Abstract

A theory for the color of the sky near the horizon for an observer in the umbral region of a total solar eclipse is presented. The model uses a Rayleigh scattering atmosphere, and the light reaching the observer is a beam of singly scattered sunlight, which, in turn, has suffered depletion by scattering in its passage from outside the shadow region. The model predicts both the red color observed in the lowest 8° of the sky for the total solar eclipse of 30 June 1973 and the enriched blue color of the sky at any elevation angle greater than the solar elevation angle. The model is also adapted to explain the reddening of the horizon sky observed during such times as when a dark cloud passes overhead or when the light from a large city is seen from the distance at night.

© 1975 Optical Society of America

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References

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  1. W. J. Humphreys, Physics of the Air (McGraw Hill, New York, 1940).
  2. H. Rohr, The Beauty of the Universe (Viking, New York, 1972).
  3. M. Minnaert, Light and Color in the Open Air (Dover, New York, 1954).
  4. S. Silverman, E. Mullen, Sky Brightness During Eclipses, AFCRL-TR-74-0363 (1974).
  5. W. E. Sharp, J. Lloyd, S. Silverman, Appl. Opt. 5, 787 (1966).
    [CrossRef] [PubMed]
  6. B. S. Dandekar, Appl. Opt. 7, 705 (1968).
    [CrossRef] [PubMed]
  7. W. E. Sharp, S. Silverman, J. Lloyd, Appl. Opt. 10, 1207 (1971).
    [CrossRef] [PubMed]
  8. D. A. Velasquez, Appl. Opt. 10, 1211 (1971).
    [CrossRef] [PubMed]
  9. J. Lloyd, S. Silverman, Appl. Opt. 10, 1215 (1971).
    [CrossRef] [PubMed]
  10. B. Dandekar, J. P. Turtle, Appl. Opt. 10, 1220 (1971).
    [CrossRef] [PubMed]
  11. W. Benyon, G. Brown, Eds., Solar Eclipses and the Ionosphere (Pergamon, London, 1956).
  12. G. Shaw, Appl. Opt. 14, 388 (1975).
    [CrossRef] [PubMed]
  13. K. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

1975 (1)

1971 (4)

1968 (1)

1966 (1)

Dandekar, B.

Dandekar, B. S.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (McGraw Hill, New York, 1940).

Kondratyev, K.

K. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

Lloyd, J.

Minnaert, M.

M. Minnaert, Light and Color in the Open Air (Dover, New York, 1954).

Mullen, E.

S. Silverman, E. Mullen, Sky Brightness During Eclipses, AFCRL-TR-74-0363 (1974).

Rohr, H.

H. Rohr, The Beauty of the Universe (Viking, New York, 1972).

Sharp, W. E.

Shaw, G.

Silverman, S.

Turtle, J. P.

Velasquez, D. A.

Appl. Opt. (7)

Other (6)

W. J. Humphreys, Physics of the Air (McGraw Hill, New York, 1940).

H. Rohr, The Beauty of the Universe (Viking, New York, 1972).

M. Minnaert, Light and Color in the Open Air (Dover, New York, 1954).

S. Silverman, E. Mullen, Sky Brightness During Eclipses, AFCRL-TR-74-0363 (1974).

W. Benyon, G. Brown, Eds., Solar Eclipses and the Ionosphere (Pergamon, London, 1956).

K. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the eclipse. Shaded area is umbra. Observer looks in the x-z plane at elevation angle k. Sun is in the y-z plane at zenith angle θ. The shadow region is approximated by vertical walls.

Fig. 2
Fig. 2

Ratio of brightness during eclipse with x1 = 100 km to brightness during normal times as a function of observer's elevation angle for various wavelengths. In all figures, λ is given in terms of microns.

Fig. 3
Fig. 3

Brightness of the sky at the horizon seen by observer during eclipse and noneclipse conditions as a function of wavelength.

