Abstract

The atmospheric transmittance and the astronomical refraction for low-elevation trajectories are discussed and quantitatively developed. The results are used to describe and calculate some of the fascinating atmospheric phenomena occurring shortly before and during sunset, such as the diminishing apparent luminance of the sun, its shape during sunset, and the green flash.

© 1980 Optical Society of America

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References

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  1. M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).
  2. D. J. K. O’Connell, S. J., The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).
  3. American Practical Navigator, An Epitome of Navigation, H. O. Publication 9, (U.S. GPO, Washington, D.C., 1966), pp. 523 and 811.
  4. A. I. Mahan, Appl. Opt. 1, 497 (1962).
    [CrossRef]
  5. J. E. A. Selby et al., “Atmospheric Transmission from 0.25 to 28.5 μm: Supplement lowtran 3B,” AFGL Report TR-76-028 (Air Force Systems Command, USAF, 1976) and previous codes.
  6. S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965), Chaps. 7 and 9.
  7. W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1963), pp. 13–17 and 156. Note that Middleton’s Eq. (8.21) has an error (see the remark by Cohen in Ref. 9 at the bottom of p. 2878).
  8. C. S. Gardner, Appl. Opt. 16, 2427 (1977).
    [CrossRef] [PubMed]
  9. A. Cohen, Appl. Opt. 14, 2878 (1975).
    [CrossRef] [PubMed]
  10. J. Warner, Infrared Phys. 19, 121 (1979).
    [CrossRef]
  11. D. P. Woodman, Appl. Opt. 13, 2193 (1974).
    [CrossRef] [PubMed]
  12. M. Bertolotti, L. Muzii, D. Sette, Appl. Opt. 8, 117 (1969).
    [CrossRef] [PubMed]
  13. P. W. Kruse et al., Elements of Infrared Technology (Wiley, New York, 1963), p. 189.
  14. H. S. Stewart, R. F. Hopfield, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic, New York, 1965), Vol. 1, pp. 127–152.
  15. D. P. Woodman, Proc. Soc. Photo-Opt. Instrum. Eng. 134, 94 (1978).
  16. G. C. Mooradian, M. Geller, P. H. Levine, L. B. Stotts, D. H. Stephens, Appl. Opt. 19, 11 (1980).
    [CrossRef] [PubMed]
  17. S. D. Gedzelman, Appl. Opt. 14, 2831 (1975).
    [CrossRef] [PubMed]
  18. The designations average and apparent apply throughout, although they are not always repeated.
  19. L. Levi, Applied Optics (Wiley, New York, 1968), pp. 233 and 514.
  20. M. P. Thekaekara, R. Kruger, C. H. Duncan, Appl. Opt. 8, 1713 (1969).
    [CrossRef] [PubMed]
  21. G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, New York, 1976), pp. 53 and 55.
  22. Although not used in this study, for completeness the azimuth (from the north) may also be mentioned: sinαN = −cosδs sinhs/cosɛr.
  23. A. K. Angström, K. H. Angström, Sol. Energy 13, 243 (1971).
    [CrossRef]
  24. D. Rawlins, Am. J. Phys. 47, 126 (1979).
    [CrossRef]

1980 (1)

1979 (2)

J. Warner, Infrared Phys. 19, 121 (1979).
[CrossRef]

D. Rawlins, Am. J. Phys. 47, 126 (1979).
[CrossRef]

1978 (1)

D. P. Woodman, Proc. Soc. Photo-Opt. Instrum. Eng. 134, 94 (1978).

1977 (1)

1975 (2)

1974 (1)

1971 (1)

A. K. Angström, K. H. Angström, Sol. Energy 13, 243 (1971).
[CrossRef]

1969 (2)

1962 (1)

Angström, A. K.

A. K. Angström, K. H. Angström, Sol. Energy 13, 243 (1971).
[CrossRef]

Angström, K. H.

A. K. Angström, K. H. Angström, Sol. Energy 13, 243 (1971).
[CrossRef]

Bertolotti, M.

Cohen, A.

Duncan, C. H.

Gardner, C. S.

Gedzelman, S. D.

Geller, M.

Hopfield, R. F.

H. S. Stewart, R. F. Hopfield, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic, New York, 1965), Vol. 1, pp. 127–152.

Kruger, R.

Kruse, P. W.

P. W. Kruse et al., Elements of Infrared Technology (Wiley, New York, 1963), p. 189.

Levi, L.

L. Levi, Applied Optics (Wiley, New York, 1968), pp. 233 and 514.

Levine, P. H.

Mahan, A. I.

Middleton, W. E. K.

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1963), pp. 13–17 and 156. Note that Middleton’s Eq. (8.21) has an error (see the remark by Cohen in Ref. 9 at the bottom of p. 2878).

Minnaert, M.

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).

