What Is the Analemma?
We're all familiar with the sun's daily motion in the sky. It rises in the east, gets higher in the sky until circa noon, then begins its hours-long descent to set on the western horizon.
You may also know of our star's more stately annual journey. For Northern Hemisphere dwellers, as summer approaches, it moves a tiny bit higher in the sky every day at noon until the June solstice, when it turns around and starts to get lower every day until the December solstice.
These motions are the clockwork of the sky. They repeat with enough precision that we base our measurements of passing time on them. But there are more than two gears to this cosmic mechanism; in addition, there are subtle and elegant—though somewhat bizarre—cogs that swing the sun's position in the sky back and forth during the year, as well as up and down.
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If you were to stand in the same spot every day and take a photograph of the sun at the same clock time (ignoring the shift during daylight saving time), after a full year, the position of the sun would form a lopsided figure eight in the sky. This phenomenon is called the analemma, and it's a reflection of Earth's axial tilt and the ellipticity of our planet's orbit. They combine in subtle ways to create the figure-eight pattern, and the best way to understand all this is to examine these effects separately.
First, what would be the sun's position in the sky if Earth had no axial tilt and its orbit were a perfect circle? In that case, the sun would trace the exact same path in the sky every day. If you took a daily photograph in the same place at the same time, the sun would always be at the same spot in the sky; the figure-eight analemma would collapse to become a simple dot. We actually have a name for this ideal motion: the mean sun ('mean' meaning average). The mean sun's motion across the sky is constant and unwavering. You could literally set your clock by this hypothetical sun—and, in fact, we do!—to define a 24-hour day that never changes. You can think of the mean sun as being a clock. It always reaches its highest point in the day at exactly noon.
Now let's fiddle with that and add Earth's tilt in. Our planet is tipped over by about 23.5 degrees relative to the plane of its orbit around the sun. This changes the sun's actual position—what we call the 'true sun'—in the sky and causes it to deviate from the mean sun's path. The true sun is what you'd use to measure time if you had a sundial, though, perhaps sadly, over the past few centuries, they've fallen somewhat out of fashion.
Earth's tilt is what causes the seasons; in the summer, when our planet's pole is tipped toward the sun, the days are longer, and the true sun's daily path across the sky is higher. The opposite is true in the winter.
This immediately explains the north-south extent of the analemma! The true sun gets higher in the summer and lower in the winter, so every day at noon, it changes its height above the horizon a little bit. Over the course of a year, the true sun will move a distance in the sky equal to twice Earth's tilt, or about 47 degrees (compared with the 90-degree angle from the horizon to the zenith, directly overhead).
But the analemma also has an east-west extent that gives it that figure-eight shape. This, too, is largely because of Earth's tilt.
Imagine you're in the Northern Hemisphere, facing south at noon on the December solstice, when the true sun is at its lowest noontime point in the sky all year. Every day for the next six months, it will move a little bit to the north, slowly at first, then accelerating, reaching a maximum rate at the equinox. Then it slows again until the summer solstice.
Because it's moving northward, it's not moving at the same speed as the mean sun. It starts to fall behind, lagging a bit to the east, taking a little more time to reach its daily apex. (It gets to its highest point a little after a clock shows noon.) Every day it lags a little bit more, taking longer and longer to peak in its daily path.
But here's where things get weird: As the true sun's lag increases, the rate at which it lags gets smaller every day. It's still falling behind the mean sun, but every day, the additional amount is smaller. Then the true sun's eastward motion stops, reverses and starts to catch up to the mean sun. It eventually pulls ahead, to the mean sun's west, and reaches its apex before mean-sun noon. This continues for a few more weeks until its westward motion slows, stops and reverses, and it starts moving to the east again. It eventually slows enough that the mean sun catches up to it and passes it, and the whole process starts again.
This back-and-forth motion, coupled with the up-and-down motion, creates the analemma's overall figure-eight shape.
The actual dependence of this motion is a complicated function of algebraic trigonometry (and spherical geometry), but in the end, very roughly, over the course of a year, the sun moves ahead of the mean sun for three months, then behind it for another three, then ahead again, then behind again. Lather, rinse, repeat.
The deviation of the true sun from the mean sun is what astronomers call 'the equation of time.' It's used to figure out how much a sundial's time is off from the mean sun's 24-hour clock, and it adds up to about an eight-degree excursion to the east and west. This is much smaller than the 47-degree north-south motion, so the analemma is narrow and tall.
This is complicated by Earth's elliptical orbit. Earth orbits at the fastest rate when it's closest to the sun, and when it's farther out, it moves more slowly. When Earth is closest to the sun (called perihelion), its orbital rate is faster than average, so the true sun lags behind the mean sun. The reverse is true at the most distant orbital point from the sun (aphelion), and the true sun moves ahead of the mean sun. If Earth had no axial tilt, its orbital ellipticity would mean that over the year the true sun would make a short line in the sky aligned east-west.
There's also a perspective effect from this; at perihelion the true sun's motion in the sky is faster than it is at aphelion, and this affects the shape of the analemma as well. Combined with Earth's tilt, this throws off the figure eight's north-south symmetry. The southern loop (made when the sun appears to be moving faster near perihelion) is larger than the northern one—and wider as well. This makes the overall shape more like a bowling pin.
This gets even more complicated; the date of perihelion (around January 4 every year) is near but not quite the December solstice (on or around December 21), which skews the figure eight a bit, knocking it catawampus, distorted from perfect symmetry.
So if all this is true for Earth, what about other planets? They have elliptical orbits and have axial tilts as well, so they, too, get analemmas, but they're shaped differently. Mars, for example, is tilted by about 25 degrees, similar to Earth. Its orbit is much more elliptical, however, and that dominates the analemma, making it more teardrop-shaped. Jupiter has a tilt of only three degrees, but its orbit is more elliptical than Earth's, making its analemma a simpler ellipse in its sky. Neptune may have the most pleasingly shaped analemma of them all: its tilt is 28 degrees, but its orbital eccentricity is very slight, so its analemma is a nearly symmetric figure eight.
I understand this is all a bit complicated and may be difficult to picture in your head; I struggled with some of the more subtle motions myself (which wasn't helped by the many incorrect descriptions of the phenomenon I found online). It's perhaps surprising that the clockwork motion of the heavens isn't as straightforward as you'd hope! But I actually find beauty in all this: the repeating cycles, the subtle motions, the glorious symmetry to it. And it's all predictable, combining physics, geometry and mathematics. There is some truth to the idea of musica universalis, the 'music of the spheres,' and, like the music of great composers, there is majesty in complexity.
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