I have read the wiki on this, but a special scenario popped into my mind.
The Mauna Kea Observatories were picked for their location in regards to proximity to the equator, low humiditiy, good weather, and elevation (9300 feet above sea level)
On the ground, I can understand our inability to see beyond the horizon, especially on the beach. the nearby water causes high humidity day or night, obscuring your view regardless of telescopes or any other viewing aid.
But above the clouds, and nearly 2 miles up, the large majority of these things no longer pose an issue. secondly, there is nothing to get in the way of continuously viewing the sun. The nearest peak the the dormant volcano of Haleakalu at 10,000ft is 50 miles away and only 700 feet above the point of our view, and only obstructing a small view to the northwest. All remaining peaks are either south or not as tall as Haleakalu, so they will not interefere with the viewing.
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At 3000 miles away, the nearest mountain that could possibility interfere with this is Mount McKinley aka Denali, at 20,322 feet. However, this will never come into play as the "shadow" cast by Haleakalu would dwarf even mount everest, being in excess of 42,000 feet. On the other side of the world on the equator is the Democratic Republic of the Congo, 11,000 miles away, and the furthest point the sun would drift on average, the shadow of Haleakalu would only become 154,000 feet tall.... less than 30 miles. So, 2970 miles above the "shadow" cast by Haleakalu, would be our sun.
The human eyes, in absolute darkness with no light pollution (won't touch that.... because.... yeah) a human can see the glimmer of a candle from 30 miles away.
(source)
http://www.livescience.com/33895-human-eye.html | You can also specifically search for the data regarding this experiment, I had it but failed at copy pasta.
Using an 8" telescope with a with x50 magnification, we now can see a candle from 400x further (12,000 miles). But wait! that also means the candle has become 400x dimmer. Good thing we aren't looking for a candle halfway across the world.
Ok, I couldn't be bothered to do the numbers myself so I just accepted the internet. Regardless, the point is more or less there.
quick numbers
200w incandescent lightbulb at 1 ft = 2.2 lumens per square inch
Sun at 3000 or 93 million miles away = 155,000,000 lumens per square inch in direct sunlight at the earth's surface
Looking for something that is in the neighborhood of 70,000,000 times brighter than a 200w lightbulb that is one foot away from you.
Scale it in a bit further maybe? the object that we are looking for is going to be 400x dimmer through that telescope, or 175,000 brighter when viewed through the telescope than the 200w incandescent lightbulb from 1 foot away.
Feel free to apply an absurdly high refraction index to that by the way.
Using the above methods, the sun would be easy to track in the sky in a FE theory.
Of course, we can eliminate all that by seeing it with an infrared telescope. Clouds don't really pose an issue to those. And even if they did, you would only need to not have a major storm between your vantage point and the coast of alaska (you are already above all cumulus, stratocumulos, and stratus clouds) at that point, you are above the 40,000 feet mark that most cumulonimbus clouds adhere to. Not to mention the further north you go, the lower the max cloud ceiling apart from Noctilucent clouds.
In FET, the point of the equator in the DCR that is due north of Hawaii is 11,000 miles away. At that point, the sun would be 2,970 miles above that point. Scaling that back to Haleakalu, the sun would be visible 13.5 miles above it's peak at it's point furthest away, or 12.7 degrees above the peak of Haleakalu
Bendy light theorists --- need a number on how much light diverges from its original course. why? If light bends in any direction, the sun will still be visible. If the light bends downards, the light that would have passed over my head on the top of Mauna Kea would become at eye level. If it goes bends up, the light that would have hit the side of the mountain would reach my eye. Ditto for bending left or right.
Spotlighters - The sun would take more of an oval shape the further it went away, especially over the extreme distances we get to play with in this.
May refine as more finite data comes in.
EDIT 1: added info regarding where the sun would appear in the sky relative to Haleakalu and the spotlighters, also, due to not caring about the diagonal, add 11% to any distances shown.
EDIT:2 added approximate degree of angling.