Background

Minute Physics has a recent video on basic optics and angular resolution.

How Far Can Legolas See?

MinutePhysics

Published on Jun 30, 2014

[ … ]

So when Legolas, who has very human-sized pupils, looked at the rider of Rohan 24 km away, diffraction tells us that everything smaller than 3 meters would have been blurred to about three meters in size - perhaps he could still count the number of horsemen, but he definitely couldn’t distinguish their heights to within a few centimeters - unless Legolas could see in ultraviolet. Shorter wavelength light diffracts less, so if he could see in the extreme UV, then he’d be able to distinguish objects 10 centimeters in size, almost enough to discern the height of a man. Except that pretty much any kind of air absorbs extreme UV light - so even if he could see UV, Legolas would have been left in the dark.

Or maybe it’s just magic.

https://www.youtube.com/watch?v=Rk2izv-c_ts

Here is the relevant passage from “The Two Towers”:

‘Let us go up on to this green hill!’ he said. Wearily they followed him, climbing the long slope, until they came out upon the top. It was a round hill smooth and bare, standing by itself, the most northerly of the downs.

[ … ]

Following with his keen eyes the trail to the river, and then the river back towards the forest, Aragorn saw a shadow on the distant green, a dark swift-moving blur. He cast himself upon the ground and listened again intently. But Legolas stood beside him, shading his bright elven-eyes with his long slender hand, and he saw not a shadow, nor a blur, but the small figures of horsemen, many horsemen, and the glint of morning on the tips of their spears was like the twinkle of minute stars beyond the edge of mortal sight. Far behind them a dark smoke rose in thin curling threads.

There was a silence in the empty fields, and Gimli could hear the air moving in the grass.

‘Riders!’ cried Aragon, 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.’

Aragon smiled. ‘Keen are the eyes of the Elves,’ he said.

‘Nay! The riders are little more than five leagues distant,’ said Legolas.

— J. R. R. Tolkien, “The Two Towers”, chapter 2, “The Riders of Rohan”, p. 22-24.

https://books.google.com/books?id=12e8PJ2T7sQC&pg=PT17

(In my edition, this is on pages 22 - 24. Yours may be different.)

There is another passage of note.

The ridge upon which the companions stood went down steeply before their feet. Below it twenty fathoms or more, there was a wide and rugged shelf which ended suddenly in the brink of a sheer cliff: the East Wall of Rohan. So ended the Emyn Muil, and the green plains of the Rohirrim stretched away before them to the edge of sight.

‘Look!’ cried Legolas, pointing up into the pale sky above them. ‘There is the eagle again! He is very high. He seems to be flying now away, from this land back to the North. He is going with great speed. Look!’

‘No, not even my eyes can see him, my good Legolas,’ said Aragorn. ‘He must be far aloft indeed. I wonder what is his errand, if he is the same bird that I have seen before. But look! I can see something nearer at hand and more urgent; there is something moving over the plain!’

‘Many things,’ said Legolas. ‘It is a great company on foot; but I cannot say more, nor see what kind of folk they may be. They are many leagues away: twelve, I guess; but the flatness of the plain is hard to measure.’

— J. R. R. Tolkien, “The Two Towers”, chapter 2, “The Riders of Rohan”, p. 15-16.

https://books.google.com/books?id=12e8PJ2T7sQC&pg=PT17

What’s a league?

Leagues were never a standardized unit of measurement. The Roman or Gallic league, for example, was 7500 pedes, the modern equivalent of 2.22 km or 1.38 miles. This would make the riders of Rohan about 7 miles away, still hard to see but more within the realm of physical possibility.

https://en.wikipedia.org/wiki/League_(unit)

Of course, from a narrative perspective, a league in Middle Earth can be as long or short as it needs to be, and in any case Tolkien’s writing was not known for consistent distances. It’s not hard to justify the idea that “leagues” in Middle Earth are shorter than the modern-day English definition of three miles. For example, a league used to be defined as the distance a person could walk in an hour.^{citation needed} Legolas, being an elf from the densely wooded Mirkwood forest, might consider a league to be a shorter distance given the difficulty of navigating it by foot.

The distances given by Tolkien in the texts are sometimes very inaccurate. Tolkien’s usage of leagues, furlongs, fathoms, and ells added to the mystique and feeling of history, but also to the bewilderment of the mapmaker.

https://forums.revora.net/topic/81762-middle-earth-cartography/

On the other hand, there is an appendix on “Númenórean Linear Measures” in “Unfinished Tales” that suggests a league is Middle Earth is very nearly three miles, with some qualifications.

