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The Longest Night |
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At the winter solstice we have the fewest hours of daylight and the longest night. Computing the daylight hours on the solstice for any given latitude is an engaging exercise in elementary trigonometry, and involves some interesting geophysical facts. |
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At the winter solstice the axis of the Earth’s rotation is tilted directly away from the Sun at an angle α of about 23.4 degrees. If we are located at a latitude corresponding to the angle θ up from the equator, then our distance from the equatorial plane is h = Rsin(θ), and our distance from the Earth’s axis of rotation is r = Rcos(θ). Lastly, the distance from the axis to the edge of the shadow is d = h tan(α). On the planar slice at our latitude the extent of the shadow is as shown below. |
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We have d = r cos(ϕ), and so |
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The hours of daylight is 24 times the angular range 2ϕ of daylight as a fraction of the total circumference, so we have |
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The complement of this is the hours of darkness, and this is given by replacing α with –α, which also gives the hours of daylight six months later when the Earth’s axis is tilted toward the Sun. Thus we have |
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The fact that Hday + Hnight = 24 corresponds to the trigonometric identity (for the principle angles) |
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for any constant k, including tan(θ). |
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The tilt of the earth relative to the plane of the Earth’s orbit around the Sun varies between about 22.1 and 24.5 degrees over a period of about 41000 years. At the present time the tilt is about 23.436 degrees. (This cyclical variation of the tilt angle is not to be confused with the precession of the equinoxes, i.e., the precession of the axis, which has a period of about 25772 years.) So we set α = 23.436(π/180) radians in the above equations. |
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To determine the hours of day and night on the winter solstice in San Diego, which is at latitude 32.7157 degrees, we set θ = 32.7157(π/180) radians, and the preceding equations give Hday = 9.844 hours and Hnight = 14.156 hours. For another example, Seattle is at a latitude of 47.6062 degrees, so we set θ = 47.6062(π/180) radians, and the preceding equations give Hday = 8.220 hours and Hnight = 15.780 hours. At the summer solstice, the hours of day and night are reversed. |
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