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:
Éomer 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.