Abstract

A new and flexible ray-tracing procedure for calculating astronomical refraction is outlined and applied to the US1976 standard atmosphere. This atmosphere is generalized to allow for a free choice of the temperature and pressure at sea level, and in this form it has been named the modified US1976 (MUSA76) atmosphere. Analytical expressions and numerical procedures are presented for calculating dry-air refractions and for the water-vapor correction. Results for all apparent altitudes are presented and compared with The Star Almanac for Land Surveyors (1951), The Nautical Almanac (1958), and the Pulkovo tables (Refraction Tables of the Pulkovo Observatory, 1985). Dependences on sea-level pressure, temperature, and temperature gradient and on humidity are discussed.

© 2003 Optical Society of America

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References

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  1. The Star Almanac for Land Surveyors (Her Majesty’s Nautical Almanac Office, London, 1st ed., 1951).
  2. The Nautical Almanac (Her Majesty’s Nautical Almanac Office, London, and Nautical Almanac Office United States, Washington, D.C., 1st combined ed., 1958, printed yearly).
  3. D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, Boca Raton, Fla., 2000).
  4. P. K. Seidelmann, ed., Explanatory Supplement to the Astronomical Almanac (University Science Books, Mill Valley, Calif., 1992).
  5. L. H. Auer, E. M. Standish, “Astronomical refraction: computational method for all zenith angles,” Astron. J. 119, 2472–2477 (2000).
    [CrossRef]
  6. V. K. Balakin, ed., Refraction Tables of the Pulkovo Observatory, 5th ed. (Nauka, Leningrad, 1985).
  7. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, UK, 1999).
  8. W. M. Smart, Textbook on Spherical Astronomy, 6th ed. revised by R. M. Green (Cambridge University, Cambridge, UK, 1977).
  9. C. Y. Hohenkerk, A. T. Sinclair, “The computation of angular atmospheric refraction at large zenith angles,” NAO Tech. Note 63 (HM Nautical Almanac Office, Royal Greenwich Observatory, Greenwich, 1985).
  10. A. T. Sinclair, “The effect of atmospheric refraction on laser ranging data,” NAO Tech. Note 59 (HM Nautical Almanac Office, Royal Greenwich Observatory, Greenwich, 1982).
  11. S. Y. van der Werf, G. P. Können, W. H. Lehn, F. Steenhuisen, W. P. S. Davidson, “Gerrit de Veer’s true and perfect description of the Novaya Zemlya effect, 24–27 January 1597,” Appl. Opt. 42, 379–389 (2003).
    [CrossRef] [PubMed]
  12. S. Y. van der Werf, G. P. Können, W. H. Lehn, “Novaya Zemlya effect and sunsets,” Appl. Opt. 42, 367–378 (2003).
    [CrossRef] [PubMed]
  13. R. D. Sampson, E. P. Lozowski, A. E. Peterson, “A comparison of modeled and observed astronomical refraction of the setting Sun,” Appl. Opt. 42, 342–353 (2003).
    [CrossRef] [PubMed]
  14. B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
    [CrossRef]
  15. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1966).
    [CrossRef]
  16. J. C. Owen, “Optical refractive index of air: dependence on pressure, temperature, and composition,” Appl. Opt. 6, 51–59 (1967).
    [CrossRef]
  17. A. A. Michailov, ed., Refraction Tables of the Pulkovo Observatory, 4th ed. (Academy of Sciences Press, Moscow, Leningrad, 1956).
  18. J. Saastamoinen, “Introduction to the practical computation of astronomical refraction,” Bull. Geod. 106, 383–397 (1972).
    [CrossRef]
  19. P. Harzer, “Gebrauchstabellen zur Berechnung der Ablenkungen der Lichtstrahlen in der Atmosphäre der Erden für die Beobachtungen am groszer Kieler Meridiankreise,” Publikation der Sternwarte in Kiel14 (1924).
  20. B. Garfinkel, “An investigation in the theory of astronomical refraction,” Astron. J. 50, 169–179 (1944).
    [CrossRef]
  21. B. Garfinkel, “Astronomical refraction in a polytropic atmosphere,” Astron. J. 72, 235–254 (1967).
    [CrossRef]
  22. A. Fletcher, “Astronomical refraction at low altitudes in marine navigation,” J. Inst. Navigation 5, 307–330 (1952).
    [CrossRef]
  23. S. Y. van der Werf, Program REF2001, unpublished. Available from the author.

