Analemma

The Earth’s orbit around the Sun is not a perfect circle, but an ellipse. This leads to a number of interesting observational effects, one of which is the ana-lemma, the apparent path traced by the Sun in the sky when observed at the same time of day over the course of a year. The path resembles a lopsided figure eight, which is sometimes printed on a globe, usually somewhere on the Pacific Ocean where there is lots of room to show it.

Suppose you were to go outside and measure the Sun’s position in the sky every day, precisely at noon, over the course of a year. You would see the Sun appear to move higher in the sky as summer approached, and then move lower as winter approached. This occurs because the tilt of Earth’s axis causes the Sun’s apparent celestial latitude, or declination, to change over the course of the year.

However, you would also notice that at some times of the year, the Sun would appear slightly farther west in the sky at noon than it did at other times, as if it were somehow gaining time on your watch. This results from the ellipticity of Earth’s orbit. According to Kepler’s second law of motion, planets moving in an elliptical orbit will move faster when they are closer to the Sun than when they are farther away. Therefore, Earth’s speed in its orbit is constantly changing, decelerating as it moves from perihelion (its closest point to the Sun) to aphelion (its farthest point from the Sun), and accelerating as it then “falls” inward toward perihelion again.

It would be difficult for watchmakers—at least, it would have been difficult for pre-digital watchmakers—to make a clock that kept solar time. The clock would have to tick at different rates each day to account for Earth’s changing velocity about the Sun. Instead, watches keep what is called mean solar time, which is the average value of the advance of solar time over the course of the year. As a result, the Sun gets ahead of, and behind, mean solar time by up to 16 minutes at different times of the year. In other words, if you measured the position of the Sun at noon mean solar time at one time of year, the Sun might not reach that position until 12:16 P.M. at another time of year.

Now all the elements are in place to explain the figure eight. The tilt of Earth’s orbital axis causes the Sun to appear higher and lower in the sky at different times of year; this forms the vertical axis of the eight. The ellipticity of Earth’s orbit causes the actual solar time to first get ahead of, and then fall behind, mean solar time. This makes the Sun appear to slide back and forth across the vertical axis of the eight as measured by a fixed clock time each day, forming the rest of the figure.

Clearly, the shape of the analemma depends upon a particular planet’s orbital inclination and ellipticity. The Sun would appear to trace a unique analemma for any of the planets in the solar system; the analemmas thus formed are fat, thin, or even teardrop-shaped variants on the basic figure eight.

Jeffrey Hall