Mooradian, G. C.

Muzii, L.

O’Connell, D. J. K.

D. J. K. O’Connell, S. J., The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).

Paltridge, G. W.

G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, New York, 1976), pp. 53 and 55.

Platt, C. M. R.

G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, New York, 1976), pp. 53 and 55.

Rawlins, D.

D. Rawlins, Am. J. Phys. 47, 126 (1979).
[CrossRef]

Selby, J. E. A.

J. E. A. Selby et al., “Atmospheric Transmission from 0.25 to 28.5 μm: Supplement lowtran 3B,” AFGL Report TR-76-028 (Air Force Systems Command, USAF, 1976) and previous codes.

Sette, D.

Stephens, D. H.

Stewart, H. S.

H. S. Stewart, R. F. Hopfield, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic, New York, 1965), Vol. 1, pp. 127–152.

Stotts, L. B.

Thekaekara, M. P.

Warner, J.

J. Warner, Infrared Phys. 19, 121 (1979).
[CrossRef]

Woodman, D. P.

D. P. Woodman, Proc. Soc. Photo-Opt. Instrum. Eng. 134, 94 (1978).

D. P. Woodman, Appl. Opt. 13, 2193 (1974).
[CrossRef] [PubMed]

Am. J. Phys. (1)

D. Rawlins, Am. J. Phys. 47, 126 (1979).
[CrossRef]

Appl. Opt. (8)

Infrared Phys. (1)

J. Warner, Infrared Phys. 19, 121 (1979).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. P. Woodman, Proc. Soc. Photo-Opt. Instrum. Eng. 134, 94 (1978).

Sol. Energy (1)

A. K. Angström, K. H. Angström, Sol. Energy 13, 243 (1971).
[CrossRef]

Other (12)

The designations average and apparent apply throughout, although they are not always repeated.

L. Levi, Applied Optics (Wiley, New York, 1968), pp. 233 and 514.

G. W. Paltridge, C. M. R. Platt, Radiative Processes in Meteorology and Climatology (Elsevier, New York, 1976), pp. 53 and 55.

Although not used in this study, for completeness the azimuth (from the north) may also be mentioned: sinαN = −cosδs sinhs/cosɛr.

P. W. Kruse et al., Elements of Infrared Technology (Wiley, New York, 1963), p. 189.

H. S. Stewart, R. F. Hopfield, in Applied Optics and Optical Engineering, R. Kingslake, Ed. (Academic, New York, 1965), Vol. 1, pp. 127–152.

M. Minnaert, The Nature of Light and Colour in the Open Air (Dover, New York, 1954).

D. J. K. O’Connell, S. J., The Green Flash and Other Low Sun Phenomena (North-Holland, Amsterdam, 1958).

American Practical Navigator, An Epitome of Navigation, H. O. Publication 9, (U.S. GPO, Washington, D.C., 1966), pp. 523 and 811.

J. E. A. Selby et al., “Atmospheric Transmission from 0.25 to 28.5 μm: Supplement lowtran 3B,” AFGL Report TR-76-028 (Air Force Systems Command, USAF, 1976) and previous codes.

S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965), Chaps. 7 and 9.

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1963), pp. 13–17 and 156. Note that Middleton’s Eq. (8.21) has an error (see the remark by Cohen in Ref. 9 at the bottom of p. 2878).

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Figures (12)

Fig. 1
Fig. 1

Snell’s law, the sine law, and various angles playing a role for the calculation of trajectories departing under high zenith angles, θ is the local zenith angle; ɛ is the local elevation; δ is the local astronomical refraction; γ is the local direction of the trajectory with respect to the original horizontal direction; β is the total earth center angle subtended by the ray; ɛr is the elevation of the real sun, and ɛd is the sun depression. Any incident angle αi is related to the local zenith angle θi+1 by Eq. (17). The following relations hold: γ0 = ɛ0; δ = γ0 − γ; ɛ = π/2 − θ; ɛ = γ + β; γ = π/2 − (δ + θ0); ɛr = ɛ0δtot= γend; and ɛd = − ɛr. The angles α, θ, , δ, and γ are local ones, θ0 and 0 = γ0 are original values, and δtot, γend, ɛr, and ɛd are final values.

Fig. 2
Fig. 2

Quantity D defined by Eq. (21) as a function of the zenith angles θ0 above 80° and of the wavelength for two visibilities, 23 and 40 km. Note the change of scale at θ0 = 89.98°.

Fig. 3
Fig. 3

Astronomical refraction in a tropical atmosphere for a wavelength of λ = 0.6 μm and for zenith angles >72°.

Fig. 4
Fig. 4

A familiar shape of the rising sun in a tranquil atmosphere. Photograph was made with a 2.4-m reflector (Newtonian focus), aperture stopped down to ~5 cm. Exposure is 1/1000 sec. Castle is some 10 km away. Reproduced with permission from Ref. 2, p. 39.

