The Science of Middle-earth -- Keen are the Eyes of the Elves - Olog-hai
'Riders!' cried Aragorn, springing to his feet. 'Many riders on swift steeds are coming towards us!'
A recent Q&A exchange on TORn considered the extent to which Elves could see further than Men, and if so, how much further. It occurred to me that the question was ill-made: what the questioner really meant to ask was not about distance, but acuity, or, as astronomers call it, resolution. Just before the passage just quoted, Aragorn "saw a shadow on the distant green, a dark swift-moving blur". This is the same object that Legolas can resolve into individual riders. Both Aragorn and Legolas see the same object, so distance is not an issue, but Legolas has far greater visual acuity.
'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.
-- The Two Towers Chapter 2: The Riders of Rohan
Thinking about this further (and doing a few quick calculations) I realized how Tolkien's apparently throw-away remarks, made to illustrate the physical capabilities of the Elves, could have had a profound effect on the way Elves would have seen and reacted to the world.
What do I mean by 'distance' and 'resolution'? Well, consider the Andromeda Nebula, the closest major galaxy to our own Milky Way. Anyone with normal vision who lives in the Northern Hemisphere can make it out on a clear night as a fuzzy patch of gray, even though it is two million light years away -- the most distant object that can be seen with the unaided human eye.
However, it is not possible to resolve any of the hundreds of millions of stars within the Andromeda Nebula as individual objects unless you have a huge (but *really* huge) telescope. This instrument wouldn't bring the Andromeda Nebula any closer: all it would do is increase resolution, that is, the degree to which you might see closely-spaced objects (stars, in this case) as individual points, rather than a hazy blur of all the stars together. The power of telescopes is rated according to their resolution, not by how far they can 'see'.
Resolution is measured as angular distance 'of arc', that is, in degrees, minutes and seconds. The Celestial Sphere is 360 degrees all the way round - a perfect circle. Each degree is divided into sixty minutes, each minute into sixty seconds. Two objects in the sky are said to be separated by an angular distance measured in degrees (or minutes, or seconds) of arc.
So just how keen are the eyes of the Elves, and how would this acuity affect their world? Let's start with the quote from The Riders of Rohan above, which offers plenty of clues. Legolas says that the riders are "little more than" five leagues away. Let us assume that they are precisely five leagues away, and also that a rider on his horse, viewed from the front, is about a yard across, and that the riders are at least a yard apart. A bit of trigonometry on the back of an envelope shows that to resolve a single rider at that distance, Legolas' keen eyes could resolve points with an angular separation of about two thousandths of a degree, or just under eight seconds of arc. We can take this as a benchmark, although Legolas had greater acuity even than this: his dismissal of Aragorn's compliment ('Nay! The riders are little more than five leagues distant') shows that he was hardly straining himself, and he could see the colour of the hair on the riders' heads, spear points, and even detect that the riders varied in height.
Traditionally, the limits of human visual acuity are tested with a pair of stars, Mizar and Alcor, a double-star system in the constellation of the Big Dipper. The Arabs, who gave the stars the names we conventionally use, called these stars the 'horse and rider'. It is possible - just - for a keen-sighted person to resolve Alcor and Mizar as two separate stars of unequal brightness, rather than as a single star. Alcor and Mizar are separated by 12 minutes of arc, so this stands as the limit of human naked-eye resolution. Objects closer together than this will only ever be seen as a blur. When you turn your telescope on the problem, it turns out that Mizar (as distinct from Alcor) is itself a double-star. The two stars are separated by 14 seconds of arc, well within the abilities of Legolas, if not Aragorn.
(To complicate the picture further, each of these stars is its itself a double, making Mizar a quadruple-star system. The stars in the brighter pair of the two doubles are separated by seven or eight thousandths of a second of arc, requiring a thousand times the acuity that Legolas showed on the fields of Rohan, a feat that might have defeated even him: for more details on the Alcor-Mizar system, see this site).
This comparison, however, gives us an impression of how the world was seen through the eyes of the Elves. Aragorn at his most acute would only ever have seen Alcor and Mizar as a double-star, but Legolas would easily have seen it as a naked-eye *triple* star (Alcor, and the two doubles that make up Mizar), an object unknown to human sight. The starry sky was of great significance to the Elves, and no wonder - they would have seen far more in it than human beings ever could.
The Elvish night sky would have been far more lively than the human one, as well as being more detailed, because greater acuity means an increased ability to detect movement. Planetary motion provides one example. The planets were first described, by the ancients, as the stars that 'wandered' against the background of the 'fixed' stars. This wandering could be charted by plotting this movement, and it was shown that the planets all describe a full circle around the sky, although at varying speeds, complicated by the fact that we ourselves live on a moving planet (which makes it look like some of the planets appear to reverse direction now and then, but we needn't worry about that here.) The planet Mars, for example, takes 1.8822 years to travel the full 360 degrees, and leaving out all other complications for the sake of simplicity, it therefore moves an angular distance of 12 minutes - equivalent to human eyesight - in just under nine hours. This means that it would take a human observer nine hours to notice that Mars had moved, with respect to the background of the sky. Given that Mars is probably not observable for nine hours together on any given night (because of the Earth's rotation), an unaided human observer would not be able to detect the motion of Mars against the background of stars in any one session.
Elves wouldn't have to wait nearly that long. Taking 8 seconds of arc as something an Elf such as Legolas was comfortable with, he would have noticed the motion of Mars in six minutes of observation, or less. To an Elf, the Sun, Moon, planets and other phenomena such as shooting stars would have fairly whizzed across the firmament. O Elbereth Gilthoniel, indeed. This argument doesn't just apply to stars. When Galadriel sadly sings of the leaves of Lórien blowing in the wind, she would have been conscious of a host of tremulous movement invisible to human eyes, making the trees seem even more alive, and giving her lament added poignancy. Keen indeed are the eyes of the Elves, a sharpness that pierces the heart.