# How far can you see?

As a result of my post on the flat-Earth believers, I was struck by their claim that when you look into the distance, you do not see the Earth’s curvature. This raised in my mind the question of, if you look out over a flat expanse, say a desert or an ocean or from a plane, how far can you see? It may seem as if we can see really far, especially since we can see distant stars, but many factors introduce a great deal of variability.

In order to ‘see’ something at a distance, what we are seeing is light from the sun that has been reflected from that object and has come in a straight line right into our eyes. So the first factor at play is the contrast between the object and its surroundings. We judge that a mountain range is distant because it looks muted compared to objects that are nearer. This is why it is so hard to gauge distances in whiteout conditions in a snowstorm or dust storm. For example, see this video of a dog being tossed out of an airplane. (Don’t worry, nothing bad happens! The reason that the snow looks like distant clouds is the lack of contrast that makes judging of distance hard.)

Another major factor is the scattering of light. The atmosphere contains oxygen, nitrogen, and water vapor molecules plus dust and other particulates. Some of the light coming to us from the distant object gets scattered by these things into other directions and no longer reaches our eyes, thus reducing the visibility of the object. But in addition, sunlight also falls on all these intervening particles between the object and us and also gets scattered, and some of that scattered light will come in the same direction and enter our eye. So this scattering (called Rayleigh scattering) affects visibility in two ways: it reduces the amount of light coming to us from the distant object, and it introduces extraneous light into that same path. All these factors reduce the visibility of the distant object and sets a limit on how far we can see.

Meteorologists have something called the extinction coefficient that enables you to calculate how far you can see. If the atmosphere is completely free of particulates, leaving only the molecules of air, then you can see a distance of about 300 km. In actual practice, if you can see up to 50 km, the atmosphere is said to be “exceptionally clear”.

(Ref: “On a Clear Day You Can’t See Forever” in the book Clouds in a Glass of Beer (1987) by atmospheric physicist Craig F. Bohren.)

1. janicot says

This comment has no relationship to air quality. But I recommend an Isaac Asimov essay you’ve probably seen --‘The Relativity of Wrong’. A sphere about the size of the earth only differs from flat by about 0.000126 per mile so it’s not obvious to the casual observer that the earth isn’t flat.

I am always amazed how many people accept their casual observations as absolute truths and defend them in spite of what just about every else tells and shows them.

2. says

A few days at sea is all you really need. Watching another ship go from hull down—all you can see is the top of her masts—to fully visible as you grow closer together dramatically makes the point.

I haven’t checked the math, but we were told in the ’70s that a six-foot tall sailor standing in a lifeboat could see a horizon that was 11 miles away.

@Janicot No. 1 Isaac was a treasure.

I think I learned more from reading his essays in F&SF than I ever did in any classroom.

3. Rob Grigjanis says

I used to live on a hill overlooking Lake Ontario. On a clear day from my apartment, you could see the land on the far side of the lake (roughly 40 miles, IIRC). On a clear night, you could see the lights from several cities. I also spent quite a bit of time at the shore. From there, on a clear day, you saw the sharp division between sky and lake, with no land whatsoever. On a clear night, no lights whatsoever.

Nitpick time:

So this scattering (called Rayleigh scattering)…

It’s only Rayleigh scattering if the scattering particles are much smaller than the wavelengths (λ) of incident light. This is the case with N₂ and O₂ molecules hit by visible light. That’s basically the blue light you see, since the scattering goes like (1/λ)^4. The whiter light you see on humid or misty days is mostly scattered from water droplets, which are much larger than visible wavelengths, and the scattering is almost wavelength-independent.

4. DonDueed says

The Apollo astronauts had difficulty judging distances, since with virtually no atmosphere there was no scattering to dim distant objects. On some of the later missions, they sometimes drove the LRV a long way toward some feature of interest before they realized it was very far away indeed.

5. Rob Grigjanis says

hyphenman @2: Pythagoras tells us that, at a height h above sea level, the distance to the horizon is about (for h much less than Earth’s radius R)

d ≅ (2hR)^(1/2)

If I did my sums right, that says a 2 metre height means a horizon 5 km away (about 3 miles).

6. DonDueed says

I can see for miles and miles and miles and miles and miles. Oh yeah.
-- Pete Townshend

7. Mano Singham says

hyphenman @#2,

It is less than that. As Rob says @#5, the distance to the horizon for a six-foot person is 3.0 miles. This handy calculator enables you to calculate horizon distances for height above sea level.

8. Mano Singham says

While thinking about this, it struck me that if the Earth were flat, why would sailors in the old days post a lookout in the crow’s nest way up high on the mast in order to see farther?

DonDueed @4: I had read about this astronaut illusion before as well. Below is a video link from an Apollo 16 mission that I grabbed from Phil Plait’s Bad Astronomy website. It shows how a large rock in the distance appears to be closer than what we initially think due to lack of atmospheric distortion, presumably. His website is full of fun facts about the moon and some of the physics that appears counterintuitive to us here on Earth.

