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The Math of a Solar Eclipse

Aug 21, 2017

Did you get a chance to see the recent annular solar eclipse? Living in California, I was lucky enough to be in its path—and the view was pretty amazing. As I watched my pinhole-camera projection of the Moon progressively blotting out a larger and larger fraction of the Sun, I couldn’t help but thinking about the very strange and fortunate mathematical and astronomical coincidence that makes solar eclipses possible in the first place. So today we’re going to talk about that mathematical coincidence, what it has to do with the different types of solar eclipses, and we’re going to make sure that you’re fully prepared for the next time you get a chance to see one.

What is an Eclipse?

Before we get too deep into the math behind solar eclipses, let’s take a minute to talk about the basic astronomy of an eclipse. This might surprise you, but eclipses don’t necessarily have to involve the Sun because the word eclipse describes any event in which one astronomical body casts a shadow on another. So that includes solar eclipses where the Moon casts a shadow on the Earth, lunar eclipses where the Earth casts a shadow on the Moon, and lots of eclipses that don’t involve the Earth or the Moon at all. For example, eclipses occur when one of Jupiter’s moons casts a shadow on the giant planet, when one of Saturn’s moons casts a shadow on another of its moons, and even when one star in a binary system (that’s two stars orbiting each other) casts a shadow upon the other.

As a short linguistic aside, an eclipse is an example of something called a “syzygy.” Fun word, right? A syzygy occurs whenever three astronomical bodies happen to be configured in a straight-line. People like to make a big deal over such things and use them to predict the end of the world and whatnot. Of course, that stuff is all nonsense. But syzygy really is a fun word.

An Astronomical Mathematical Coincidence

Let’s now focus on the most spectacular type of eclipse that you and I can experience here on Earth—a total solar eclipse. If you’re in the right location when such an eclipse occurs, you’ll witness what is reportedly an awe inspiring sight as the Moon completely blocks the light from the disk of the Sun, a bizarre twilight suddenly overtakes the day, and the Sun’s normally invisible two-million degree corona—it’s outermost layer—streams out from behind the Moon’s disk. And while that certainly sounds like a spectacular sight (I’ve yet to witness one myself), it’s also a spectacular (or should I say astronomical) coincidence. Why? In short, it’s because the math works out so perfectly.

Here’s what I mean: The diameter of the Moon is about 3,475 km and the diameter of the Sun is about 1,391,000 km. Which means that the diameter of the Sun is about 1,391,000 / 3,475 or almost exactly 400 times larger than the diameter of the Moon. That means that if the Moon and the Sun were equally far away from the Earth (which, of course, they’re not), the Sun would appear in the sky to be 400 times wider than the Moon. And, since the Moon would be tiny compared to the Sun, total solar eclipses would not be possible! From this observation we can see that in order for total solar eclipses to be possible, the apparent size of the Moon that we see in the sky must be at least as big as the apparent size of the Sun (since the Moon has to “cover” the Sun). And, coincidentally, it is…almost exactly!

The Math Behind Total Solar Eclipses

As a result of a bit of lucky math, we get to experience the wonder of total solar eclipses.
 

As painters have known about and often struggled with for millennia, the apparent size of an object changes depending upon how far away it is from you. In the graphic arts, this is known as perspective. As a simple example, if you compare the apparent height of two identical trees that are 50 and 100 meters away, you’ll find that the more distant tree appears to be 50/100 or half as tall. And it’s exactly this same principle that applies to the apparent sizes of the Sun and the Moon. In particular, as we saw earlier, the diameter of the Sun is about 400 times larger than the diameter of the Moon. Using the principle of perspective that we used to determine the relative sizes of trees, we see that the apparent size of the Moon will be larger than the apparent size of the Sun—and thus that total eclipses will be possible—if the distance to the Sun is at least 400 times more than the distance to the Moon.

 

So, how does the math work out? Well, the distance to the Sun ranges between 147,000,000 km and 152,000,000 km, and the distance to the Moon ranges between 360,000 km and 406,000 km. Given these numbers, we can now figure out how much the apparent size of the Sun is reduced in comparison to the apparent size of the Moon due to its greater distance from us. When the Sun is at its farthest point and the Moon is at its point, the Sun is 152,000,000 km / 360,000 km or about 422 times further away than the Moon. In other words, at that time, the factor of 400 difference between the diameters of the Sun and Moon and the factor of 422 difference between the distances to each of them conspire to make the apparent size of the Sun and Moon nearly identical. Which is great news because it means that as a result of a bit of lucky math, we get to experience the wonder of total solar eclipses.

When are the Next Solar Eclipses?

If you’re wondering when your next chance to see a total solar eclipse will be, the answer is that it depends on where you live…and on how far you’re willing to travel. If you’re in Australia (or somewhere out in the middle of the Pacific ocean), you’re in luck because there’s going to be a total solar eclipse on November 13, 2012. People in the United States have reason to get excited because there’s going to be a spectacular total eclipse cutting diagonally across the country from the Northwest to the Southeast on August 21, 2017. That’s still five years away, so you have plenty of time to plan your trip!

Wrap Up

Okay, that’s all the math (and astronomy) we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via FacebookTwitter, or by email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!