Sunrise and Sunset: The Analemma

Why the winter solstice isn't the darkest morning—or evening—of the year

Imagine you set up a camera to snap a picture of the sun every couple of weeks at exactly the same time of day, let's say 10:00 am. After a year you combine all the pictures into a single image.

photo of analemma What you would see is an odd figure-eight pattern called the analemma. The sun slowly creeps around the analemma over the course of a year. No matter what time of day you choose to "freeze" the sun, you'll see it drift along the same figure-eight pattern.

Where does the analemma come from?

The tilt of Earth's axis constantly changes the angle at which we view the sun—moving it higher in summer, lower in winter, and slightly left and right in the process—and it's this changing perspective that produces a figure eight. On top of this, the Earth's elliptical orbit constantly changes the speed at which we travel around the sun, and this adds additional left and right movement making the lower half of the eight fatter.

Visualizing just how axis tilt and varying orbital speed conspire to make the sun drift in such a peculiar pattern is a herculean exercise in spacial perspective and I will not attempt to explain it here. If you want to learn a little bit more about it, I have a slightly expanded explanation of the analemma.

Fortunately it's a bit easier to see how the figure eight—once you accept that it exists—causes the year's latest sunrise and earliest sunset NOT to occur on the winter solstice, the shortest day of the year.

Let's look at some rise and set snapshots around the winter solstice. The image below shows morning and evening analemmas with the sun moving along them on each of three days, as viewed from Chicago. Note the fixed time of day for each snapshot. You can imagine the analemma itself, with the sun attached, rising and setting at the same time every day of the year—it's the sun's varying position on the analemma that causes sunrise and sunset times to change.

The key is that the analemma is tipped when viewed from the mid latitudes. Because of this the bottom of the figure eight (where the sun lies on the winter solstice) is NOT the lowest point relative to the horizon. In addition, the lowest point relative to the eastern horizon and that relative to the western horizon are completely different.

Note that your perspective changes depending on whether the analemma is in the east or west. You can mimic the morning (eastern) analemma with your left hand by holding a finger—tipped left—in front of your face. Then, without moving your finger, move your head forward until your finger is behind you. Then turn and look back—now your finger appears tipped to the right, like the evening (western) analemma.

the solstice analemmas

December 8: In the morning the sun is still quite high on the analemma curve, having risen at 7:06 AM. But in the evening, with the analemma flipped, the sun is at its lowest point relative to the horizon and therefore sets the earliest of the year, at 4:19 PM. Remember the sun is in the same spot on the figure eight morning and evening—it's just that the analemma's relationship to each horizon is different.

December 21: The sun's morning movement has been more or less vertically down resulting in a loss of eight minutes of morning light. Now compare this with the evening motion: the sun has moved upward, but less so because its path relative to the western horizon had a bigger horizontal component—so the evening gain is only four minutes. As a result, the day length has shortened by a net four minutes. This is the shortest day of the year because the sun is at its lowest average point relative to both horizons.

Another way to think of it is this: freeze the sun at any given time of day and measure the distance to its rising point and the distance to its setting point. Add the two distances together; the total distance is smallest on the winter solstice.

January 3: The morning sun has reached its lowest point and has lost another four minutes of morning daylight. But due to the strong vertical gains in the evening the day has gotten longer overall.

It might help to run the morning (or evening) sequence like a movie in your mind: visualize the sun drifting down and to the left (morning) or up and to left (evening) as the days march on.

What if...

  • ...I view the analemma from the north pole? It appears perfectly vertical.
  • ...I view the analemma from the equator? It lies on its side.
  • ...the earth's orbit were a perfect circle? The analemma would be there, but it would be symmetrical top to bottom, and the solstice effect described here would be more subtle.
  • ...the earth had a circular orbit AND an untilted axis? The analemma would collapse into an unmoving dot...and this website would be about frogs.

Want more? A full explanation of what produces the analemma can be found at www.analemma.com.

©2000 Jeremy Kohler