10. DonDueed says

While thinking about this, it struck me that if the Earth were flat, why would sailors in the old days post a lookout in the crow’s nest way up high on the mast in order to see farther?

Congratulations, Mano! You’ve done it, you’ve found the ultimate trump card for the flat earthers! Now no one will ever believe in silly things again. Unless… maybe they were sent up the mast so they wouldn’t be distracted by all the grog-swilling and shanty-singing down below.

Seriously, though, the real problem with this argument is that the average person these days has never been far out to sea. It doesn’t hold on land, where an elevated position is useful even on a flat earth (because of local topography).

11. grasshopper says

Is it just planet Earth which is flat?
@5 Using the imperial system with your equation lets us simplify a little further to d = (1.5h)^1/2. So a height of 6 feet gives the horizon at 3 miles. Which is what you said 🙂 The metric system confuses me. And NASA.

12. grasshopper says

Fine print --> These simple formulae for distance to the horizon don’t allow for the effects of atmospheric refraction and are provided for educational purposes only.

13. mordred says

grasshopper@12:

The failure of the simple formula is depressingly often used by the flat earthers as an argument: “Standing here at x I can see y n miles away which, according to science should not be possible so the earth is flat.”

Refraction of course does not exist, just like gravity and the fact that you still can’t see z which is n+m miles away is completely ignored.

14. blf says

Heh. For some reason I’d gotten it into my head that from a typical ship’s crows nest one could see about 50(km?). That is wrong, using the ideal equation and a crows nest of 25m (roughly that of the Titanic), it’s a bit less than 18km. (Actual height above sealevel will vary due to, e.g., the current waterline on the ship and the height of the lookout.) I’d been carrying around — and possibly spewing — that bullshite 50 “fact” for an embarrassing number of yonks now. You would actually need to be around 200m high to “see” 50km. Thanks for encouraging me to check my “facts” !

15. Quirky says

Mano, As the New Scientist points out the earth is actually shaped like a potato, if one removed the water..
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https://www.newscientist.com/article/dn20335-earth-is-shaped-like-a-lumpy-potato/
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The water is said to create the shape of an oblate spheroid.
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So why is it that the moon, other moons, and other planets all appear round yet we are told that there is no water on them?
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How would they, in the same time frame, form into nearly perfect spheroids without the presence of any significant amount of water?
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And how would a potato shaped planet with water continually sloshing, eroding and potentially reshaping itself remain shaped like a potato?

16. Rob Grigjanis says

Quirky @16: You’re misreading the article. The water doesn’t “create the shape of an oblate spheroid”. The shape of the Earth very closely approximates an oblate spheroid, with or without water. As do all planets, because they rotate, and have (or had) fluid interiors.

So, Earth has an oblateness (or flattening) of about 0.003
The Moon has an oblateness of about 0.0012
Mars has an oblateness of about 0.006
Venus has an oblateness of nearly 0, because it’s rotation is so slow (1 Venus day is about 117 Earth days).

The article is talking about the geoid;

the shape the Earth’s surface would be if it was entirely covered by water and influenced by gravity alone.

That still closely approximates an oblate spheroid, but not because of the water.

17. Owlmirror says

On our round Earth, north and south are directions that indicate the axis of spin of the earth, and east and west are the directions on the round earth that the earth spins to and from. The sun appears to rise above the horizon, and set below the horizon, as the turning sphere of the earth rotates.

On the flat earth, north is still the direction of spin of the earth (or of the sun, moon, & sky, if they posit a motionless earth), and east and west are directions that the sun or sky moves — but the sun should never appear to set below the horizon or rise above it. The sun should make a circle in the sky (or an ellipse, depending on the angle of view) that goes from east to west, and then from west back around to the east. The only way it could possibly get dark in one part of the earth is if there were a magic lampshade on the sun, or maybe an “off” state of the sun. But the darkened sun should still never set and never rise.

So every sunrise and sunset falsifies the flat earth.

There should also never be an eclipse of the moon, since the sun would never be behind the earth to cast a shadow on the moon.

If the sun did in fact set or rise over the disc of the flat earth, it would necessarily do so in the south, since “south” is all of the outermost rim of the flat earth disc. It might do so in the southeast part of the disc for some, but that would look like the southwest part of the disc for those on the opposite side of the disc, and for those directly under the sun’s path, it would always be straight south.

Terry Pratchett got the names of the Discworld directions right, with there being a hub and rim, and a turnwise and widdershins (anti-turnwise). He may well have spent too much time pondering how a flat earth would actually work when he was younger.

18. Quirky says

@ Rob Grigjanis #17,
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Rob, I was reading that wrong. The title and the picture are misleading.. Thanks for the explanation.

19. Matt G says

I’m at the point in “20,000 Leagues Under the Sea” where they visit the South Pole. Apparently in Verne’s day they thought what we today call Antarctica was far smaller than it is.