Look out to sea at night from the beach at Portobello in Edinburgh, and you’ll often see lighthouses blinking in the blackness. Here’s a picture of those lights, taken in such a way as to convert time into space so that the different flashing sequences become apparent.
I pointed my camera out to sea, on a tripod, and slowly panned upwards over what was effectively a long exposure of about a minute. The movement of the camera smears out any light source into a near-vertical streak (the tripod wasn’t very level). The two thick streaks are ships at anchor in the Firth of Forth.
A flashing light leaves a broken streak; this reveals the lighthouses. At the left is Elie Ness on the Fife coast, flashing once every 6 seconds. Second from the right is the Isle of May in the Firth of Forth, flashing twice every 15 seconds. And at far right we have Fidra, off North Berwick, flashing 4 times every 30 seconds.
And the green streak? A passing dog with a luminous collar!
Lighthouses are lovely, romantic, things. But what’s extra lovely about the lighthouses in this image is that you shouldn’t be able to see two of them!
Both the Elie Ness and Isle of May lighthouses are, geometrically, beyond the horizon. If light travelled in straight lines, we wouldn’t be able to see either of them. As it is, whether we can see them or not depends upon the weather: if I’d taken the picture in cold weather, only the Fidra light would have been visible.
The reason that we can see them at all is that light travelling through the atmosphere is refracted by the air: its path is (usually) slightly curved downwards. This means that we can see slightly “over the hill”, a bit further than simple geometry would suggest. Things near the horizon appear higher in the sky than they really are.
It’s not a small effect. For example, when we see the Sun sitting with its lower edge just touching the horizon, the geometric reality is that all of it is actually below the horizon.
The refraction happens because the air gets thinner as you go upwards. Just as light passing through a prism is bent towards the thicker end of the prism, so light passing through the atmosphere is bent towards the thicker (denser) part of the air.
The amount of atmospheric refraction depends upon the weather. It depends upon the pressure and temperature, and the temperature gradient (how quickly the temperature drops as you go upwards). When it’s cold, the Elie Ness and Isle of May lights disappear. If it warms up, they pop back into view again. And when it’s really cold and the air is colder than the sea, the refraction can be reversed, and we see mirages along the coast. But that’s for another time.
How horizons work, and how we see things that are beyond the horizon
The horizon exists because the Earth’s surface is curved. In these diagrams the circular arc represents the surface of the Earth. If my eye is at the point O, I can see the Earth’s surface up to the point H (where my line of sight just grazes the surface) but no further. The point H is on my horizon. Its distance depends upon the height of O. It can be surprisingly close: if I stand at the water’s edge, my horizon is only 4.5 km away.
If the thing I’m looking at (a lighthouse L for example) is raised above the surface of the Earth, I’ll be able to see it when it is further away. In the diagram below, I’ll just be able to see the lighthouse. The rays from the lighthouse will just kiss the wave tops in the darkness on their way to my eye. The point H is on the lighthouse’s horizon as well as mine. The higher the lighthouse (or me) is, the further away it can be and I’ll still see it.
But if it’s too far away (below), I won’t be able to see it. There’ll be a region of the sea that is not in sight from either my eye or the lighthouse. This “no-man’s land” is the region between my horizon 
and the lighthouse’s horizon 
. For example, there’s about 6 km of no-man’s land between me on the promenade at Portobello and the Isle of May lighthouse.
All of these diagrams assume that rays of light are perfectly straight. But rays of light passing through the air aren’t perfectly straight: they have a downward curvature because of atmospheric refraction. This means that rays from an object that is geometrically beyond the horizon might actually reach my eye. That’s why I can sometimes see the Elie Ness and Isle of May lights. 
The curvature of the rays of light is hugely exaggerated in this diagram. Otherwise it would be imperceptible. The rays deviate from straightness by only a few metres over a journey of several tens of kilometres.
The curvature of the rays varies according to temperature, pressure etc. It’s a happy accident (for me) that both lighthouses are only marginally out of view over the horizon, so that they can disappear and reappear according to the whims of the weather.