Measures of distance are converted as nearly as possible into modern terms. “League” is used because it was the longest measurement of distance: in Númenórean reckoning (which was decimal) five thousandrangar (full paces) made a lár, which was very nearly three of our miles. Lár meant “pause,” because except in forced marches a brief halt was usually made after this distance had been covered [see note 9 above]. The Númenóreanranga was slightly longer than our yard, approximately thirty-eight inches, owing to their great stature. Therefore five thousand rangar would be almost exactly the equivalent of 5280 yards, our “league:” 5277 yards, two feet and four inches, supposing the equivalence to be exact. This cannot be determined, being based on the lengths given in histories of various things and distances that can be compared with those of our time.

https://www.sf-fandom.com/vbulletin/showthread.php?34834-Why-did-Narsil-break/page2

http://askmiddlearth.tumblr.com/post/99488083026/units-of-measurement-in-middle-earth

https://books.google.com/books?id=lEfGx16U7FgC

A sense of scale

If we take a league to be three miles, then “five leagues distant” is fifteen miles away, which does sound challenging. An eagle can supposedly spot a rabbit two or three miles away, but fifteen miles is pushing it.

A pair of extraordinary eyes gives the eagle its legendary vision — it is estimated that an eagle’s visual acuity is at least four to eight times that of a human. In practical terms that means a grazing rabbit may be visible to an eagle flying as far as 2 miles (3.2 km) away.

https://books.google.com/books?id=O5J8JK07QykC&pg=PA11#v=onepage&q&f=false

An eagle eye has two focal points (called “fovea” [singular] or “foveae” [plural]) one of which looks forward and the other to the side at about a 45 degree angle. These two foveae allow eagles to see straight ahead and to the side simultaneously. The fovea at 45 degrees is used to view things at long distances. An eagle can see something the size of a rabbit running at three miles away.

https://www.nationaleaglecenter.org/faq-items/how-far-can-an-eagle-see/

Thinking in terms of horizontal distance, the distance to the horizon for a person with an eyeline at a 6 feet elevation is about a league (3 miles).

You have: sqrt(2 earthradius 6 feet)
You want: mile
    * 2.9995311
    / 0.33338545

Here we are using the approximation for an observer close to the earth, \(d = \sqrt{2 R h}\).

https://en.wikipedia.org/wiki/Horizon

Since Legolas is at the top of a hill, this isn’t an issue for visibility, but it does help give a sense of scale.

To use another length scale, suppose you are in the observation deck of the Sears Tower (a.k.a Willis Tower) in Chicago, 412.4 meters above the ground. In principle, the horizon at the top of the Skydeck is 45 miles away.

You have: sqrt(2 earthradius 412.4 meter)
You want: miles
    * 45.043236
    / 0.022200892

This is corroborated by the website’s FAQ.

On a clear day, you can see four states – Illinois, Indiana, Wisconsin and Michigan. Visibility from the Skydeck is approximately 40-50 miles (65 – 80 kilometers).

https://theskydeck.com/for-kids/fun-facts/

This depends heavily on visibility, naturally.

http://www.cleardarksky.com/c/Chicagokey.html?1

The Lincoln Park Zoo is about 3 miles away; Wrigley Field is about 5 miles away.

Wrigley Field, home of the Chicago Cubs, and Sears Tower are really nowhere near each other. But I was visiting both on the same day so when I was in the observation deck on the top of the Sears Tower in the morning I asked someone to point out where Wrigley Field was to me. I couldn’t identify it with my naked eye, but I pointed my camera that way anyway, put it on maximum zoom, and pushed the button. After a generous crop I had myself the above picture.

http://traveljapanblog.com/wordpress/2009/05/wrigley-field-from-the-sears-tower/

O’Hare International Airport (ORD) is about 15 miles away. Here’s what the Sears Tower looks like close up:

https://commons.wikimedia.org/wiki/File:Sears_Tower1.JPG

Note the antennas. They are apparently 12 feet in diameter at the base.

ROOFTOP TOWER STRUCTURES – TECHNICAL INFORMATION

• West antenna tower height = 294’-5” (from 109 roof level to top of strobe light)
• East Antenna tower height = 290’-7” (from 109 roof level to top of strobe light)
• Tower Bases = 12’ diameter solid wall steel cylinders
• The two main antenna tower structures consist of 80’ tall, 12’ diameter cylindrical steel bases, with open frame triangular tower sections atop each that house various television and FM transmission antennas

https://www.willistower.com/history-and-facts/antennas

What does the Chicago skyline look like from O’Hare?

