# The Limits of Elven Vision

Kate’s Tolkien re-read has reached Rohan, and her latest re-read post includes a reference to a comment I made about Legolas’s improbable visual acuity:

Re: Legolas seeing the Riders: I have since been advised by the resident physicist that the size of a pupil is a limiting factor on the resolution possible–if I understood properly, basically it boils down to how much light can come in–and under the laws of physics as we know them, it is not actually physically possible for Legolas to have resolved that level of detail at 5 leagues, regardless of how good his brain is at decoding images or whatnot.

The passage in question is this, from The Two Towers:

“Riders!” cried Aragorn, springing to his feet. “Many riders on swift steeds are coming towards us!”

“Yes,” said Legolas,”there are one hundred and five. Yellow is their hair, and bright are their spears. Their leader is very tall.”

Aragorn smiled. “Keen are the eyes of the Elves,” he said.

“Nay! The riders are little more than five leagues distant,” said Legolas.”

So, how reasonable is this, in physics terms?

The key idea here is the notion of resolution, that is, how well can you distinguish between two objects separated by some distance. The limiting factor here is the wave nature of light– light passing through any aperture will interfere with itself, and produce a pattern of bright and dark spots. This means that even an infinitesimally small point source of light will appear slightly spread out, and two closely spaced point sources will begin to run into one another.

The usual standard for determining whether two nearby sources can be distinguished from one another is the Rayleigh criterion, which takes the mathematical form:

That is, the sine of the angular separation between two objects is equal to 1.22 multiplied by the ratio of the wavelength of the light being considered to the diameter of the (circular) aperture through which the light passes. To get better resolution, you need either a smaller wavelength or a larger aperture.

We can make a ballpark estimate of what Legolas’s eyes would need to look like, given the rest of the data from the passage. He says that the riders are “little more than five leagues distant.” A league is something like three miles, which would be around 5000 meters, so let’s call it 25,000 meters from Legolas to the Riders (that’s a bit low, but we’ll assume he’s bragging). Visible light has an average wavelength of around 500nm, which is a little more green than the blond hair of the Riders, but close enough for our purposes.

The sine of a small angle can be approximated by the angle itself, which in turn is given, for this case, by the size of the separation between objects divided by the distance from the objects to the viewer. Putting it all together, these figures suggest that, in order to distinguish between two point sources separated by one meter, Legolas’s pupils would need to be 0.015m in diameter. That’s a centimeter and a half, which is reasonable, provided he’s an anime character. I don’t think Tolkien’s Elves are described as having eyes the size of teacups, though.

We made some simplifying assumptions to get that answer, but relaxing them only makes things worse. Putting the Riders farther away, and using yellower light would require Legolas’s eyes to be even bigger. And the details he claims to see are almost certainly on scales smaller than one meter, which would bump things up even more.

So, we can say with some confidence that Tolkien was neither a physicist nor an astronomer.

(This is actually used as the basis for a problem in one of the Six Ideas That Shaped Physics textbooks– the Quantum volume, I think, but I don’t have the books here– which is where I first encountered it.)