# What's the minimum height of a mountain resolvable to the human eye from 800 miles on a flat Earth?

The Taniquetil / Oiolossë (Mount Everwhite) on the eastern shore of Aman is the tallest mountain on Arda. Far higher than the other Pelóri, it could be seen from the mountain Meneltarma on Númenor which, according to Karen Fonstad, is 800 miles (1290 km) away from the shore of Aman. Before the change of the world, Arda was flat but 800 miles is far enough that only the Taniquetil was large enough to be seen to the naked eye.

Númenoreans probably have better eyes than the men from Middle-earth (such as Rohirrim) but I suppose any human would see the Taniquetil from Meneltarma. Let's assume just any human would look to the west from Númenor and see the Taniquetil 800 miles away with the naked eye, how tall should the Taniquetil be to be visible to a human on Meneltarma? (Meneltarma by far isn't as tall as the Taniquetil or the other Pelóri; the King of Númenor was terrified when he saw the Taniquetil from close)

• What are the atmospheric conditions? I can barely see the end of the garden some days (20 metres) because of low-lying mist (we live on an ex-swamp), most days the limit's about 15 miles here (so my dad tells me, a qualified-ish pilot). Also, you should be aware, this is perilously close to being closed as about a third-party world, but I think you got away with it this time. Commented Nov 20, 2021 at 9:44
• @ARogueAnt. Isn't it clear I mean ideal conditions or is your comment a joke? Commented Nov 20, 2021 at 9:54
• Oh, no joke. Ideal conditions it is then. Commented Nov 20, 2021 at 10:05
• For the 3rd party world issue - something to be aware of. Commented Nov 20, 2021 at 10:08
• @ARogueAnt. Per Monica's post this should be on-topic. I'm a Numenorean btw so it's actually my world kind of/sort of. Commented Nov 20, 2021 at 10:33

Under real-world optics, it's impossible. The upper limit of visibility through an Earth-like atmosphere in best possible conditions is about 300 km. Wiki gives a technical explanation of how this is calculated, which I'll summarise:

As light passes through the air, it interacts with stuff and gets absorbed or scattered (think "deflected"). That "stuff" can be particulates (smoke, fog etc. in the air) but even in perfectly clean air, light will interact with the nitrogen and oxygen molecules of the air itself.

The result of that air interaction (Rayleigh scattering) is that about 1.32% of light is scattered for every kilometer between the object and its observer.

This scattering reduces contrast, and past about 300 km so much contrast is lost that it's impossible to distinguish between objects and their surroundings.

What if the mountain is so high that it sticks up above the atmosphere? Most of the Earth's atmosphere is in the bottom 10 km or so. Taking that as an approximate cutoff, a little bit of trigonometry indicates that if our line of sight to the top of the mountain passes through no more than 300km of atmosphere, then the top of the mountain needs to be about (1290/300)*10 = 43 km high, or about five times as high as Everest.

IRL, geology suggests that Everest is about as high as a mountain on Earth can get before erosion and its own mass pulls it down.

The above assumes that our observer is at sea level. What if we're standing on top of one Everest-like mountain, looking at another?

Air density at the peak of Everest is about 0.30 x that at sea level. (Based on atmospheric pressure; should probably be a little higher due to colder temperatures, but let's ignore that.) If we assume a constant density of 0.30 x surface for the line of sight between the two peaks, then we might be able to extend as far as 300/0.30 = 1000 km, but this still falls a little short of the 1290 km required.

If you're willing to ignore scattering (perhaps the same magic that reshaped the world also changed the way optics works?), then L. Dutch's answer based on resolution of 1 minute of arc is good.