(If you want to try this yourself, make sure you wait for a clear day. Any kind of clouds will make the skyline invisible.)

The big antennas are clearly visible, but the narrow antennas less so.

Thinking in terms of vertical distance, commercial airlines fly at about 7 miles.

typical “cruising altitude” – that is, the highest altitude reached during a flight and sustained between the ascent of takeoff and the descent of landing – is around 35,000 feet. That’s nearly 7 miles up in the air. However, the number generally varies from about 33,000 feet to 42,000 feet.

https://traveltips.usatoday.com/altitude-plane-flight-100359.html

Anecdotally, when I’ve seen the ground from a commercial flight, I can easily see large features like irrigation rings in cropland, and make out roads, but not see the vehicles on the road unless the airplane is close to landing. Making out people at twice that distance would be daunting.

Diffraction-limiting and the Rayleigh criterion

Now that we’ve considered prior knowledge and physical intuition, let’s consider the physical resolution limits of human vision. The MinutePhysics video used a pupil diameter of 5mm, a wavelength of 500 nm, and found a resulting angular resolution limit of ≈0.007°, or ≈0.4 arc minutes.

You have: 1.22 500 nm / (5 mm)
You want: degree
    * 0.0069900851
    / 143.05977
You have: _
You want: arcminute
    * 0.41940511
    / 2.3843296

Here, we are using the Rayleigh criterion for angular resolution.

The mysterious 1.22 is a dimensionless constant derived from the first zero of the Bessel function \(J_1\). Specifically, the first zero is at 3.83, and we divide by \(2\pi\) to switch from wavenumber to wavelength, then multiply by 2 to switch from radius to diameter.

You have: 3.83 * 2 / (2 pi)
You want:
    Definition: 1.2191269

(We’re using overlapping Bessel functions to define what it means to resolve two different light sources. See p.444 of Eugene Hecht’s Optics for details.)

https://books.google.com/books?id=0qEMPwAACAAJ&dq=editions:UsdZZQEFt4AC

The 0.4 arc minutes dovetails neatly with optimal visual acuity as measured by laser interferometers.

If the optics of the eye were otherwise perfect, theoretically, acuity would be limited by pupil diffraction, which would be a diffraction-limited acuity of 0.4 minutes of arc (minarc) or 6/2.6 acuity. The smallest cone cells in the fovea have sizes corresponding to 0.4 minarc of the visual field, which also places a lower limit on acuity. The optimal acuity of 0.4 minarc or 6/2.6 can be demonstrated using a laser interferometer that bypasses any defects in the eye’s optics and projects a pattern of dark and light bands directly on the retina.

https://en.wikipedia.org/wiki/Visual_acuity

To apply angular resolution to the size of distant objects, we simply multiply the distance away. For a distance of 24km (5 leagues = 15 miles), we can use the small-angle approximation.

You have: 0.0069900851 degree 24 km
You want: meter
    * 2.928
    / 0.34153005

If we want to look at the lower limit, let’s give Legolas a generous pupil diameter of 9mm (assuming he can dilate them without damaging his eyes) and use a wavelength for violet (400 nm). We arrive at a resolution of 0.2 arcminutes, which at 5 leagues is a little more than a meter.

You have: 1.22 * 400 nm / (9 mm)
You want: arcminute
    * 0.18640227
    / 5.3647416
You have: 1.22 * 5 leagues * 400 nm / 9 mm
You want: meter
    * 1.3089331
    / 0.76398097

But there’s is a subtlety we have not accounted for: what is the proper measurement of d in this equation? Is the resolution actually limited by the diameter of the pupil?

The order of optical components in the human eye is:

• cornea
• aqueous humor
• pupil
• lens
• vitreous humor
• retina

Optical components after the pupil, such as the lens and vitreous humor, are diffraction-limited by the diameter of the pupil. However, the optical components before the pupil, such as the cornea, can potentially gather more light and increase the effective aperture. So realistically, we should be using the diameter of the cornea, or rather the diameter of the cornea that is exposed by the eyelid.

This is, after all, how telescopes and binoculars work. The objective lens is generally the widest part of a refractive telescope, because the objective lens diameter determines the resolution limit, not the eyepiece diameter, the beam waist, or the pupil diameter.

Actually the eye of the average person is not able to resolve objects less than about 1 minute apart, and the limit is therefore effectively determined by optical defects in the eye or by the structure of the retina.

