Over at Tor.com, Kate has a Lord of the Rings re-read post about the Battle of the Pelennor Fields, which includes a shout-out to me that I missed because I was driving to NYC:
Ãomer is "scarely a mile" away when the standard unfurls and is clearly seen to bear the White Tree, Seven Stars, and a high crown. If I were at home, I could ask the resident scientist to tell me how big these elements would need to be to be visible at a mile, but I'm finishing this post on the train down to New York City (vacation! Woo! I'm going to try and write the next post while I'm there, too, so as to make up for getting so far behind) and he's joining me later. Also, he's busy. But even without doing the math, I can well believe that at a mile, magic would be involved in the visibility.
This is a reference to our earlier discussion of the resolving power of Legolas's eyes. Using the same analysis, I can answer this question pretty easily: assuming that Eomer's pupils are 2mm in diameter, and treating the elements of the flag-- say, two of the stars-- as point sources, at one mile distance, they would need to be separated by about half a meter to be resolved. So, yeah, magic would need to be involved.
Now, that's maybe being excessively literal about the resolving-- it's probably easy to distinguish between Aragorn's flag and whatever standard Sauron might use even if you can't make out all the details, and Eomer could've filled in the rest knowing whose flag it was. It's also a little generous as to the pupil size, though, particularly if it's supposed to be daylight, so it's probably a wash.
Follow up question: would the pupils actually contract if the eyes were good enough to be diffraction limited, or would some other mechanism to limit light levels evolve?
Looking at the comments at your original discussion, it seems that no-one thought to put the discussion into information terms. Roughly, suppose that Legolas can store multiple copies of what he can resolve in the general region of the riders. He can see the general area he wants to analyze, so he records a million images of the small region of interest in a tenth of a second (assuming there's enough light for sensor noise not to be too much of a problem?). For each image he records the position of his head accurately. Suppose each image provides 300 bits of information about the region of interest. Given the head position information and enough information about the response function of the cells of his retina, he can invert the transformation from real world at 25000 meters to what he has recorded. He has to make assumptions about the motion of the riders and about any heat haze, but he's already provided coarse compensation by tracking the motion. Given that Legolas has 300 million bits of information about the riders, he can in principle do better than the Rayleigh limit. This superior information processing also accounts for his ability to say that there are 105 riders.
Discovering that there is a planet in orbit around a star at many light-years distance precisely does this kind of processing, well below the Rayleigh limit. Record the intensity of a star for long enough and do some fancy calculations using the information available and a few assumptions about what might be causing the changes of intensity. Similarly, when TV cop shows enhance a video of a number plate, up to a point it could be done by processing enough out of focus images.
It's usually more effective to adapt the optics to the conditions rather than to apply data processing, but we generally use a mixture of the two methods. Of course, the amount of processing power we're talking about to be able to do this fast enough (instead of evolving eyes a couple of meters across) may be pretty much magic --- at least in Arthur C. Clarke's advanced civilization sense.
Though I wonder if it could just be a color matter -- if Sauron uses a black standard and Aragon's is a light or bright color, then the half-meter would just need to be the length of the standard (or less). I'd also assume that a bright color would be desirable for just this reason -- black may be intimidating, but it's also close in color to your horses.
Aragorn's banner is black with a white tree and seven stars emblazoned on it and above that a crown. Sauron's forces probably had multiple banners (eye, ghastly moon, what each of the leaders of the Haradrim or Easterners carried). I also suspect the banner is quite large, perhaps 2 meters by 1 meter.
As for Eomer recognizing it, I think unlikely since no one has used it on a banner since before Rohan was founded. He had seen the banner before but only furled (at the fords of Isen when the Dunedain found Aragorn). The elite guard of Minas Tirith wore the emblem but had Eomer ever been to Minas Tirith before and seen the guard (the Steward's own banner was plain white)?
Now does anything change on visibility if the stars, tree, and crown are glittering in the sunlight?
The Haradrim carried a black serpent on scarlet. Since you asked. =>
And they are definitely glittering in the sunlight.
2 mm sounded a bit small to me - you are using diameter
quick google shows 3-4 mm is more typical in bright light,
maybe 5 mm for young eyes.
Dark adapted human pupils go to 9 mm
makes it a bit more plausible.
Battle banners ought to be designed to be easily distinguishable from distance, it is the rationale for man classic flag designs
Might also solve your Legolas Problem - if elf lenses are irises are bit wider than human - not anime level, just less white - AND if elves, after however many millennia, have conscious control over their pupil size and the pupil can widen to full width of the stop - ie the iris under the lens
then 15 mm for Legolas is not inconceivable
Tolkien was a signal officer during WWI. I would give him some credit.
Magic? Basically, but note all the recent articles about how to beat diffraction limits with special techniques, special materials, etc. The ones I've seen are in microscopy, but the very phrase "beat the diffraction limit" looks weird if you've been brought up on the fuddy-duddy orthodoxy. I don't think anyone says they're breaking real laws of physics (note: LOP not equal to laws on nature, the latter being the real ones and not what we think.) But it's odd that previous assumptions are being challenged. (And not just in banal ways, like having detectors so close to the emitters. Heck, we could put glowing little nuggets 1/10 lambda wide on a sensitive surface and record darkening, that's no big deal.)
See for example http://en.wikipedia.org/wiki/Diffraction-limited_system
Where were all you people in high school, when I thought I was so alone...
Seriously, awesome display of LOTR geek here. I salute you all.
Shouldn't the title be "The Limits of Rohirric Vision"?
Why should we assume that LOTR physics is the same as Real World Physics?
@11 JRRT's Middle-earth is our world. It is not another dimension or another planet. The stories take place in an imaginary time period, but it's the same world we know. So our physics is legit.
@10 You are right!
Forgive me, for I'm no sort of physicist at all. But I do know Tolkien. Tolkien was writing mythology, and certain portions of his books are in a heroic mode. The story arc to do with Rohan fits that mode. It would be quite right to propose that the Men (and Women) of Rohan â described as a young and vigorous race â had senses and strengths somewhat "brighter," more acute, than what we must settle for in these latter days. Their eyes worked the same way ours do, had the same structure, etc., but without doubt, a little better. Not as good as Elves, or even (probably) Numenoreans. If that helps.