How good are your eyes at seeing really tiny things that are really far away? How much better is the Hubble Space Telescope at doing the same thing? What do these two questions have to do with the mathematical concept of angular size? Keep on reading to find out!
What Is Angular Size?
Once we know how to measure an angle, we can begin talking about the angular sizes of objects. What do we mean by an object's angular size? Well, if you draw lines from you to each side of an object in the distance, the angle between those two lines is that object's angular size. In other words, the angular size of an object tells you how big it appears in terms of its angular extent from your vantage point.
There is a handy trick that you can use to estimate the angular size of something using nothing more than your hand. All you need to know is that if you hold your hand at arm's length, the distance across the end of your pinky finger subtends (a fancy math word meaning "spans") an angle of about 1 degree. How is that useful?
Apparent Size of the Sun and Moon
Now that you know how to estimate angular sizes, one thing you can do is come up with a quantitative answer to the question we kicked things off with: How big does the Sun look to you? We can now understand this question as: What's the angular size of the Sun? If you use your handy-dandy pinky trick, you'll find that your pinky finger can completely cover the Sun. And that's because the Sun has an angular size of approximately 1/2 degree or 30 arcminutes.
Now, here's an astronomical fun fact. If you play this little game with the Moon, you'll find that it's angular size is almost exactly the same as the Sun's. Of course, that's not because the Moon is as big as the Sun. Far from it! In fact, the diameter of the Moon is about 400 times smaller than the diameter of the Sun. But the Sun happens to be about 400 times further away, which means that their angular sizes are almost identical.
Amazingly, this cosmic coincidence is what makes solar eclipses possible. If the Moon's angular size were less than that of the Sun, total solar eclipses would be impossible (since the Moon could not cover the disk of the Sun and blot out all of its light).
Human Eyes and Space Telescopes
I mentioned earlier that 1 arcsecond is a really tiny angle, but just how small is it? Well, if you were to set a penny on its end and walk about 2.5 miles away from it, that penny would subtend an angle of about 1 arcsecond from your vantage point. Of course, there's no way you could actually see that penny because your human eyeballs can't resolve details smaller than a few arcminutes in size. Which means you'd have to move that penny to about 100 feet away from you before you could actually make out enough detail to realize it's more than just a dot.
Amazingly, the Hubble Space Telescope can resolve details as small as 1/20 of an arcsecond. That means the Hubble Space Telescope would be able to see a penny from a distance of 50 miles! Which is one of the very important reasons why all of those beautiful pictures we get from it look a lot better than what you see with your unaided eye.
In the end, it's just math.
OK, that's all the math we have time for today.
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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!
Telescope image courtesy of Shutterstock.