How I took the picture
The picture isn’t actually a single long exposure. I thought that I might end up with a pale sky rather than a black one if I did that. Instead, I took a one-minute video, slowly panning the camera upwards. I then wrote a program in Python to produce a composite image derived from all of the video frames. For each pixel position in the image, the program inspected that pixel position in all of the video frames and chose the brightest value.
Calculating the distance of the horizon
We can use Pythagoras’ theorem to work out how far away the horizon is. In the diagram below, the circle represents the surface of the Earth, with centre at C and radius r. You, the observer, are at O, a height h (greatly exaggerated) above the surface. Your horizon is at H, a distance d away.
The triangle OHC is right-angled at H. Applying Pythagoras’ Theorem, we get
and so
Where h is very small compared to r, as it will be for the heights we’re dealing with, 
, so we can neglect the term 
and get, to a very good approximation (within centimetres)
Calculating the visibility of the lighthouses
There is a Wikipedia article with formulae for the curvature of near-horizontal light rays in the atmosphere.
I’m only going to give a summary of results here, principally because although I’ve got the spreadsheet that does the calculations, I can’t find the notebook where I worked out the geometry. Here are the details of the lighthouses. Heights are the height of the lamp in metres above mean sea level.
| Lighthouse | Distance (km) | Height above sea level (m) |
|---|---|---|
| Isle of May | 44 | 73 |
| Elie Ness | 31 | 15 |
| Fidra | 24 | 34 |
Consider the next table as an example, based on roughly typical weather conditions for this time of year (March). The figures assume that I’m standing on Portobello promenade. The Ordnance Survey map shows a 3 metre spot height marked on the prom, so that would make my eye about 4.5 metres above mean sea level.
The first three numerical columns of the table shows how high above the horizon each lighthouse would be in the absence of refraction, what the estimated refraction is, and thus how high above the horizon the light should appear. The heights are expressed as angles subtended at your eye. There’s a lot of uncertainty in the estimated refraction (because of uncertainty about the input values such as temperature gradient), which is why the middle two columns are given to only one significant figure.
| Lighthouse | Angle above horizon with no refraction (degrees) | Estimated refraction (degrees) | Estimated angle above horizon with refraction (degrees) | Water depth over direct line of sight (m) |
|---|---|---|---|---|
| Isle of May | -0.040 | 0.049 | 0.01 | 5.3 |
| Elie Ness | -0.051 | 0.035 | -0.02 | 6.8 |
| Fidra | 0.030 | 0.027 | 0.06 | -4.1 |
Thus we see that the Isle of May and Elie Ness have negative heights above the horizon without refraction, ie they’re geometrically below the horizon. In the conditions given, refraction is enough to raise Isle of May into visibility, but not Elie Ness – the angle with refraction is still negative. This accords with my experience: I’m more likely to be able to see Isle of May than Elie Ness. Fidra is above the horizon, refraction or no refraction.
Note that the angles above and below the horizon are tiny. For comparison, an object 1 mm across held at arm’s length subtends an angle at your eye of about 0.1 degrees. Most of the angles in the table are less than half that.
The rightmost column is there to help understand how the tide can affect things. Saying that the Isle of May and Elie Ness lights are beyond the horizon is saying that there’s water between my eye and their lamps. Imagine that the light from the lamps could travel through the water completely unimpeded and in a perfect straight line to my eye. This column shows how far under the water the light rays would be at their deepest. As you can see, they are single-digit numbers of metres. Now the tidal range in the Firth of Forth is about 4 metres. What this shows us that that the state of the tide could easily make the difference between seeing or not seeing a given lighthouse. It also brings home how slight a curvature of the rays is produced by refraction: in Isle of May’s case, there’s just enough curvature to get the rays over a 5-metre bump in a 44-kilometre journey.
Image credits
Isle of May: Jeremy Atherton; Elie Ness: Francis Webb. Both under CCA license.