Fig. 5
Fig. 5

Astronomical refraction δ tot and the atmospheric transmittance D values, for conditions as described by the parameters, as a function of the chosen thickness of the spherical shells in the computer program.

Fig. 6
Fig. 6

Height ΔHh, direction angle γ′ (both with respect to the original horizontal direction at the observer’s site), the logarithm of the atmospheric transmittance in absolute measure: DAR, astronomical refraction δ′ and altitude h along the trajectory for an observed elevation of ɛ0 = 0.2°.

Fig. 7
Fig. 7

Influence of ozone absorption expressed by the quantity ΔDoz(0.6), as a function of the zenith angle θ0 (ozone absorption has its maximum at a wavelength of λ = 0.6 μm). Proportions j = ΔDoz/DAR are also depicted.

Fig. 8
Fig. 8

Sun luminance from some 12 min before its setting onward for two visibilities. Locations of the observed and the real sun during this time are indicated. The blinding threshold Lbl high sky and moon luminances L ˆ sky and L ˆ m, and a typical sky luminance at sunset near the direction where the sun sets, Ls,o, are indicated.

Fig. 9
Fig. 9

Spectral luminance of the sun for zenith angles θ0 from 5 to 90° and visibilities of 23 and 40 km. Toward setting, the sun color turns orange and reddish.

Fig. 10
Fig. 10

As the sun depression ε d , tip is 8.95 mrad, the red sun just disappears behind the horizon [see Eq. (24)]. The astronomical refraction for the yellow, green, and blue being slightly greater, segments of these colors are still visible. For any color of λ < 0.6 μm, an observed segment having an arrow ɛ0(λ) is still above the horizon. The magnitude of the spatial angle ΔΩ(λ,ɛd) depends on the depression of the sun’s upper rim, its angular diameter ψ s, and the shrink constant cs, per Eqs. (49)(51).

Fig. 11
Fig. 11

Spectral illumination as caused by the setting sun from some 5 sec before the last color of the sun sets onward. Time tf = 0 corresponds to a real sun depression of 8.77 mrad. The vanishing color (of highest wavelength λthr) at any time tf is indicated.

Fig. 12
Fig. 12

Dominating or central color of the sun as dependent on time ts,υ (in seconds) before the last color disappears behind the horizon.

Tables (3)

Tables Icon

Table I Various Wavelength-Dependent Factors in the Visible and the Near Infrared as Defined in the Text

Tables Icon

Table II Constants Describing the Tropical Model Atmosphere

Tables Icon

Table III Typical Times (in seconds) for the Real Sun to Descend the Last 0.26-mrad Before its Complete Disappearance Below the Horizon

Equations (60)