2003

2000

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

1972

J. Saastamoinen, “Introduction to the practical computation of astronomical refraction,” Bull. Geod. 106, 383–397 (1972).
[CrossRef]

1967

B. Garfinkel, “Astronomical refraction in a polytropic atmosphere,” Astron. J. 72, 235–254 (1967).
[CrossRef]

J. C. Owen, “Optical refractive index of air: dependence on pressure, temperature, and composition,” Appl. Opt. 6, 51–59 (1967).
[CrossRef]

1966

1952

A. Fletcher, “Astronomical refraction at low altitudes in marine navigation,” J. Inst. Navigation 5, 307–330 (1952).
[CrossRef]

1944

B. Garfinkel, “An investigation in the theory of astronomical refraction,” Astron. J. 50, 169–179 (1944).
[CrossRef]

Auer, L. H.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, UK, 1999).

Ciddor, P. E.

Davidson, W. P. S.

Edlen, B.

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Fletcher, A.

A. Fletcher, “Astronomical refraction at low altitudes in marine navigation,” J. Inst. Navigation 5, 307–330 (1952).
[CrossRef]

Garfinkel, B.

B. Garfinkel, “Astronomical refraction in a polytropic atmosphere,” Astron. J. 72, 235–254 (1967).
[CrossRef]

B. Garfinkel, “An investigation in the theory of astronomical refraction,” Astron. J. 50, 169–179 (1944).
[CrossRef]

Green, R. M.

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

Harzer, P.

P. Harzer, “Gebrauchstabellen zur Berechnung der Ablenkungen der Lichtstrahlen in der Atmosphäre der Erden für die Beobachtungen am groszer Kieler Meridiankreise,” Publikation der Sternwarte in Kiel14 (1924).

Hohenkerk, C. Y.

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

Können, G. P.

Lehn, W. H.

Lide, D. R.

D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, Boca Raton, Fla., 2000).

Lozowski, E. P.

Owen, J. C.

Peterson, A. E.

Saastamoinen, J.

J. Saastamoinen, “Introduction to the practical computation of astronomical refraction,” Bull. Geod. 106, 383–397 (1972).
[CrossRef]

Sampson, R. D.

Sinclair, A. T.

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

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

Smart, W. M.

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

Standish, E. M.

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

Steenhuisen, F.

van der Werf, S. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, UK, 1999).

Appl. Opt.

Astron. J.

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

B. Garfinkel, “An investigation in the theory of astronomical refraction,” Astron. J. 50, 169–179 (1944).
[CrossRef]

B. Garfinkel, “Astronomical refraction in a polytropic atmosphere,” Astron. J. 72, 235–254 (1967).
[CrossRef]

Bull. Geod.

J. Saastamoinen, “Introduction to the practical computation of astronomical refraction,” Bull. Geod. 106, 383–397 (1972).
[CrossRef]

J. Inst. Navigation

A. Fletcher, “Astronomical refraction at low altitudes in marine navigation,” J. Inst. Navigation 5, 307–330 (1952).
[CrossRef]

Metrologia

B. Edlen, “The refractive index of air,” Metrologia 2, 71–80 (1966).
[CrossRef]

Other

A. A. Michailov, ed., Refraction Tables of the Pulkovo Observatory, 4th ed. (Academy of Sciences Press, Moscow, Leningrad, 1956).

P. Harzer, “Gebrauchstabellen zur Berechnung der Ablenkungen der Lichtstrahlen in der Atmosphäre der Erden für die Beobachtungen am groszer Kieler Meridiankreise,” Publikation der Sternwarte in Kiel14 (1924).

S. Y. van der Werf, Program REF2001, unpublished. Available from the author.

V. K. Balakin, ed., Refraction Tables of the Pulkovo Observatory, 5th ed. (Nauka, Leningrad, 1985).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, Cambridge, UK, 1999).

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

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

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

The Star Almanac for Land Surveyors (Her Majesty’s Nautical Almanac Office, London, 1st ed., 1951).