With a given objective in a telescope, the angular size of the image as seen by the eye is determined by the magnification of the eyepiece. However, increasing the size of the image by increasing the power of the eyepiece does not increase the amount of detail that can be seen, since it is impossible by magnification to bring out detail which is not originally present in the primary image. Each point in an object becomes a small circular diffraction pattern or disk in the image, so that if an eyepiece of very high power is used, the image appears blurred and no greater detail is seen. Thus diffraction by the objective is the one factor that limits the resolving power of a telescope.

— Francis A. Jenkins and Harvey E. White, “Fundamental of Optics”, 3rd edition, 1957, pages 304-305

https://archive.org/stream/FundamentalsOfOptics/JenkinsWhite-FundamentalsOfOptics_djvu.txt

For example, I have personally seen the dark spot at the center of the Ring Nebula by using a small optical telescope. The Ring Nebula has an angular diameter of something like 1.2 or 1.4 arc minutes.

It has an apparent visual magnitude of 8.8 and its angular diameter is 1.4x1.0 arc-minutes.

http://astropixels.com/planetarynebulae/M57-01.html

It has a diameter a little under one light-year and is 3000 light-years from Earth (angular size 1.2 arc minutes).

https://www.noao.edu/image_gallery/html/im0350.html

This would be extremely difficult to see with the naked eye, but not difficult at all with a 16” telescope. (In practice, there is not much difference in diameter between the cornea and the pupil, but it’s still important to make this distinction.) If you are curious about your own eyes’ resolving power, here’s a fun experiment you can try at home.

http://stokes.byu.edu/teaching_resources/resolve.html

Other considerations

However, maybe we should consider the Rayleigh criteria as being overly generous, since it treats the eye as a diffraction-limited system and does not account for the various aberrations of vision (astigmatism, chromatic aberration (coma), spherical aberration, etc.) This is to say nothing of the imaging difficulties introduced by dust in the atmosphere, refraction of the air, heat shimmer, and so on.

The air’s light-bending power, or refractive index, depends on its density and therefore its temperature. Wherever air masses with different temperatures meet, the boundary layer between them breaks up into swirling ripples and eddies that act as weak, irregular lenses. You can see this where hot air from a fire or a sunbaked road mixes with cooler air above; those ordinary heat waves are astronomers’ poor seeing writ large. Our windy, weather-ridden atmosphere is almost always full of slight temperature irregularities, and when you look through a telescope you see their effect magnified.

https://www.skyandtelescope.com/astronomy-equipment/beating-the-seeing/

Furthermore, the Rayleigh criteria corresponds to resolving two point sources of light in an otherwise dark environment, which provides maximal contrast. If there is additional background light or otherwise less contrast the task of resolving the light sources becomes more difficult. Distinguishing point sources is much easier than trying to discern individual horsemen and their height, especially when they are riding close together in a group.

On the other hand, the Rayleigh limit applies to monocular systems, but humans have binocular vision. Could this improve matters?

For example, could we use our eyes like twin radio telescopes? If so, instead of being limited by pupil diameter (≈ 5 mm), we could resolve objects separated the baseline, which for humans is the interpupillary distance (≈ 60 mm). This would be an order of magnitude improvement, and it would mean that our binocular vision could function similarly to an astronomical interferometer. While this may seem intuitively plausible, there is the problem of a much shorter wavelength in the optical range, and hence a much shorter coherence length than found for radio waves. Put another way, the eyes can compare brightness and frequency between left and right, but not phase.

At the shorter wavelengths used in infrared astronomy and optical astronomy it is more difficult to combine the light from separate telescopes, because the light must be kept coherent within a fraction of a wavelength over long optical paths, requiring very precise optics.

https://en.wikipedia.org/wiki/Astronomical_interferometer

Aperture synthesis is possible only if both the amplitude and the phase of the incoming signal are measured by each telescope. For radio frequencies, this is possible by electronics, while for optical frequencies, the electromagnetic field cannot be measured directly and correlated in software, but must be propagated by sensitive optics and interfered optically. Accurate optical delay and atmospheric wavefront aberration correction is required, a very demanding technology that became possible only in the 1990s. This is why imaging with aperture synthesis has been used successfully in radio astronomy since the 1950s and in optical/infrared astronomy only since the turn of the millennium.

https://en.wikipedia.org/wiki/Aperture_synthesis

Having tried to think about this, an immediate problem is that the sensor in an optical telescope only captures the magnitude of the signal and does not capture its phase.

https://physics.stackexchange.com/questions/168432/what-prevents-digital-interferometry-in-an-optical-telescope-array

Maybe binocular vision could help in some other way, but as near as I can tell from the following paper, binocular vision as it exists in human eyes does not help exceed the Rayleigh limit.

