Sunday 7 February 2016

67) “The distance across the Irish Sea from the Isle of Man’s Douglas Harbor to Great Orm’s Head in North Wales is 60 miles. If the Earth was a globe then the surface of the water between them would form a 60 mile arc, the center towering 1944 feet higher than the coastlines at either end. It is well-known and easily verifiable, however, that on a clear day, from a modest altitude of 100 feet, the Great Orm’s Head is visible from Douglas Harbor. This would be completely impossible on a globe of 25,000 miles. Assuming the 100 foot altitude causes the horizon to appear approximately 13 miles off, the 47 miles remaining means the Welsh coastline should still fall an impossible 1472 feet below the line of sight!”

Light bends.

Let's reverse our formula for horizon distance. The horizon you see is the lowest point. So, how far away can we see the Great Orme's Head? Great orme has an elevation of 207 m. The hill has an elevation of 30.48 m. As the object far away is higher up, you can see it better. Intuitively, I surmise that we sum the heights. Thus, we have h=237.48mwhich leads to a distance of 55 km. Clearly, we aren't able to see it directly - something is interfering.

Still, Wikipedia notes that it is visible, although they clearly say that it is the summit, not the head:
Due to its location, the Isle of Man and the Lake District can be seen from the Orme's summit on clear days.
This might be the right place to talk about what is interfering. Also, we'll edit our distance formula slightly, taking into account refraction [Good description here].

Mirages are optical phenomenon due to the different densities of air causing different bendings of light [
Wikipedia/Mirage]. A well known mirage effect is that the sun appears flattened when setting. They are the better known one, but this sort of thing happens all the time. Looming is fun too [Wikipedia/Looming]. I first saw that in the plane to London, when I saw a lot of boats forming a sky armada. Rather than believing my eyes and believing the Empire was about to strike, I thought about refraction for a bit and continued to read my book. The green flash is good evidence, too.

I won't redo the calculation, but 
it has been done. The estimate for the horizon distance becomes 3.86h, due to light bending. There's a lot of assumptions going into that, largely because refraction depends on atmospheric results - basically, the weather.

On a clear day doesn't mean that the weather is nice, but is actually a reference to the optimal conditions for far-seeing. We can see that described nicely [

When conditions are unusual, this approximation fails. Refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water. In extreme cases, usually in springtime, when warm air overlies cold water, refraction can allow light to follow the Earth's surface for hundreds of kilometres. Opposite conditions occur, for example, in deserts, where the surface is very hot, so hot, low-density air is below cooler air.
Well, that makes sense.

I don't know the particular atmospheric conditions that are present on 'a clear day' , i.e. a day on which you can see it, but I propose that refraction is the answer.

Denial of refraction is a hard one. Above, I've already pointed to a few points of evidence. Other things that depend on it are telescopes, glasses, microscopes and cameras. Probably a lot more, too. The underlying theory of light is so very, very settled by now that I can't imagine someone denying it. Of course, even if I can't imagine it, I know someone will, anyway. The underlying theory is the wave theory of light, from which Snell's law can be derived. And below that lie Maxwell's equations. Which are consistent with quantum mechanics, and on which a lot of work is based.

Incomprehension of the model

Great Orme's elevation is 207m

Only looking at google maps, one waterfront road in Douglas Harbour has an elevation of 70m. I'm sure if you looked at a topographical map, you could find an even higher point of observation, but let's stick with this one.
The drop due to curvature over 60 miles (97km) is roughly 740m:

Refraction coefficients of more than 0.13 are not only entirely possible (even up to+16), but they are also very likely especially for sunny days over open water:

Even just a value of 0.18 will reduce your visible drop by 130m for this given distance. Let's add these 130m to one of the two locations. You have one with 207m, its sight line meets the horizon at 51.4km:

The other one with (refraction-induced virtual) 70+130 = 200m meets the horizon at 50.5km. That gives us a sum of 101.9km. If the distance between these two locations is 97km that means that THEY CAN SEE EACH OTHER under clear conditions, it doesn't even have to be a hot summer day.

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