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by Jerry Oltion

What the Heck Is an Analemma?


Have you ever looked at a world globe and noticed that figure-eight out in the Pacific ocean? Ever wonder what it's for? The legend often says something like: "Analemma, showing the sun's declination for every day in the year. Also the equation of time." That's about as helpful as a software instruction manual, right?

Good news: I've puzzled it out, and it turns out the analemma is really cool. Let's have a look.




The term "declination" provides a clue. Declination is basically how far up or down something is in the sky. We all know that the Sun is high in the sky in summer and low in winter, and a quick look at the analemma on a globe shows that the figure-eight is divided up into months and it peaks out in June and bottoms out in December. So the upward-downward axis of the analemma follows the Sun up and down the sky with the seasons.

You might also note that the tips of the analemma just reach two dotted lines: the tropic of Cancer in the northern hemisphere and the tropic of Capricorn in the southern hemisphere. This is not a coincidence: Those are the points farthest north and south where the Sun can be directly overhead at noon. Outside that range the Sun never quite makes it to the zenith (the point straight up from wherever you're standing).

What makes the Sun go up and down in the sky, and why does it stop at the tropics?


Axial Tilt


The Earth rotates once a day and it goes around the Sun once a year. If the axis of both of those motions pointed in the same direction, the Sun would simply shine straight down on the equator all the time. Those two axes aren't parallel, though. The Earth's rotational axis is tilted 23.5° away from its orbital axis. That means that as the Earth goes around the Sun in its yearly journey, the rotational axis sometimes points more toward the Sun than it does at other times. If you've got a globe and a flashlight handy, you can demonstrate this easily yourself: Tilt the globe if it isn't already, then walk around it while aiming the flashlight straight at it from the side. Note that for half your orbit, your flashlight shines mostly on the northern hemisphere, and for the other half of your orbit, your flashlight shines mostly on the southern hemisphere. If you're really careful holding that flashlight level, the middle of the beam will just kiss the tropics of Cancer and Capricorn.

Now suppose you mounted a camera pointing up into the sky and took a picture once a week or so at noon, never moving the camera between shots. After a year, what would you expect to see? A series of images of the Sun rising up into the sky in the spring, topping out at the beginning of summer, then dropping downward until they bottom out at the beginning of winter, right? If you forgot to correct for daylight saving time, your line would have a kink in it in March and November, but your series of photos would indeed show the Sun changing declination with the seasons.


Not So Fast


If you stacked those photos on top of one another so you could see every image of the Sun at once, you'd notice something odd. Instead of making a straight line up and down in the sky, your images would swing around in a lopsided figure-eight.

Not surprisingly, people have done this. It really works.

So what's going on here? Why do we get a figure-eight rather than a straight line?

An analemma photographed in Hawaii. It tilts because it was taken in the afternoon. Photo © Rob Ratkowski.


Axial Tilt Again


No, it's not orbital eccentricity, despite what you'll read on about half the web pages dedicated to analemmas. Orbital eccentricity plays a part, and we'll get into that in a moment, but the same axial tilt that causes the range in declination is actually responsible for the lion's share of the left-right motion, too.

Your globe probably has another line on it called the "ecliptic." It has dates on it in little tiny print. Get some reading glasses and you'll notice right away that the ecliptic crosses the equator on March 21 and September 23, and it peaks out—hey, look at that!—at the tropic of Cancer on June 22 and bottoms out at the tropic of Capricorn on December 22. That looks suspiciously like the path of the Sun in the sky! That's because it is.

Now notice the angle that the ecliptic takes when it crosses the equator: 23.5°. If you imagine the Sun moving along that line and use the date scale to measure its progress, you'll see that it takes 16 days to cross 15 degrees of latitude when the ecliptic is angled steeply in spring and fall, but it only takes 14 days to cross the same distance in summer and winter. That's because the Sun travels the same distance along the ecliptic every day (just a smidgen under one degree per day), but in the spring and fall it's moving along at an angle. It's like flying from New York to New Orleans: You travel 1200 miles, but in east-west terms you've only flown about the distance to Chicago (725 miles).

So in March and September the Sun lags behind in its east-west motion because it's moving at an angle. But in June and December it's moving nearly directly east-west, so it speeds up again.