Perhaps the simplest description of Panum’s area is the spatial resolution limit that is based upon the similarity between monocular spatial resolution thresholds and binocular diplopia thresholds measured with low spatial frequencies. The similarity between these thresholds suggests that spatial resolution of coarse detail is limited cortically at or beyond the site of binocular interaction. Monocular spatial resolution thresholds for fine detail are lower than binocular diplopia thresholds at high spatial frequencies which indicates that binocular sensory fusion may be processed by spatial channels coarser than 22 arc min.

— Clifton Schor, Ivan Wood, and Jane Ogawa, “Binocular sensory fusion is limited by spatial resolution”, 1984

https://doi.org/10.1016/0042-6989(84)90207-4

Ophthalmologists measure the resolution of the human eye for a living. One ophthalmology textbook notes that the naked eye can do better than the Rayleigh criterion in some circumstances.

For example, the size of the Airy disc image of a point of light for the human eye under photopic conditions may be determined as follows. If f = 17 mm (focal length of eye), p = 4 mm (average photopic pupil), l = 0.00055 mm (median wavelength in visible spectrum of 0.0004-0.0007 mm), then the diameter of the Airy disc, \(D_\lambda\) is given by equation 2-1.2.

Equation 2-1-2

\begin{equation*} D_\lambda = \frac{(1.22)(17)(0.000555)}{4} = 2.8 \mathrm{\mu m} \end{equation*}

Note that the size of the Airy disc can vary with the focal length of the eye, the wavelength of light, and the pupil size. Also note that 2.8 μm is close to the size of the average foveal cone (1.5-2.0 μm). In comparison, the eagle has a large photopic pupil (about 6 mm); its foveal cones are thinner than those of the human and the eagle eye’s resolution is finer.

[ … ]

Animal	Pupil Width(mm)	F-Number
Net-casting spider	1.325	0.08
Cat	14	0.89
Flour moth	0.02	1.2
Tawny owl	13.3	1.3
Housefly	0.0025	2.0
Human	7-8	2.1-2.4
Pigeon	0.2	4.0

[ … ]

The second concept, the angle of resolution, is related closely to the Airy disc. The Airy disc is the physical distance, on the retina or a screen, between two points that are just resolvable. The angle of resolution, \(AS\), is another way to describe just resolvable points in physical space (equation 2-1-3; see Fig. 2-1-3).

Equation 2-1-3

\begin{equation*} AS = \frac{1.22 \lambda}{p} \end{equation*}

The angle of resolution, AS, for two distant stars viewed by a healthy, average human eye with a pupil of 8 mm in diameter is given by equation 2-1-4. However, it is known that the human eye can resolve two separate points in 1 minute or even less.¹¹ This discrepancy is explained as follow. The Raleigh criterion for resolution demands that the maxima of one point source must intersect the minima of the second point source (see Fig. 2-1-3),¹³ which allows a patch of no light (high-contrast image) between the two maxima. However, in the case of the healthy young human eye, contrast determinations can be made for targets of lower contrast. Thus, many human eyes are able to distinguish two point sources or two black bars when the diffraction patterns overlap (see Fig. 2-1-3).

Equation 2-1-4

\begin{equation*} AS = \frac{(1.22)(0.00055)}{8} \end{equation*}

\begin{equation*} = 0.000084 \textrm{radians} \end{equation*}

\begin{equation*} = 2.5 \textrm{minutes} \end{equation*}

For example, if it is assumed that the human separation criterion is one half the width of the Airy disc, then the angle of resolution is close to 1 minute of the arc. If the contrast enhancement know to be built into the neural processing of the human visual system is considered, it becomes apparent how some subjects have a resolution angle of less than 1 minute of arc.^{10, 14}

[ … ]

10. Jenkins FA, White HE. Fundamentals of optics. New York: McGraw Hill; 1950:290-3
11. Emsley HH. Visual optics. London: Hatton Press; 1950:47.
12. Blatt FJ. Principles of physics. Boston: Allyn and Bacon; 1987.
13. Lythgoe JN. The ecology of vision. Oxford: Clarendon Press; 1979.
14. Snyder AW, Bossomaier JR, Huges A. Optical image quality and the cone mosaic. Science. 1986;234:499-501

https://books.google.com/books?id=pv8cvsTB6fsC&pg=PA27

It’s not clear to me how the neural processing overcomes the Rayleigh limit, but it seems to be a known phenomenon.