• But if the mountain is tall enough, part of it will be in thinner to perhaps non-existent atmosphere. Thus the question becomes an exercise in trigonometry: how high does the mountain have to be so that the portion of the line that passes from observer to summit is less than 300 km? Commented Nov 21, 2021 at 3:56
• @jamesqf Good question - have edited to answer this, but it's roughly 40 km.
– G_B
Commented Nov 21, 2021 at 7:30
• I ain't sure if you are correct about the 300 km (186 mi) value when determining the way of the light through the atmosphere. 9/10 of the (Middle-)Earth's atmosphere is within 10 mi (16 km) of altitude, and 3/4 of it are as you say within 10 km, so we'd need to determine how much light would travel through the thick atmosphere, the part where light scattering becomes a given. E.g. what's the limit of visibility at 10 mi (16 km) altitude, where air pressure is 1.47 psi? Add to this that the Taniquetil was visible from another mountain the Meneltarma (which may be about 10,000 ft (3 km) tall). Commented Nov 21, 2021 at 9:01
• This answer on Quora states the Alps have been photographed from the Pyrenees at a distance of 443 km during sunrise. quora.com/… So the 300-km-figure doesn't apply in case the Sun shined from behind the Taniquetil. Commented Nov 21, 2021 at 13:29
• @Giovanni The reason for this is that when the sun is overhead, visibility is dominated by single scattering events-- if you trace a ray from your eye to the mountain, at each point along that ray light coming off the mountain scatters away and light from the sun scatters into it, both of which reduce contrast of the mountain with the background. However, if the sun is behind the object you're imaging, then contrast is reduced only through double scattering events-- for a ray going to the mountain from your eye, it can only be connected to a ray leaving the sun by an intermediate line. Commented Nov 21, 2021 at 15:38

The resolving power of an human eye is about 1', or 1/60 of a degree.

The minimum height of a mountain to appear 1' at 800 miles of distance would be given by inverting the formula $$tan \alpha = h/800$$, giving $$h=800 tan \alpha = 0.23$$. Since it's a flat Earth, we can neglect any curvature effect in hiding the peak.

A mountain 0.23 miles high would be at resolvable by a human eye at 800 miles distance.

• This seems a bit too high Commented Nov 20, 2021 at 10:14
• @studentstephan, OP explicitly says it's a flat Earth
– L.Dutch
Commented Nov 20, 2021 at 10:26
• On depictions the Taniquetil's summit is portrayed as being in space. Outer space is called "Ilmen" in Quenya and the hall of Manwë and Varda on the peak of Taniquetil is called Ilmarin, so 13.3 miles sounds reasonable, for at that altitude the daytime sky is black and you see the brightest stars, and you'd need a spacesuit. The snow referred to in Taniquetil's name probably isn't meant to be on the very summit, unless it could also mean frost. Commented Nov 20, 2021 at 10:40
• @studentstephan is right, the formula you gave makes sense but the answer is wrong-- I think you input $\alpha = 1^\circ$ rather than $\alpha = 1'$. The latter gives a height of .23 mi. In reality though, visibility of far off mountains is gonna be determined much more by atmospheric scattering of light than angular resolution. Commented Nov 20, 2021 at 14:18
• @Giovanni It might seem surprising, but on an airless planet (maybe a flat Moon?) 0.23 miles would be about right. 0.23 miles from 800 miles is about the same size as a standard 1/96" pixel seen from a meter away, if that makes it a little easier to believe.
– G_B
Commented Nov 23, 2021 at 23:19

In a sphere world, about 1 foot high per 1.3 mile, some sources suggest slightly over or under this. Check out this distance visible to the horizon guide.

You also have to take into consideration the time of day, correspondingly, how much light is on the retina will determine how far you can see.

• Arda was flat at that time, your diagram is for a spherical planet. Commented Nov 20, 2021 at 9:57
• I think you can still use this chart (as its based on the physics of human sight), except you leave out the last step of adding the height of the mountain to the height of your eye from the ground. Thus, you can see farther (than if world was sphere), provided there aren't other obstructions such as the weather Commented Nov 20, 2021 at 10:12
• Yes it's helpful nonetheless. I didn't downvote btw. Commented Nov 20, 2021 at 10:33
• thank you. I agree the diagram clearly cant be applied to a flat world, but I think it helps to learn about the laws of a round world in trying to figure this out. Commented Nov 20, 2021 at 14:09