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.
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:
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!
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