The Analemma

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People ask me all the time what my name means, but while it is very elegant mathematically and astronomically, there is no simple way to explain what an analemma is. Here is my feeble attempt to put it into words. You really need to draw lots of pictures and stand with flashlights and basketballs strategically placed to see what's going on. Please Note: This is northern hemisphere centric. If you are in the southern hemisphere please reverse the seasons of the solstaces and equinoxes. Suppose the earth's orbit around the sun were a perfect circle. Then we would always be the same distance from the sun, and always travelling around it at the same speed. We would expect the earliest sunrise and latest sunset of the year on the summer solstace; the latest sunrise and earliest sunset of the year on the winter solstace. The sun would rise at 6am and set at 6pm on the equinoxes. The sun would be directly overhead at noon every day. Life would be great! Alas, the universe is not so simple. The earth's orbit is an ellipse, a slightly flattened circle. Contrary to popular opinion, we are closest to the sun (perihelion) in early January, about two weeks after the winter solstace. We are farthest from the sun (apehelion) in early July, about two weeks after the summer solstace. As anyone who remembers high school physics can attest, the earth moves faster along its orbit when nearer the sun, and more slowly when farther from the sun. This means that when we are near perihelion there are slightly less than 24 hours between when the sun is directly overhead (astronomical noon) on two consecutive days. When we are far from the sun it is slightly more than 24 hours. Over the course of the year it evens out, 24 hours is the average length of a solar day. The sum of these slight offsets causes the sun to become "ahead" of our clocks for part of the year, and "behind" our clocks for part of the year. The sun is usually not directly overhead at noon, and sunrise and sunset are usually not perfectly centered on 6 o'clock. Combining this offset with the angle north or south of the equator that the sun is at on a given day and the length of the day due to the season yields the earth's analemma. It is almost a perfectly symmetric figure 8 because apehelion and perihelion are close to the solstaces. If they were close to the equinoxes it would be a very flattened figure 8. The more squashed the elliptical orbit, the wider the analemma would be. Essentially, the analemma is a figure that shows where in the sky the sun would be at noon throughout the course of the year for an observer at a fixed point. By taking a picture of the sun from a camera in the same place at fixed intervals, we can make photographs of the analemma as you can see at the top left of the page. Here is the equation of time (the east-west portion of the analemma), given as a function of N, the number of the day of the year (with 1 January is day 1): D(N) = 3.98892[a*(N-80) - arctan(.917408*tan(a*(N-80))) + 1.915169*sin(a*(N-2))] Where a = 0.985653 For a more thorough explanation check out Analemma.com. You will need the Quicktime player to make use of all of the videos and interactive demonstrations, but it is still very informative without. Updated 5 May 2004 Graphics from www.analemma.com used with permission.