Seeing Far Away Mountains

When you're on a hill on one island and you look out to sea, sometimes you can see mountains on other islands. On days when humidity is low, it is possible to see mountains that are rather far away. But how far can you see?
Let's look at the maths, assuming a spherical earth and ignoring the bending of light due to refraction or indeed general relativity.

Suppose you're at a height of \(h_a\) above sea level and you look out to sea.
We'll let \(d_a\) be distance from the observer to the horizon, where the sky appears to meet the sea. We'll measure this distance along the surface of the earth at sea-level. And we'll let \(a\) be the angle at the centre of the earth between the observer and the horizon.
We know that the arc-length is the radius multiplied by the angle in radians, so: \[d_a = r a\] where r is the radius of the earth.
We'll now let \(d_b\) be the distance from the horizon point to the object being observed (along the surface of the earth at sea level), so: \[d_b = r b\] And let d be the full distance from the observer to the object being observed (along the surface of the earth at sea level), so: \[d = d_a + d_b = r ( a + b )\] We also know that: \[ cos(a) = \frac{r}{r + h_a}\] \[ cos(b) = \frac{r}{r + h_b}\] Where \(h_b\) is the minimum height of the object in order for it to be visible to the observer.
Rearranging those equations, we find the distance to the horizon is: \[d_a = r \hspace{2mm} cos^{-1} \left( \frac{r}{r + h_a} \right) \] and \[h_b = r \left( \frac{1}{cos(b)} - 1 \right)\] where \[ b = \frac{d}{r} - cos^{-1} \left( \frac{r}{r+h_a} \right)\] So, given our current height \(h_a\) and the distance to the object \(d\), we can work out \(h_b\) the minimum height the object needs to be in order to be visible.
Let's try that out with some numbers.

Inputs
Observer height above sea level		meters
Distance to object		km
Results
Distance to horizon		km
Height required to be visible		meters

If you change one of the inputs above and hit enter, then the results will be updated.


If you're on a hill south of Dublin, is it possible to see the mountains of Wales? Well mostly the answer is no, because the humidity in the air obscures the view. Anyway, let's find out what our calculator tells us about the visibility on an exceptionally clear-skyed day.
For example, Killiney Hill is 153 meters above sea level and Snowdon in Wales is about 138 km away. Using the calculator above, we see that Snowdon would need to be at least 692 meters tall to be visible.
In fact it is 1,085 meters high and so we should be able to see the top few hundred meters of the mountain. That is indeed the case, as the photo below shows:

That's the view across the Irish sea of Snowdon (Yr Wyddfa) in Wales, from Killiney Hill in south Dublin, Ireland.
Holyhead (in Wales) is closer to Killiney (99km) than Snowdon is to Killiney (138km), but you won't be able to see Holyhead, because there aren't any hills in Holyhead high enough to be visible. The minimun height required would be 236 meters.
If you travel south from Killiney to County Wicklow and go up one of the higher mountains you might expect to be able to see more of the hills and mountains in Wales. But once again, on most days there will be too much humidity and indeed too many clouds to be able to see Wales, but on a good day, with an really clear sky, you'll be able to see lots.