin α = cos ψ · sin β .
cos ψ = sin 2 α + cos 2 α · cos 2 γ = sin 2 α + cos 2 α · tan 2 α tan 2 β = sin 2 α + sin 2 α · cos 2 β sin 2 β .
cos ψ · sin β = sin 2 α · sin 2 β + sin 2 α · cos 2 β = sin α ,

Metrics