The Nautical Almanac (Her Majesty’s Nautical Almanac Office, London, and Nautical Almanac Office United States, Washington, D.C., 1st combined ed., 1958, printed yearly).

D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, Boca Raton, Fla., 2000).

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

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Figures (7)

Fig. 1
Fig. 1

Arbitrary path in Cartesian coordinates and in the spherical coordinates R, ϕ, and β, on which the ray-tracing method is based.

Fig. 2
Fig. 2

Saturated vapor pressure of water. The data are from The Handbook of Chemistry and Physics.3 The curves represent the different formula’s: the power-law expression (PL2), the two- and four-parameter Clausius-Clapeyron forms CC2 and CC4, discussed in the text.

Fig. 3
Fig. 3

Curvature, 1/r = (1/n)dn/dh in units of 1/R Earth, of a locally horizontal ray for the US1976 standard atmosphere. The procedure to cope with the discontinuities, illustrated in the inset, is discussed in the text.

Fig. 4
Fig. 4

Dry-air refractions for low apparent altitudes, calculated on the basis of the MUSA76 atmosphere for sea-level temperatures -25 °C, -5 °C, 15 °C (standard), 35 °C, and 55 °C. The sea-level pressure is 1013.25 hPa in all cases. The wavelength has been chosen as λ = 0.574 μm. For comparison the standard Nautical Almanac refraction is shown.

Fig. 5
Fig. 5

(a) Dependence of the refraction at 0° apparent altitude on P 0, at T 0 = 288.15 °K. (b) Its dependence on T 0 at P 0 = 1013.25 hPa. (c) Apparent altitude scaling factors, relative to the standard T 0 = 288.15 °K, P 0 = 1013.25 hPa, for different sea-level temperatures and pressures: (a) t 0 = -25 °C, P 0 = 1013.25 hPa, (b) t 0 = -5 °C, P 0 = 1013.25 hPa, (c) t 0 = 35 °C, P 0 = 1013.25 hPa, (d) t 0 = 55 °C, P 0 = 1013.25 hPa, (e) t 0 = 15 °C, P 0 = 1073.25 hPa, (f) t 0 = 15 °C, P 0 = 953.25 hPa. The fits, discussed in the text, are shown as dashed curves.

Fig. 6
Fig. 6

Difference between refractions for dry air and for 100% humidity for the different equations for saturated water vapor, PL2, CC2 and CC4. The basis is the MUSA76 atmosphere for temperatures (a) -5 °C, (b) 15 °C (standard), (c) 35 °C, and (d) 55 °C at a sea-level pressure of 1013.25 hPa in all cases. λ = 0.574 μm.

Fig. 7
Fig. 7

Left, different temperature profiles in the troposphere for P 0 = 1013.25 hPa, T 0 = 288.15 K and temperature gradients, ranging (from left to right) from -0.0465 °C/m to 0.0335 °C/m in steps of 0.010 °C/m. The temperature profiles are obtained as in Eq. (52), with H C = 1 km. Right, the corresponding dry-air refractions, increasing as the sea-level temperature gradient increases.

Tables (5)

Tables Icon

Table 1 MUSA76 Atmospherea

Tables Icon

Table 2 US1976 Standard Atmosphere Refractionsa

Tables Icon

Table 3 Comparison with the Star Almanac Atmospherea

Tables Icon

Table 4 Stepwise Change from Star Almanac to MUSA76

Tables Icon

Table 5 Comparison with the Nautical AlmanacAtmospherea

Equations (55)

