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One person took experiment from Montenegro and with distance 201 km he saw clearly montains that are 1521 meters high. He was on 189 meters above the sea level. Official calculator says you cannot see less than 1811 meters from that distance. My question is: Why he saw the mountain? Is there any reference that can help me understand that. I'm a mathematician and I calculated that the radius of the Earth should be 7260 km for that to be possible. I need to understand which physical laws are related to this problem and please give me a reference.

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  • $\begingroup$ John, I need reference on that problem to understand it clearly. Is there any textbook or some research paper that explain it? $\endgroup$ – Alem Nov 16 '17 at 4:24
  • $\begingroup$ @allen your calculator is how far away the horizon is from a given height - the other mountain is high and so you need the sum of the horizon distances for each peak - as in John's answer $\endgroup$ – Martin Beckett Nov 16 '17 at 5:36
  • $\begingroup$ Martin...calculator calculates the height of an object that can be seen...here is the link dizzib.github.io/earth/curve-calc/… $\endgroup$ – Alem Nov 16 '17 at 6:36
  • $\begingroup$ Martin...So calculations are ok, they are not how far away is horizon. I made a formula for what I need and numbers and identical to those on that link. Only in that link picture is a little wrong. We dont take air distance, but curve distance...It took me about three hours to build it and program it... $\endgroup$ – Alem Nov 16 '17 at 6:40
  • $\begingroup$ The Wikipedia article on the horizon has formulas for the distance to the horizon which take atmospheric refraction into account. However, you need to have figures for the optical density of the atmosphere at various heights, and those figures depend on the atmospheric pressure, temperature and humidity. $\endgroup$ – PM 2Ring Nov 16 '17 at 6:40
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These are selective quotes from the link:

The atmosphere of earth consists of different layers of air. Out of these some air layers are hot which have low densities, while the others are cold which have high densities. The hot layers of air behave as optically rarer medium for light rays, whereas the cold layers of air behave as optically denser medium for light rays. So, when an object emits light rays in the atmosphere, these light rays pass through the atmosphere having different air layers of different densities and get refracted by atmosphere. So, the refraction of light caused by the atmosphere of earth is called atmospheric refraction.

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Looming is also an optical illusion which occurs usually in very cold regions. In looming, a distant object such as a ship moving in polar areas appears to be hanging in midair due to atmospheric refraction and the total internal reflection of light rays.

LOOMING

loomrefr

In Polar Regions the layers of air near the surface of earth are very cold and hence behave as optically denser medium. Whereas, the upper layers of air are comparatively warm and hence behave as optically rarer medium. Now, a ray of light coming from point S of ship goes from denser to rarer medium along the path SB and bends away from the normal, at every layer due to atmospheric refraction. But, at a particular layer, when the angle of incidence becomes greater than the critical angle, the total internal reflection occurs, and the totally reflected ray travels along the path BE and reaches the observer. As we have already discussed that we can see the light only in straight line path, so the reflected ray BE appears to be coming from the point S’ to the observer. Due to this, the observer sees a virtual and erect image of the ship at position S’, which is much above the actual position of the ship in the sea.

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  • $\begingroup$ "Looming is also an optical illusion which occurs usually in very cold regions." So looming is for very cold regions? Montenegro and Italy are not cold regions, as I know. We have Mediterenian climate there. Please tell me if I'm wrong. $\endgroup$ – Alem Nov 28 '17 at 4:24
  • $\begingroup$ The temperature differences are smaller and spread to large heights, so we do not see floating ships in greece either. But the higher one goes vertically the colder it is, and that even at the equator. So we see things higher than they are , but with dimensions for refraction planes from colder air at kilometers height , not hundred meters. $\endgroup$ – anna v Nov 28 '17 at 4:31
  • $\begingroup$ the same happens with sunsets when the sun is still visible due to atmospheric refraction. en.wikipedia.org/wiki/Atmospheric_refraction I just discovered looming :) and thought the floating shipwas a fun example . $\endgroup$ – anna v Nov 28 '17 at 4:35
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Per @John Forkosh, the refraction phenomena is called "looming":

https://en.wikipedia.org/wiki/Looming_and_similar_refraction_phenomena

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