Abstract

Regardless of profile, whether flat topped or pointed, to the summit observer all mountain peaks cast triangular shadows when the sun is low. A theory for such anomalous shadows is developed. The shadow apex angle is shown to depend only on the ratio of the breadth of the mountain to its height.

© 1979 Optical Society of America

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References

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  1. M. Minnaert, I. Licht en Kleur in het landschap (Thieme and Cie, Zutphen, 1968).
  2. , Mt. Washington Obs. Bull. 14, 52 (1973).
  3. , Mt. Fuji, Japan Times, Ltd.Tokyo (1970).
  4. H. Arakawa, Weather 16, 220 (1961).
    [CrossRef]

1973 (1)

, Mt. Washington Obs. Bull. 14, 52 (1973).

1961 (1)

H. Arakawa, Weather 16, 220 (1961).
[CrossRef]

Arakawa, H.

H. Arakawa, Weather 16, 220 (1961).
[CrossRef]

Minnaert, M.

M. Minnaert, I. Licht en Kleur in het landschap (Thieme and Cie, Zutphen, 1968).

Mt. Washington Obs. Bull. (1)

, Mt. Washington Obs. Bull. 14, 52 (1973).

Weather (1)

H. Arakawa, Weather 16, 220 (1961).
[CrossRef]

Other (2)

M. Minnaert, I. Licht en Kleur in het landschap (Thieme and Cie, Zutphen, 1968).

, Mt. Fuji, Japan Times, Ltd.Tokyo (1970).

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Figures (9)

Fig. 1
Fig. 1

Photographs of the shadow of (a) Baboquivari by K. DeGioia, (b) Kitt Peak, and (c) Pico del Teide by M. Cohen. To the left are the respective profiles that give rise to their shadows.

Fig. 2
Fig. 2

The remarkable Kage-Fuji, or shadow of Mt. Fuji (Japan Times photograph).

Fig. 3
Fig. 3

Coordinate systems: For derivations (upper) and for the observer (lower).

Fig. 4
Fig. 4

Shadow width (χ) as a function of dip below the horizon (γ) in the observer’s coordinates.

Fig. 5
Fig. 5

Examples of how the convergence function transforms various input geometries.

Fig. 6
Fig. 6

The sunrise shadow of White Mountain, California, as viewed from the high altitude research laboratory. This is the special case of a flat-topped summit, which is effectively infinite in extent.

Fig. 7
Fig. 7

Diagram used for deriving the effective width of the sun as a light source.

Fig. 8
Fig. 8

Sunset shadow of Mauna Loa as seen from the weather station, approximately 16 km from the summit.

Fig. 9
Fig. 9

Two views of the shadow of Kitt Peak as modified by the presence of the 4-m telescope dome (photograph by G. Ladd).

Tables (1)

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Table I Comparison of the Calculated and Observed Shadow Apex Angle A

Equations (11)

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tan α = H ( x ) / H ( x ) .
z = y tan α .
ω ( y ) = sin - 1 [ W ( y ) r ] = sin - 1 [ W ( y ) ( x 2 + y 2 + Z t 2 ) 1 / 2 ] ,
ω ( y ) [ W ( y ) ] / ( y 2 + Z t 2 ) 1 / 2 .
γ = Z t / y ,
ω ( γ ) W ( γ ) r W ( γ ) ( Z t 2 + Z t 2 / γ 2 ) 1 / 2 .
C z t ( γ ) = 1 Z t 1 ( 1 + 1 / γ 2 ) 1 / 2 ,
ω ( γ ) = W ( γ ) C Z t ( γ ) .
A 2 tan - 1 ( W ¯ / Z t ) .
I ( γ ) = 1 - 1 π cos - 1 ( R - γ R ) - ( R - γ ) π R 2 ( 2 R γ - γ 2 ) 1 / 2 ,
d I d γ = 1 π 1 [ R 2 - ( R - γ ) 2 ] 1 / 2 + ( 2 R γ - γ 2 ) 1 / 2 π R 2 - ( R - γ ) 2 π R 2 ( 2 R γ - γ 2 ) 1 / 2 .

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