Fig. 4
Fig. 4

Same as Fig. 3 but for k = 0.10.

Fig. 5
Fig. 5

Same as Fig. 3 but for k = 0.20.

Fig. 6
Fig. 6

Schematic diagram for pathlength of light through the atmosphere when the solar elevation angle is less than the observation elevation angle. Notice that pathlength 1 > 2 > 3 > 4.

Fig. 7
Fig. 7

Contribution to the brightness as a function of distance x along the scattered beam for λ = 0.5 μm during an eclipse.

Fig. 8
Fig. 8

Same as Fig. 7 but during noneclipse conditions.

Fig. 9
Fig. 9

Wavelength of maximum intensity, neglecting the factor F0(λ) as a function of x with θ = 0°.

Fig. 10
Fig. 10

Same as Fig. 9 but with θ = 60°.

Equations (21)

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σ = K ρ / λ 4 ,
dB = F ( r , k , λ , θ ) P exp [ 0 r σ dr ] dr ,
tan ( k ) = z / x = k = sin ( k ) and r = x .
x r = 0 h [ 1 ( R R + h n 0 n ) 2 ] 1 / 2 dh .
n 2 = 1 + 2 ( n 0 1 ) ρ / ρ 0 ,
ρ = ρ 0 exp ( h / H ) ; H = R g T / g .
h x 2 2 ( R 1 n 0 1 H ) .
F ( x , k , λ , θ ) = F 0 ( λ ) exp [ KH ρ 0 sec ( θ ) λ 4 exp ( kx / H ) ] .
F 0 ( x , k , λ , θ ) F 0 ( λ ) = 1 π a 2 [ π a 2 4 x x 1 2 a ( a 2 y 2 ) 1 / 2 dy ] ,
F 0 ( x , k , λ , θ ) F 0 ( λ ) 2 ( x x 1 ) π a , x > x 1 .
B ( k , λ , θ ) = F 0 ( λ ) p K ρ 0 λ 4 x 1 Q exp ( kx H K ρ 0 H k λ 4 { 1 [ 1 ksec ( θ ) ] · exp ( kx / H ) } ) dx ,
Q = 2 ( x x 1 ) π a during an eclipse , Q = 1 otherwise .
B e ( 0 , λ , θ ) = F 0 ( λ ) p 2 λ 4 π a K ρ 0 exp { K ρ 0 λ 4 [ x 1 + H sec ( θ ) ] } .
B e ( k , λ , θ ) = F 0 ( λ ) p H 2 K ρ 0 2 k 2 λ 4 π a exp ( K ρ 0 H / k λ 4 ) exp ( k x 1 / H ) × ( 1 + n = 2 exp [ k x 1 ( n 1 ) / H ] { K ρ 0 H k λ 4 [ 1 ksec ( θ ) ] } n 1 ( n ) ( n ! ) ) .
B n ( k , λ , θ ) = F 0 ( λ ) P [ 1 ksec ( θ ) ] 1 × { exp [ K ρ 0 H sec ( θ ) / λ 4 ] exp ( K ρ 0 H k λ 4 ) } .
λ max 4 = K ρ 0 H k { 1 exp ( kx / H ) [ 1 ksec ( θ ) ] } .
1 = ( x max x 1 ) k H { 1 + K ρ 0 H k λ 4 [ 1 ksec ( θ ) ] exp ( k x max / H ) } .
x max = x 1 + ( λ 4 / k ρ 0 ) ,
x max = x 1 + ( H / k ) ,
λ max 4 K ρ 0 [ π H 2 ( 1 R n 0 1 H ) ] 1 / 2 as x
B c = F 0 ( λ ) P exp ( k ρ 0 H k λ 4 ) [ 1 ksec ( θ ) ] 1 × ( exp { k ρ 0 H k λ 4 [ 1 ksec ( θ ) ] exp ( k x 1 / H ) } 1 ) .

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