What Makes the Figure Eight?

The earth circles the sun. Or, from our earthbound point of view, the sun circles the earth. The sun takes one full year to orbit the earth. If you could see the background stars behind the sun, you would see the sun drift east to west across the stars, more or less going around the Earth's Equator and returning to its starting point one year and one full circle later.

But because the earth's axis alternately tilts the northern and southern hemispheres towards the sun, the sun also drifts north and south in its "orbit." It doesn't appear to travel in a straight east-west line, rather its orbital path (called the ecliptic) is more like a sine wave.

But the sine wave effect has nothing to do with the actual orbit. The orbit is a flat circle. The sine wave we see is merely an artifact of our shifting perspective—we are being moved up and down relative to the plane of a perfectly flat circle as the year progresses. In fact, you could cancel out this effect just by driving a few thousand miles south (down) each winter and back north (up) again for the summer. Our shifting north-south perspective is what makes the sun drift up and down.

photo of analemma

But up and down alone does not make a figure-eight.

Remember the true sun completes a straight circular orbit in exactly one year, but what we see is an apparent sun completing a wavy and therefore longer path in the same amount of time.

The apparent sun has a quite a problem here: it needs to somehow keep up with its true self (which is on the short straight path) even as it appears to be wandering north and south.

So there is another artifact resulting from our seasonally changing perspective: as the apparent sun arcs north or south of its true path, it's apparent speed changes.

Strangely enough, this affects the length of our day.

Twenty-four hours is the time it takes for the sun to go from a point in the sky and return to the same point. This includes the motion due to the earth's rotation plus the average daily east-west orbital motion. Because the apparent sun is orbiting at a slightly different speed each day, it takes a different amount of time to go point to point each day. This means that 24 hours is the average length of our day—or more precisely, our day length is 24 hours +/- a few minutes, depending on the season. Only on the equinoxes is the day exactly 24 hours long.

So on most days if you take a picture of the sun at 24 hour intervals, it's going to be a bit east or a bit west of where it was 24 hours earlier. This small east-west shifting combined with the north-south wandering is what makes a figure eight.

Now wait a minute! Why is the analemma fatter on the bottom? Its because in addition to the apparent speed changes described above, there are real speed changes attributed to a completely separate issue: the earth's orbit is elliptical. The earth physically accelerates as it gets closer to the sun, and decelerates as it gets farther away. This adds some additional east-west shifting of the sun, distorting an otherwise symmetrical figure eight.

Nothing is simple in this Universe. If you want to dive into the mathematics and explore 3-D animations check out www.analemma.com.

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©2000 Jeremy Kohler