Finally, we should consider that there are other cases where the human eye is known empirically to exceed the resolution limit, such as in Vernier acuity.

Visual Acuity in practice is closely related to the Nyquist limit. With your cones spaced 0.5 arc minutes apart, you need three cones to determine that there is a gap. Two of the cones need to be “on”, and you need one to be “off” in between them. The smallest detail that can be resolved is the distance between the centers of those two “on” cones. A cone spacing of 0.5 arc minutes gives you visual acuity of 1.0 arc minutes, just as the Nyquist limit says.

[ … ]

But for some reason that edge aliasing feature is much stronger. It’s a phenomena known as Vernier Acuity, and it is a form of Hyperacuity. In fact, we can detect slightly misaligned thin lines with 5x to 10x more precision than Visual Acuity.

What about signal theory? How is this effect possible? Signal theory is very well studied, and it tells us that we should not be able to resolve features that are smaller than 1 arc minute (assuming we have 20/20 vision with cone spacing of 0.5 arc minutes). Yet, somehow we can detect aliasing well below that threshold.

Is signal theory wrong? Did Nyquist and Shannon make a mistake in the sampling theorem? Of course not. The problem is that the Nyquist-Shannon theorem has one major caveat: NO PRIOR KNOWLEDGE! The sampling theorem is based on the assumption that you are trying to reconstruct frequencies from samples and nothing else. But if you have prior knowledge about the structure of the scene, then you can determine information about your scene that exceeds the resolvable resolution of your samples.

http://filmicworlds.com/blog/visual-acuity-is-not-what-the-eye-can-see/

We haven’t even addressed the fact that the eyes can move around and oversample a stationary object. This oversampling is only useful when the system is limited by the sensor resolution, but it does help get closer to the diffraction limit.

What’s more, cones themselves are 30-60 arcseconds across – between 5x and 10x times the size of the smallest gap you can see.

So that’s theoretically impossible… Or it would be if your eye was just a simple camera. But it’s not. Your retina is actually a CPU all by itself, and does a lot of processing for you. It also has some pretty specialized elements – like the design of the cones themselves.

[ … ]

So here’s the hypothesis. The ocular microtremors wiggle the retina, allowing it to sample at approximately 2x the resolution of the sensors.

http://accidentalscientist.com/2014/12/why-movies-look-weird-at-48fps-and-games-are-better-at-60fps-and-the-uncanny-valley.html?

If human eyes can accomplish this, maybe Legolas’ feat isn’t quite as far-fetched as it initially sounds.

Conclusion

Is Tolkien’s description of elven eyesight physically unrealistic? There are other possibilities:

• Legolas over-estimated the distance he was standing from the riders.

• In Middle Earth, leagues are significantly shorter than three modern-day miles.

• Elves have physically different eyeballs which can gather more light than human eyes, increasing their effective aperture.

  Eagles, for example, have slightly larger pupils, a thicker cornea, and thinner foveal cones.

  (This would improve matters a little, but not dramatically. Eagles can see a long way, but not 15 miles.)

• Elven eyes can use binocular vision to enhance their effective optical resolution in some unknown manner.

  This is an intriguing possibility, but note that unlike a diffractometer, Legolas is able to not only distinguish objects but also color; he can see the “yellow” hair.

• Elven eyes can oversample a distant object, gathering light over a longer period of time.

  This would be analogous to increasing the shutter time or frame rate of a camera; I’ve seen a paper call it “neural integration time”.

  This would not work as well for a moving target, however, and even the best eyes would still have trouble seeing details at the distance described in the book.

Related discussion:

The video states that although Legolas would in principle be able to count 105 horsemen 24 km away, he shouldn’t have been able to tell that their leader was very tall.

https://physics.stackexchange.com/questions/122785/could-legolas-actually-see-that-far

In J. R. R. Tolkien’s The Lord of the Rings (volume 2, p.32), Legolas the Elf claims to be able to accurately count horsemen and discern their hair color (yellow) 5 leagues away on a bright, sunny day. Make appropriate estimates and argue that Legolas must have very strange-looking eyes, have some means of nonvisual perception, or have made a lucky guess. (1 league ≈ 3.0 mi.)

https://perpetualstorm.tumblr.com/post/659963990791815168/theory-nobody-who-writes-a-physics-textbook-gives

https://www.google.com/books/edition/Six_Ideas_That_Shaped_Physics_Unit_Q_Par/t-9AAQAAIAAJ