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November-December 2015

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Weather Queries

My question isn't related to weather, but perhaps you can answer it anyway. On the winter solstice (usually around December 21), I found out that in Bettles, Alaska, the sun rises at 12:13 p.m. and sets at 1:56 p.m., giving one hour and 43 minutes of daylight. But Bettles is slightly north of the Arctic Circle. I thought that on the winter solstice, the shortest day of the year, the sun did not rise north of the Arctic Circle. How can this happen?

Will Becker

Dallas, Texas

Latitudes and longitudes are commonly given in degrees, minutes, and seconds. A minute is 1/60 of a degree; a second is 1/60 of a minute. This is similar to the division of hours into minutes and seconds. The Arctic Circle is the latitude circle that marks the southernmost location where the true position of the center of the sun does not go below the horizon on the longest day of the year, nor does it rise above the horizon on the shortest day of the year. In 2015, the Arctic Circle is 66 degrees, 33 minutes, and 46 seconds (66°33′46″) north of the Equator. (The position wanders very slightly with time.) It is commonly assumed that there will be continuous sunlight north of the Arctic Circle on the longest day of the year and no sunlight on the shortest day of the year.

This is not true for two main reasons: (1) The sun is not a point source of light but rather a disk having an angular width of 32′. You can get some idea about angular width by extending your arm fully and holding up your little finger. At arm's length, it subtends an angle of about one degree. (2) Atmospheric refraction bends light rays from the sun, making it appear higher in the sky than its true position, particularly when it is near the horizon.

Technically speaking, sunrise and sunset occur at the instant when the geocentric zenith distance of the center of the sun's disk is 90°50′. The geocentric zenith distance is the angle, measured at the center of the Earth, between a line to the center of the sun and a line to the point on the earth's surface where the time of sunrise or sunset is desired. See Figure 1. Practically speaking, sunrise occurs when the upper rim of the sun is first visible above the horizon; sunset occurs when the upper rim of the sun sinks below the horizon.

Figure 1.  The geocentric zenith distance for sunrise/sunset.

The sun is so far away (93,000,000 miles) compared with the radius of the earth (about 4,000 miles) that the direction of the sun from the point where sunrise/set is to be observed and its direction from the center of the earth are essentially the same. The true position of the center of the sun is thus 50′ below the horizon at the instant of sunrise/set.

Part of the 50′ comes from the angular width of the sun. If there were no atmosphere, the upper rim of the sun would appear or disappear when the center of the sun is 16′ below the horizon because 16′ is half the sun's angular width. Refraction (bending) of light rays occurs when the rays pass through regions of changing density. Look obliquely into a swimming pool at an object on the bottom. The object will appear to be less deep than it actually is because light rays bend significantly when they pass from the air to the water, because water is much denser than air. When the sun's rays pass through the atmosphere, the density continually increases from top to bottom, so the rays are curved, but only slightly. On average, the bending angle is only 34′ at sunrise/set and in the same direction as the curvature of the earth. Depending upon the vertical temperature and moisture profiles at the time of sunrise/set (these determine the density at each level), atmospheric refraction may be more or less than 34′. Figure 2 illustrates the effect of the sun's angular width and atmospheric refraction on the time of sunrise.

Figure 2.  The sun's angular half-width and the average angle of atmospheric refraction must be added together before calculating the time of sunrise and sunset.

Finally, we can explain why Bettles, Alaska, north of the Arctic Circle, gets more than an hour of sunshine on the shortest day of the year. The latitude of Bethel is 66°53′12″, which is 19′26″ north of the Arctic Circle. Without atmospheric refraction, Bethel would not quite see the sun on the shortest day. With it, the solar disk can rise to where its lower rim nearly grazes the horizon.

The same geometry explains why at least part of the sun stays above the horizon at Bethel from June 4–July 10, 2015.

Weatherwise Contributing Editor THOMAS W. SCHLATTER is a retired meteorologist and volunteer at NOAA's Earth System Research Laboratory in Boulder, Colorado. Submit queries to the author at weatherqueries@gmail.com, or by mail in care of Weatherwise, Taylor & Francis, 530 Walnut Street, Suite 850, Philadelphia, PA 19106.       

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