The Sun Gets Loopy


Now imagine your year-long photo series of the Sun, taken at exactly the same time of day once a week. If the Sun lags behind in the spring and fall, its path is going to bend in an arc. When it starts catching up, say in November or July, that arc will flatten out and the line will be nearly vertical for a few weeks. Then the Sun will start to run ahead of its average speed of one degree per day and the line will curve the other way. It's still moving sideways as it peaks out in June and December, so the curve loops make a figure-eight.


Orbital Eccentricity


But why is the analemma squished at the top? Now we get to orbital eccentricity.

If the Earth's orbit around the Sun were a perfect circle, the analemma would be a perfectly symmetrical figure-eight. But we move in an ellipse. That means we edge closer to the Sun on one side of our orbit and farther away on the other side. Many people think that's why we have seasons, but that's not true, as you can readily prove by the simple realization that the southern hemisphere has winter while the northern hemisphere has summer.

If you read my column in our May/June 2019 issue on orbits, you know that the closer you get to the body you're orbiting, the faster you move. (And now you know it even if you missed that column.) So since it's the Sun's speed around the ecliptic that causes the analemma to deviate sideways, you would expect the width of the analemma to be wider when we're closest to the sun and narrower when we're farther away. And hey presto! We're closest to the Sun in January and farthest in July.

Not December and June? Nope. It's off a little bit, which is why if you make a really accurate analemma you'll notice that it's skewed a little bit to the side.


The Equation of Time


So what's all this good for? Correcting sundials, of course!

No kidding. Before we had accurate clocks, we used sundials to tell the time. That gave us what's called "solar time," and it was perfectly fine for our needs. When the Sun was at its highest point for the day, that was noon, and we didn't notice that noon came a little bit earlier each day for half the year and a little bit later the other half. But when we developed clocks, we noticed it pretty quickly because the error is as much as sixteen minutes on one side and fourteen on the other. That's a half-hour's difference overall, and it must have driven clock-makers crazy until they figured out what was happening. After clocks became common, though, people who still used sundials had to correct for the Sun's deviation in the sky or they would miss their flights to New Orleans!

Note the little horizontal scale on your globe's analemma. That's marked in minutes, and there might even be a legend that tells you when your sundial is slow and when your sundial is fast. (Mine tells me when a clock is slow and when it's fast, a distinction that warms my heart—and tells you the age of my globe.) The width of the analemma on any particular day gives you the correction for that day.


Analemmas on the Ground


You can make your own analemma without need of a camera. All you need is an accurate watch, something to cast a shadow, and an open space to mark it in. And a year of time, of course.

The thing that casts the shadow needs to be solid. It can't move even a smidgen all year. The top of a pole driven in the ground works well, as does a sticker in a sun-facing window. Bear in mind that in the wintertime the shadow is going to fall quite a ways away from the source, so it's going to be pretty blurry. Use something that will cast a fairly definitive blob that you can estimate the center of. (A tennis ball on the end of a 10-foot pole works well, as does a Post-it note stuck to a window.)

Then at the same time every day—it doesn't have to be noon, but it has to be within a minute or so of whatever time you choose— mark the shadow's center on the ground or the floor. (You might want to use tape on the floor rather than make a permanent mark on it.) Remember to do it an hour later during daylight saving time, because the Sun doesn't know you've changed your clock. Over the course of the year you'll be able to watch the shadow make its way around in a narrow figure-eight.

And yes, by marking the dates, drawing a center line, and measuring carefully, you can use your homemade analemma to correct a sundial!


Analemmas on Other Planets


You might expect that every planet has an analemma shaped like the Earth's, but that's not so. Mars has an axial tilt about the same as Earth's, but its orbital eccentricity is so extreme that the narrow end of its analemma is completely pinched off, making it look more like a cartoon raindrop. Saturn's eccentricity is just enough to pinch the top of the analemma down to a tiny little loop. Mercury and Venus rotate so slowly that their day length is a significant fraction of their year (Venus's day is actually longer than its year), so they don't even have analemmas.

Mars's analemma is teardrop-shaped

I leave it as an exercise for the reader to figure out what an analemma of a planet orbiting a double star looks like.


Jerry Oltion has been a science nut since he was old enough to spell "curious." He has written science fiction almost as long, and has done astronomy somewhat less. He writes a regular column on amateur telescope making for Sky & Telescope magazine, and spends many, many nights a year out under the stars.

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