# Is apparent horizon curvature lesser due to refraction of light in the atmosphere?

I have encountered this claim while searching for sources answering " Can we see the curvature of earth from the top of world's tallest building? ".

Wikipedia article on horizon claims (with no references), that compared to a geometrical model, the the apparent curvature of Earth is less than expected due to refraction of light in the atmosphere.

Thinking about the problem, I fail to see why this could be so. It is easy to see the refraction affects a distance of the apparent horizont, but I fail to see how it could affect its curvature, as the refraction seems to have the same effect in all directions, because your distance to horizon is the same in all directions.

Is that claim just bogus and there is no such effect, or is the effect real?

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## migrated from skeptics.stackexchange.comJul 21 '11 at 8:50

This question came from our site for scientific skepticism.

You should ask this on Physics. Should I migrate this for you? – Sklivvz Jul 21 '11 at 8:29
Yes, please. Sorry for posting it here, just a habit. – Suma Jul 21 '11 at 8:38
See the comments by @Philip Gibbs in this question: physics.stackexchange.com/questions/11021/… It's not exactly the same situation but the relevant physics is the same. – qftme Jul 21 '11 at 11:01
No, I am sorry, it is not (this is discussed in the question liked in more detail). It is almost flat, but not absolutely. A simple counterexample are NASA photos. Or do you want to claim there is some magical boundary at 199 km, above which the horizony is curved, but below which it is flat? – Suma Jul 21 '11 at 13:32

Yes -- because refraction influences the apparent distance to the horizon, it also has an effect on the curvature.

To visualize this, it might help to think in extreme cases, for example in the case where due to refraction the horizon is at an apparent distance of only 1 meter. In this case, the curvature of the horizon would be extreme (it would be a circle of radius 1 meter around you).

In reality the curvature effect is much smaller of course, and I doubt that it is visible.

(As an aside: one result of atmospheric refraction that is observable is a phenomena called 'the green flash'. Because refraction is colour-dependent, the red and yellow part of the sun could already have set, while a small part of the 'green sun' is still above the horizon. This can be observed by the naked eye, preferably when the horizon is sharp.)

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I understand your reply as: refraction makes Earth to appear 15 % larger (cf. mintaka.sdsu.edu/GF/explain/atmos_refr/dip.html), and the apparent curvature is affected accordingly. Can you confirm this is what you meant? That would make sense to me. – Suma Jul 21 '11 at 14:19
That's a somewhat more concise way of putting it, yes ;) – Tim Jul 21 '11 at 14:24
(download) Fig 1 Atmospheric Temperature Profiles from Sunset Images, Dan Bruton, Dissertation 2600K PostScript File – Helder Velez Jul 21 '11 at 14:56

Sometimes.

When you are standing on the ground, you are in air at roughly one atmosphere of pressure; so are the points on the earth that you see on the horizon; and so is the whole path of the light from the horizon to you. In this case there is no significant boundary where the refractive index (RI) changes, nor is there an appreciable deviation in RI over the path of the light you are seeing. Therefore there will be no bending of the light rays, and no effect on the perceived curvature of the horizon.1

When you are at significant altitude, you are in air at significantly lower pressure than the air at the surface of the earth. Therefore the light from the horizon moves through a region with a pronounced gradient in RI. This causes the light rays to curve, making the radius of the horizon appear larger (and thus the curvature smaller) than it otherwise would. It is this scenario that the linked Wikipedia page is talking about when it states:

At an altitude of 10 km ... the apparent curvature is less than that due to refraction of light in the atmosphere

1: There will of course be small deviations in air pressure (and therefore the RI) along the path of the ray, but these would be random and time-varying, so they will manifest as a slight blurring or "wobble" of the image of a point seen on the horizon, not a shift in it's observed location.

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the Earth curves at the rate of 157mrad per km travelled. The refractive index of dry air at sea level is 1.00029, but at a height of 1km, the air pressure is 12% less and (neglecting temperature density and humidity), the refractive index would be 1 + 0.00029 * 88%. The difference, 0.000035 means that light at 1km altitude travels 35mm further for every km, curving the path of light by 35mrad per km travelled, about 20% of the Earth's curvature.. allowing us to see past the theoretical horizon a bit .. an extra 20% distance!

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