6. (Problem Solving)

Resolution

The minimum separation between two sources of light so that they can be distinguished as separate lights is $$s_{\min} = 2.44{\lambda L\over a}$$ or, in Angle Measurements, $$\theta_{\min} = 2.44{\lambda\over a},$$

where

$\lambda$ is the wavelength of the light (visible light is roughly $\lambda\approx 500\u{nm}$
$a$ is the diameter of the aperture (for a human eye, $a=\hbox{2-4}\u{mm}$ depending on lighting conditions; the pupil usually grows larger in darkness)
$L$ is the distance from the observer to the source

If a question asks you "Two sources are a distance $s$ (or angle $\theta$) apart, can they be distinguished?" then you want to write the inequalities

$$ s > s_{\min} \quad\hbox{OR}\quad \theta > \theta_{\min}.$$ If the inequality is satisfied then the two sources can be distinguished, otherwise they look like a single blur.

Example

In the Lord of the Rings, Legolas demonstrates his keen elvish eyesight by counting the number of horsemen in a group five leagues distant, while Aragorn sees nothing but a blur. Is it possible for someone to have eyesight that keen? We'll assume that Legolas is viewing them in visible light ($\lambda\approx 500\u{nm}$) and that elves' pupils are about as wide as human eyes ($a=3\u{mm}$). Five leagues is fifteen miles or 24 kilometers, so the absolute minimum separation Legolas should have been able to distinguish is $$\begin{align} s_{\min} &= 2.44{(500\ten{-9}\u{m})(24\ten3\u{m})\over 3\ten{-3}\u{m}}\\ &\approx 10\u{m}\\ \end{align}$$ Since a person on a horse is less than 4 meters high, it is physically impossible for any normal eye to see any details of the horsemen at that distance. How did Legolas do it, then?

A picture of Moana smiling up at the camera, with approximate guesses at her head width (roughly 18mm) and her pupils (13mm). — The keen eyesight of Disney princesses?

If Elvish eyes had very large pupils (and thus a larger $a$), then the size of details they could see would be smaller. For instance, Moana's pupil size is about 13mm, meaning she can see details that are a quarter the size of a normal human, or about $2.5\u{m}$. Perhaps Elves look like Disney princesses? (If not all the time, perhaps they can make their pupils expand as needed.)
If Elves can see in light with a smaller wavelength $\lambda$, such as ultraviolet, then they will be able to see finer detail. Some animals can see near-ultraviolet light at $300\u{nm}$; at those wavelengths they can make details that are half the size.
Magic.

Example

How far away is it possible for a human to read text that is 1 inch (or 72 points) high?

To read we need to be able to make out the features in each character. In the letter E, for example, we need to be able to see that the three horizontal branches are separate from each other. Since those lines are 0.5in or 1.25cm apart, we'll use that as our desired separation. Visible light has a wavelength of $500\u{nm}$, and the human pupil is about $a=3\u{mm}$ across, so we need

$$\begin{align} s&>s_{\min}=2.44{\lambda L\over a}\\ 1.25\u{cm} &> 2.44{(500\u{nm})L\over 3\u{mm}}\\ \implies L &< {(1.25\ten{-2}\u{m})(3\ten{-3}\u{m})\over 2.44(5.00\ten{-9}\u{m})}\\ \implies L&<3\u{km}\\ \end{align}$$

So it would be impossible to read 72-font at a distance greater than 3km. Of course, this is an absolute maximum distance. In reality, we would need to be much much closer to read text of that size, partly due to the imperfections in our own eyes, partly due to fluctuations in the Earth's atmosphere which can have a lensing effect.