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σ A R = R ( λ ) exp ( h / H r ) + A ( λ ) exp ( h / H a ) ,
H r = 7.26 ; H a = 1.14.
T i = T i , 0 + a t h ; H i h < H i + 1 ,
P i = P i , 0 ( T i / T i , 0 ) c t .
σ 2 ( ln 50 ) / V 2 ,
0.02 = W ( λ ) exp [ σ 2 ( λ ) V 2 ] d λ / W ( λ ) d ,
σ 2 ( λ ) = ( ln 50 ) ( 0.56 / λ ) n υ / V 2 n υ = 0.42 V 2 1 / 4 } .
σ A R = R ( λ ) exp ( h / H r ) + w ( h ) A ( λ ) exp ( h / H a ) ,
V 2 26 / ( w 0 + 0.1 ) ,
w ( h ) = w 0 0 h 5 km w ( h ) = 1 h > 5 km } .
O T p ( λ ) = 0 σ A R d h = R ( λ ) H r + f ( w 0 ) A ( λ ) H a f ( w 0 ) w 0 / ( 1 + H a ln w 0 / 5 ) } .
D p ( λ ) | log τ p ( λ ) | = 0.434 O T p ( λ ) ,
D A R ( λ , θ 0 ) | log τ ( λ , θ 0 ) | = 0.434 M [ R ( λ ) H r + A ( λ ) H a ] , θ 0 80 ° ,
δ Δ n 0 tan θ 0 .
δ 0.27 tan θ 0 ; θ 0 80 ° .
B C = ( R 0 + h i + 1 ) sin β i / sin θ i ,
n i sin α i = n i + 1 sin θ i + 1 ,
n i ( R 0 + h i ) sin θ i = const = ( 1 + Δ n 0 ) R 0 sin θ 0 ,
Δ O T = B C σ ¯ B C = B C ( σ i σ i + 1 ) / [ ln ( σ i / σ i + 1 ) ] ,
n = 1 + Δ n = 1 + 10 6 ( a n + b n / λ 2 ) P / T a n = 77.46 ; b n = 0.459 } ,
D A R ( θ 0 , λ, V 2 ) 0.434 O T A R ( θ 0 , λ , V 2 ) = | log τ ( θ 0 , λ , V 2 ) | ,
ε r = ε 0 δ tot ( θ 0 ) = ε d ,
δ tot ( 0.6 ) = 7.67 + 0.142 ε d ε 0 ( 0.6 ) = 7.67 0.858 ε d } .
ε d , t ( 0.6 ) = 8.95 ,
σ o z ( λ , h ) = A υ ( λ ) K ( h ) ,
Δ D o z ( λ , θ 0 ) = 0.434 0 σ o z ( λ , θ 0 , h ) d L ,
Δ D o z ( λ , 90 ° ) 1.7 A V ( λ ) · cm ,
Δ D o z ( λ , θ 0 ) = [ A υ ( λ ) / A υ ( 0,6 ) ] Δ D o z ( 0.6 , θ 0 ) .
τ tot = τ A R 10 Δ D o z .
O T A R ( 90 ° ) = 0 [ R ( λ ) exp ( h / H r ) + A ( λ ) exp ( h / H a ) ] d l ,
O T A R ( λ , 90 ° ) = ( π R 0 / 2 ) 1 / 2 [ H r 1 / 2 R ( λ ) + H a 1 / 2 A ( λ ) ] .
M r , a ( 90 ° ) = ( π R 0 / 2 H r , a ) 1 / 2 ,
M r ( 90 ) = 37.1 M a ( 90 ) = 93.7 O T r ( 0.5,90 ) = 5.93 O T a ( 0.5,90 ) = 17.52 D A R ( 0.5,90 ) = 0.434 ( O T r + O T a ) = 10.2 } .
n = 1 + Δ n 0 exp ( h / H n ) ,
n 1 + Δ n 0 ( 1 h / H n ) .
H n T 0 / a t ( c t 1 ) .
ε i = ε 0 + p L / R 0 ,
p 1 Δ n 0 R 0 / H n .
h = p L 2 / 2 R 0 + ε 0 L ; β = L / R 0 ; γ = ε 0 q β δ = q β ; h u r = q L 2 / 2 R 0 ; Δ H h = ε 0 L h u r
q 1 p = Δ n 0 R 0 / H n .
L hor ( r ) = ( 2 R 0 h 0 / p ) 1 / 2 .
L hor ( r ) / L hor ( u ) = p 1 / 2
ε ( r ) / ε ( u ) = p 1 / 2 .
δ ( λ 1 ) / δ ( λ 2 ) 1 + ( 1 / λ 1 2 1 / λ 2 2 ) / 168 .
L s = 680 N ( λ , T s ) τ ( λ, θ 0 , V 2 ) V ( λ ) d λ
cos θ r = sin ϕ sin δ s + cos ϕ cos δ s cos h s ,
sin ε r = 0.80 sin ( t d / 240 ) ° ,
ε ˙ = 1 300 × 1000 π 180 = 0.058 ( rad sec ) .
ε 0 = a s + b s / λ 2 c s ε d a s = 7.43 b s = 0.0475 c s = 0.848 ,
Δ Ω ( λ, ε 0 ) = c s ψ 2 s [ sin 1 ( 1 u 2 ) 1 / 2 u ( 1 u 2 ) 1 / 2 ] / 4 × 10 6 ,
u 1 2 ε 0 ( λ ) / c s ψ s .
E ( λ, ε d ) = 680 N ( λ, T s ) τ ( λ , ε d , V 2 ) V ( λ ) r ( λ , 1 ) Δ Ω ( λ, ε d )
N ( λ , d ) = N ( λ , 0 ) ( 1 0.237 d 2.4 / λ ) ,
N ( λ , d ) = N ( λ , T s ) r ( λ , d ) r ( λ , d ) = ( 1 0.237 d 2.4 / λ ) / ( 1 0.1136 / λ ) } ,
r ( λ , 1 ) = ( 1 0.237 / λ ) / ( 1 0.1136 / λ ) .
ε d = ε d , t h r + ε ˙ t f ,
Δ Ω ( λ, ε 0 ) 4 ( 2 ε 3 0 ψ s / c s ) 1 / 2 / ( 3 × 10 6 ) = 4.42 × 10 6 ε 3 / 2 0 .
ε 0 = b s / λ 2 ε ˙ c s t f = 0.0475 / λ 2 0.0493 t f .
ε ˙ r ( δ s , ϕ ) = ± ( cos 2 δ s sin 2 ϕ ) 1 / 2 ( 1000 π ) / ( 240 × 180 ) ,
Δ t ( ϕ , δ s ) = Δ t ( 37,0 ) cos 37 ° / ( cos 2 δ s sin 2 ϕ ) 1 / 2 .

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