Conquering the Horizon

And now for something completely different. The horizon; how far away is it? How different would it look on another planet? We're used to horizons on Earth but if we're writing a story set on an asteroid or on a small moon or planet the horizon will be closer because the planet's surface falls away more quickly.

Distance to the Horizon

When I talk about the distance to the horizon, what I mean is if you're somewhere flat, what's the furthest you can see (including with the aid of binoculars or a telescope etc)? On Earth a good example of this would be how far away the point where the sea meets the sky is, when you're standing on a jetty. Once we have trees and buildings in the way it can get a bit more complicated. Luckily, the sort of extraterrestrial locales where the horizon is going to be most different to Earth's are least likely to have a large abundance of trees. Convenient.

Working out how far away the horizon is takes a little bit of trigonometry. I've drawn a sketch below of all the relevant distances and whatnot.

R is the radius of the planet/moon, h is the height of your person (well, of their eyes) or if they're in a building, it can be how high up they are, d is the straight line distance to the horizon, s is the distance along the surface of the planet/moon and θ is an angle that will be useful in some calculations.

The important thing to that you need to know about your non-Earth planet is how big it is or, more specifically, it's radius which is labelled R in the image above. It's a reasonable assumption that you'll have at least a rough idea of how tall your characters are. If you don't, it doesn't really matter, you can just guess something close since there's not going to be much difference between a tall person's horizon and a short person's (the differences really come into play in non-horizon situations, such as crowds). For the purposes of my calculations later on, I'm going to set h = 1.7 meters. Because I can.

Now, that's a right angle between the line I've labelled d and the left hand radius line. Since we know R and h we can now use Pythagoras's Theorem to work out the distance d. Don't worry if you don't remember any maths, I'm just going to tell you the answer.

Chances are, your planet is significantly larger than your person, so you will usually be able to ignore the h² but not always (if on a small asteroid, for example). If in doubt, leave it in. It won't make your answer worse.

This is not an unhelpful result. However, I can't help but feel that when people stand in a tall tower and say things like "They're ten miles away but gaining ground!" they don't mean ten miles from their eyes, but ten miles from the bottom of the tower (if nothing else, they'd probably be estimating based on land marks and those are definitely relative to the ground distance).

So how do we find the distance along the ground, s? Unsurprisingly, with more maths. We use the fact that s = Rθ and then work out θ so we can substitute for it and not have to actually calculate it directly. Using the same triangle as before, we can find s in two different ways:

If you're wondering, cos and tan are trigonometric functions all scientific calculators (including the ones hiding in all your computers) can do. The ^-1 indicates that's it's the inverse of the function which you can usually access by pressing shift/2nd or something like that, depending on the calculator.

For planets/moons which are much, much larger than a person, s and d will be very close; it's the tiny, tricksy moons or asteroids are where it'll really make a difference.

So how far?

Some examples now for a person 170 cm tall and for a ten storey building (30 metres high):

•  On Earth, ignoring atmospheric effects which actually extend the apparent horizon thanks to bending light, the horizon is 4.7 km away. From a ten storey building it's 19.7 km.
• On the moon or Io, which are similar in size, a standing horizon is just under 2.5 km and the ten storey building horizon is about 10 km.
• Ganymede and Titan (moons of Jupiter and Saturn, respectively) are a bit bigger than those two, with standing and ten storey building horizons of 3 km and 12.5 km.
• Mars is about one and a half times the size of Ganymede and a bit more than half the size of  Earth. It has horizons 3.3 km and 14 km for standing and building respectively.
• Deimos, Mars's moon, is rounder than Mars's other moon, Phobos, but still not that round. If you stand on a fortuitously round bit, the horizon will be 140 meters away (that's right, metres not kilometres—Deimos is actually an oblong with dimensions only 15⨉12.2⨉10.4 km across. Its average radius is 6.2 km). If you somehow managed to put a 10 storey building on it... well you'd see about 600 meters away.
• Ceres is a large, round asteroid (or dwarf planet) in the asteroid belt. It was one of the bodies that, when Pluto's planethood was called into question, was up for being classified as a planet if Pluto got to stay. (If you're wondering, it is considerably larger than Deimos, with a radius of 471 km.) A person would see the horizon 1.3 km away (pretty close if you think about how far a kilometre looks when you're driving, for example) and a ten storey building would see the horizon drop off 5.3 km away.
• And speaking of Pluto, Pluto's largest moon (it also has two tiny ones), Charon, is a bit bigger than Ceres and has a standing horizon of 1.4 km and a building horizon of 6 km.

Seeing things beyond the horizon

The horizons I've talked about above are the limiting distances for seeing things on (or close to) the ground. Things are a little bit different if we want to work out from how far away we can start to see the top of a tall building, for example.

The diagram below shows that although my little stick figure can only see the ground up to d distance away, s/he can see the top of a building which is d+b distance away. Huzzah!

My drawing skillz know no bounds. Close up of previous diagram with a building of height H, that the stick figure can just start to see the top of, added in. The building is b distance further away than the ground point being cut off by the curvature of the planet.

So what if we want to work out how from how far away we start to see the top of a building/monument/spaceport/volcano? Easy. All we have to do is work out the distances to the horizon for both the person and the building/monument/spaceport/volcano and add them together. The distance, D, from which the person starts to see the top of the building/whatever is then approximately given by:

You may have noticed that if we want to work out the distance from which someone can start to see a ten storey building, all we have to do is add the horizons I worked out above together. How convenient! Just quickly, the distances at which the building will start looming out of the ground are:

• Earth: 25.5 km (assuming you can find an isolated ten storey building in the middle of a 25 km circle of flat ground...)
• Moon/Io: 12.5 km
• Ganymede/Titan: 15.5 km
• Mars: 17 km
• Deimos: 740 meters if you can manage an ideal situation (I strongly suspect that you can't, but these calculations do give you a good idea of how small Deimos is... the edge of Deimos would look so close!)
• Ceres: 6.6 km
• Charon: 7.4 km

There you have it. Note that with Earth being the largest rocky body in the solar system, it by far has the widest plains (or planes, if you prefer to be mathematical about it) around. The larger-but-not-as-bit-as-Earth planets and moons (Mars, the moon, Io, Ganymede and Titan) give us similar results for their horizons, which suggests to me that someone travelling between them wouldn't notice much of a difference. Our small-but-still-roundish bodies (Ceres, Charon) both give results about half that of the larger bodies (and hence a quarter that of Earth). I only included Deimos for fun, but it is important to remember that standing (or floating, as the case may be) on or near the surface of this moon (or any similarly sized asteroid, of course) would, visually, be a very different experience to any other body I discussed.