Circular reasoning makes me radiant

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Number crunching

Because I think math is cool, and I think that because it is

Every now and again the Internet delights me. It’s rare, of course, but shines like a diamond in the muck when it occurs.

Last Friday’s xkcd comic was one such time. On March 10, 2023, Randall Munroe posted this:

Ha! If you’re not familiar with radians, they’re an angular measurement, and can’t be used to measure an actual length. We use them in astronomy all the time: We say the Moon is about half a degree across, but we mean its apparent size in the sky. For comparison, there are 90° from horizon to zenith, so about 180 Moons would fit in that space. And we know the distance to the Moon (about 380,000 km on average) so we can convert that, using a bit of simple trigonometry, to an actual physical size (3,475 km).

But then you wouldn’t multiply that size by 180 and say, “The sky is 625,500 km from horizon to zenith”. That wouldn’t make any sense.

So, as always, it’s a funny comic.

The part that got me, though, is when you mouseover (or long tap if you’re on a mobile device) the comic. A second punchline comes up (this is true for every xkcd, you should know):

Woohoo! An xkcd shoutout! Full disclosure: Randall and I have been friends a long time. But still. Cool! Oh, he mentions this exact thing in one of his what-if comics, too.

But… what does all this mean?

Well, you know I can’t leave something like this alone. So let me elaborate*.

A radian is what we call a dimensionless unit, in that it doesn’t have a physical dimension like length or mass or time associated with it. That makes sense; a circle can be any size, but a radian is defined as the length of the arc along the circle that equals the circle’s radius. Remember the circumference of a circle is the radius times π times 2, so 2 π radii can fit along the circumference. If the circle is bigger that arc is longer in, say, centimeters, but the apparent size of it as seen from the center is still 1 radian. It still occupies 1/2 π ᵗʰ  of the circle’s circumference.

The physical distance from the center of the circle to the edge never comes into play, so the radian is dimensionless. We can convert it to the more familiar unit of degrees by remembering there are 360° in a circle, and that’s equal to 2 π radians. So a radian = 360 / (2 pi) degrees = about 57.3 degrees.

But that doesn’t mean the radius of a circle is 57.3°! That doesn’t make physical sense. But — and note I’m not contradicting Randall here, because that would be folly — if you look at the third circle in the xkcd comic, you can say that from the point at the upper part of the arc, the radius drawn would subtend an angle of 57.3°. That’s totally fair.

And this is where the popup comes in. Years and years (and years) ago I dreamed up a way of calculating the area of the sky in square degrees (just like you might figure out the area of a room in square meters). If a radian is 57.3 degrees, and the area of the surface of a sphere is 4 x π x radius², then we can assume that the area of the sky is 4 x π x 57.3^2 = 41,253 square degrees.

That answer is correct. But I kinda cheated to do it, assuming the “distance” to the sky is 57.3 degrees, which is not really true.

But I realized there’s a way to do this more rigorously:

The circumference of a circle = 2 x π x radius (call that radius r).

That means r = circumference / (2 π)

But the area of a sphere is 4 x π x r². Substituting in my new definition of r we get

Area = 4 π x (circumference / 2 π)² = circumference² / π

But we know that the circumference is 360°, so area = 360² / π = 41,253 square degrees**. AHA!

I never used the radius as a length here, and it divides out, so I think this is a better way to do it. Either way you get the same answer, which is what Randall is saying in his popup text. Yay!

These units (and ways of thinking) are a little hard to get used to, but astronomers use them all the time. We talk about the apparent size of an object in degrees — or arcminutes, where 60 arcmin = 1 degree, or even smaller units called arcseconds, where 1 arcmin = 60 arcsec. Heck, for really good telescopes that see very small objects in the sky we use milliarcseconds, a thousandth of an arcsec.

To give you an idea of these, the full Moon is about 0.5° in the sky, or 30 arcmin, or 1,800 arcsec, or 1.8 million milliarcsec. Some cameras on Hubble Space Telescope have pixels that are a couple of dozen milliarcseconds on the sky. Teeny. Using a sophisticated technique called intereferometry, some objects can be seen down to only a handful of milliarcseconds across.

All of this can be pretty handy. For example, an object that is as far from you as its size is one radian or ~57° across. So, a tree 10 meters tall and 10 meters away from you would appear to be about 57° in apparent size. A finger held at arm’s length is roughly a degree wide. If you see something (a house, say) that’s the same apparent size as your finger held out, then that house is 57x farther away than its true size. If it’s two stories, say, then that’s roughly 6 meters high, so it’s 57 x 6 = 340 meters away (keep these numbers rough and you’ll be fine; don’t seek too much accuracy here!).

OK, well, that’s handy for me at least. I use this approximation relatively often. Going back to my Moon example from the beginning of this article, it’s half a degree in size. That means it’s 57 x 2 (because there are two Moons per degree) times its physical size away from us: 3,475 km x 57 x 2 = ~400,000 km, which is pretty close.

The most important astronomical use for all this is that it’s how we figured out the distance to the nearest stars… but that’s for another issue. Remind me sometime and I’ll wax on about that, too.

* Did you know there’s a site that explains nearly every xkcd comic? It’s not affiliated with Randall, but I’ve found it useful on occasion.

** I expect that if there really were such a professional as dimensional analyst they’d probably still hate this.

Et alia

You can email me at [email protected] (though replies can take a while), and all my social media outlets are gathered together at about.me. Also, if you don’t already, please subscribe to this newsletter! And feel free to tell a friend or nine, too. Thanks!

Et alia

You can email me at [email protected] (though replies can take a while), and all my social media outlets are gathered together at about.me. Also, if you don’t already, please subscribe to this newsletter! And feel free to tell a friend or nine, too. Thanks!