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## Analemma

June 30, 2012

Today is a day of new things, since I finished my last day at my job yesterday and I’m going to math camp tomorrow. It’s exciting, and I’m going to kick off this first day of new things with a silly but fun thing I recently learned about the earth and the sun.

Some people know this already, but some people don’t, so sorry in advance if I bore you, but it’s super interesting the first time you think about it.

Namely, have you ever noticed, on your globe, a weird figure eight looking thing?

Nobody could be blamed for their curiosity, because there are so many important looking notches and then of course there’s the phrase “Equation of Time” next to it looking both pompous and intriguing. What is that thing??

After a few moments of contemplation, you’ve probably noticed there are months indicated, and since it’s a closed loop it’s probably describing something that is periodic with a one year period. Plus there are two axes, the vertical axis looks to be measured in degrees and the horizontal is called the “scale of time”.

Whenever I see north/south degrees I think of the earth’s tilt, and when I see something about time, it makes me think about how we measure time, which is vague to me, but probably has something to do with the sun, and orbiting around the sun, and spinning while we do it, again at a tilt. And if I want to be expansive at a time like this I’ll remember that the (pretty much circular) orbit of the earth lies on some plane where the sun also lives.

Now as soon as I get to this point I get nervous. What is time, anyway? How do we know what time it is? What with time zones, and daylight savings time, we’ve definitely corrupted the idea of it being noon when the sun is at its highest in the sky or anything as definitive as that.

So let’s imagine there are no time zones, that you are just in some specific place on the earth. You never move from that spot, because you’re afraid of switching time zones or what have you, and you’re’r wondering what time it is. If someone comes by and tells you it’s daylight savings time and to reset your clock, you tell them to go to hell because you’re thinking.

From this vantage point it’s definitely hard to know when it’s midnight, but you can for sure detect three things: sunrise, high noon, and sunset. I say “high noon” to mean as high as it gets, because obviously if you’re way north or way south of the equator the sun will never be totally overhead, as I noticed from living in the northeast my whole life.

But wait, even if you’re at the equator, the sun won’t be directly overhead most of the time. This goes back to the tilt of the earth, and if you imagine your left fist is the sun and your right fist is an enormous earth, and you tilt your right fist and stick your finger out and make it move around the sun (with your finger staying stuck out in the same direction because the tilt of the earth doesn’t change). As you imagine the earth spinning, you realize a point on the equator is only going to be directly in line to see the sun straight overhead about twice a year, and even then only if things line up perfectly.

Similarly you can see that, for any point between the equator and some limit latitude, you see the sun straight overhead twice a year – at the limit it’s once.

Going back to the point of view of a single person looking for high noon at a single place, we can see the height of the sun when it reaches its apex, from her perspective, is going to move around every day, possibly passing overhead depending on her latitude.

This is starting to sound like a periodic loop with a one-year period – and it makes me think we understand the x-axis. But what’s with the y-axis, the so-called “scale of time”?

Turns out it’s a definition thing, namely about what time noon is. Sometimes it takes the earth less time to spin around once than other times, and so the definition of “noon” can either be what we’ve said, namely “high noon,” or when the sun is at its highest in the sky, or you could use a clock, which has, by construction, averaged out all the days of the years so they all have the same length (pretty boring!). The difference between high noon and clock noon is called the equation of time.

By the way, back when we used sundials, we just let different days have different lengths. And when they first made clocks, they adjusted the clocks to the equation of time to agree with sundials (see this). It was only after people got picky about all their days having the same length that we moved away from sundial time. So it’s really just a cultural choice.

But why are some days shorter than others in terms of high noon? There are actually two reasons.

The first one, quaintly named “The Effect of Obliquity,” is again about the tilt. Imagine yourself sitting at the equator, looking up at the sun. It might be better to think of your position as fixed and the sun as going around the earth. And for that matter, we will assume the orbit of the earth around the sun is a perfect circle for this part.

Then what is being held constant is the spin of the tilted earth, or in other words the speed of the sun in the sky from the point of view of an observer on earth (this point is actually not obvious, but I do think it’s true because we’ve assumed a fixed tilt and a perfectly circular orbit. I will leave this to another post).

You can decompose this motion, this velocity vector, at a given moment, into two perpendicular parts: the part going in the direction of the equator (so the direction of some ideal sun if there were no tilt to the earth) and the part going up or down, i.e. in a right angle to the equator. Since we already know the sun doesn’t stay the same height all year, we know there has to be some non-zero part to the second part of this vector.

But since we also know the total vector has constant length, that means that the first vector, in the direction of the equator, is also not constant. Which means the length of the days actually varies throughout the year. The extent to which it does vary is approximated by a sin curve (see here)

The second reason for a varying length of a day, also beautifully named “The Effect of Orbital Eccentricity,” is that we don’t actually have a circular orbit around the sun- it’s an ellipse, and the sun is one of the two foci of the ellipse.

The thing about the earth being on an elliptical orbit is that it goes faster when it’s near the focus, which also causes it to spin more, due to the Conservation of Angular Momentum, which also makes an ice skater spin faster when her legs and arms are close to her body. Update (thanks Aaron!): no, it doesn’t cause it to spin more, although that somehow made sense to me. It turns out it just traverses a larger amount of angle with respect to the sun that we would “expect” because it’s moving faster. Since it turns as it moves faster, the day is shorter than you’d expect (this only works because of the way the earth spins – it’s counterclockwise if you’re looking down at the plane on which the earth is orbiting the sun clockwise). We therefore have faster days when we are closer to the sun.

When you add up these two effect, both approximated by sin curves, you get a weird function.

This is the “x-axis” of the analemma.

You can take a picture of the analemma by shooting a picture of the sun every day at noon, like these guys in the Ukraine did.

And by the way, you can use stuff about the analemma to figure out when sunrise and sunset will be, and why on the longest day of the year it’s not necessarily the day of the earliest sunrise and latest sunset.

And also by the way, there are lots of old things written about this stuff (see here for example) and there’s an awesome CassioPeia Project video (here uploaded on YouTube) explaining how all of this stuff varies over long periods of time.

Categories: musing
1. June 30, 2012 at 11:17 am

Ptolemy may have been wrong, but the argument that Copernicus and Kepler simplified things is a stretch.

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2. June 30, 2012 at 1:55 pm

Those that enjoy this post might like this book (note the illustration on the cover, and watchout for spoilers further down the page — sadly the Amazon link has a different cover art): http://en.wikipedia.org/wiki/Anathem

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3. July 4, 2012 at 1:29 am

I’m not sure what you mean by “the earth is one of the two foci of the ellipse”.
I thought the earth was on the ellipse, its orbit.

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• July 4, 2012 at 6:37 am

whoops! I meant “the sun”. I’ve changed it. Thanks!

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4. May 22, 2013 at 9:25 pm

Hey Mathbabe – how do you measure the eccentricity of the earth’s orbit using an anlemma?

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• May 22, 2013 at 9:28 pm

Wait, are you setting me up for a joke? Am I the straight man here?

I don’t know, how?

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• May 23, 2013 at 8:42 pm

Ba dump bum, crash! Unfortunately NOT a joke! I spent a year tracking the analemma and now I have to (among other things) calculate the eccentricitiy of the earth’s orbit using nothing but the analemma. Any thoughts?

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5. March 13, 2014 at 1:21 pm

Cool post, thanks! :)

Nice pic, too. But how did those guys get the clouds to be in the same place in the sky every day at noon? ;)

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