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1r=d2ydx21+dydx23/2=-R2+R d2Rdϕ2-2dRdϕ2R2+dRdϕ23/2.
β=arctan1RdRdϕ,
dβdϕ=-R2+R d2Rdϕ2-2dRdϕ2R2+dRdϕ2+1.
R2+dRdϕ21/2=R1+tan2β1/2=R/cosβ,
dRdϕ=R tanβ,
dβdϕ=1+1rRcosβ.
nR cosβ=constant,
tanβdβdϕ=1R+1ndndRdRdϕ.
dβdϕ-1=RndndR,
1r=cosβ1ndndR.
1r=1nνˆ · n,
rˆ|r|=kˆ×1nn,
dRdϕ=R tanβ,
dβdϕ=1+RndndR.
ξ=Pathdβ-dϕ.
ξ=β0βs1-dϕdβdβ=β0βsRdn/dhn+Rdn/dhdβ,
dPdh=dPDdh+dPWdh=-mDghkPDhTh-mWghkPWhTh,
gh=g0REarthREarth+h2.
N=Avogadros number, R=Nk=8314.472 J kmol-1K-1 universal gas constant, MD=NmD=28.964 kg mass per kmol of dry air, MD=NmW=18.016 kg mass per kmol of water vapor, RE=6356766 m the Earths mean radius at 45 °N, g0=9.7803561+0.0052885 sin2ϕ-0.0000059 sin22ϕm/s2 ϕ=latitude,
nλh-1=1ThADλPDh+AWλPWh.
ADλ=10-88342.13+2406030130-1/λ2-1+1599738.9-1/λ2-1288.151013.25,
ADλ=10-85792105238.0185-1/λ2-1+116791757.362-1/λ2-1288.151013.25.
ADλ=10-828760.4+162.88λ2+1.36λ4273.151013.25.
AWλ=10-824580.4+162.88λ2+1.36λ4273.151013.25.
AWλ=1.022×10-8295.235+2.6422λ2-0.032380λ4+0.004028λ6293.1513.33.
dPDdh+mDghkPDhTh=-dPWTdTdTdh-mWghkPWThTh.
PWsatT=T247.118.36,
PWsatT=expa-b/T,
PWsatT=expAT2+BT+C+D/T,
PD0=P0-PWT0=P0-RH×PWsatT0.
hj=h-h1jdim. m,aj=1T1jdTdhjdim. m-1,cj=mDgh1jkT1jdim. m-1,bj=1RE+h1jdim. m-1,ηj=-ajcjaj-bj2dimensionless,ζj=-bjaj-bjdimensionless.
1PhjdPhjdhj=-cj1+ajhj1+bjhj2
Phj=Phj=01+ajhj1+bjhjηj exp-cjζjhj1+bjhj.
Phj=Phj=0ThjT1jγj if aj0,
Phj=Phj=0exp-mDgh1jhjkT1j if aj=0.
dPDdT/T0-ηDPDT/T01-ζT/T02=-dPWdT/T0+ηWPWT/T01-ζT/T02,
PDThom.=CT/T0ηD1-ζηD expηD1-ζ×1-ζT/T0-ηD exp-ηD1-ζT/T0,
ZηD, ζ, T/T01-ζηD expηD1-ζ×1ζT/T0-ηD exp-ηD1-ζT/T0.
PDTspec.=DPWT=DPWT0T/T0δ with PWT0=RHT0/247.1δ,
PWT=RH×T0/247.1δT/T0δZδ, ζ, T/T0.
PDT=CT/T0ηDZηD, ζ, T/T0+DT/T0δZδ, ζ, T/T0,
D=PWT0ηW-δδ-ηD,
C=P0-PWT0-D.
PDT=C1-ζηD expηD1-ζ×T/T01-ζT/T0ηD exp-ηD1-ζT/T0+fTPWT0expb1T0-1T.
dfdT=ηDT1-ζT/T02-bT2fT+ηWT1-ζT/T02-bT2.
fT=-1+ηD-ηWηD 1F11, ηD+1; bT,
PDT=C1-ζηD expηD1-ζ×T/T01-ζT/T0ηD exp-ηD1-ζT/T0-1-ηD-ηWηD 1F11, ηD+1; bT×PWT0expb1T0-1T.
C=P0-ηD-ηWηD 1F11, ηD+1; bT0PWT0.
dndh27nh+½Δh-nh-½Δh-nh+³/₂Δh-nh-³/₂Δh24Δh.
ξP1, T1=P1P0κT0T1λξP0, T0,
dndh=-ADPT2dTdh+B,
ξP1, T1, β=P1P0κT0T1λξP0, T0, β.
tanβ=P1P0μT0T1ν tanβ.
R=Nk=8314.36 universal gas constant,Md=Nmd=28.966 molar weight of dry air,Mw=Nmw=18.016 molar weight of water vapor,REarth=6,378,120 m average radius of the Earth,g=9.7841-0.0026 cos2ϕ ϕ=obs. latitude.
Th=T0-0.0065h+dTdh0+0.0065×hHT-hHT1+h/HC,

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