Assuming a flat world and no obstacles, how far could you see?

Here's a first-order approximation based on a fundamental limit: diffraction and angular resolution.

How far someone can see of course depends on the size of the object they're looking at, because the main limitation we have is one of angular size. The issue comes down to something called the Rayleigh criterion, which is a limitation telescopes have to deal with - and it has nothing to do with the medium light is traveling through. Any circular opening will produce a diffraction pattern, with peaks and troughs of intensity. The best angular resolution you can get depends on how far apart the two inner peaks are. If the aperture of your telescope (or eye) is $D$, and you're observing at a wavelength $\lambda$, then your angular resolution is $$\theta\approx1.22\frac{\lambda}{D}$$ For a human eye, $D\approx9\text{ mm}$ when pupils are maximally dilated, so even if you're observing at really red (and therefore long) optical wavelengths - say, $\lambda\approx700\text{ nm}$ - then you can achieve an angular resolution of $\theta\approx9.5\times10^{-5}$ radians. If the size of an object is $h$, then the distance at which your eye can resolve $h$ is $d\approx h/\theta$. If we choose $h=2\text{ m}$ (roughly human-sized), then $d\approx21\text{ km}$. We can repeat the calculation for a bluer object (say $\lambda=300\text{ nm}$), and we get $d\approx49\text{ km}$.

Is this a reasonable model? I've made the assumption here that we're dealing with a diffraction-limited system. Any astronomer can tell you that in reality, if you're looking through several kilometers of air, your resolution is actually going to seeing-limited - that is, limited by turbulence and other atmospheric disturbances. Attenuation due to scattering and absorption is going to be a problem.

That's why I call this a first-order approximation. Calculating scattering and optical depth is, I think, beyond what I can easily do. Taking atmospheric effects into account will reduce the numbers I have here - and the amount of reduction depends on the atmospheric conditions. So consider this an extreme upper limit.

posted about 4 years ago

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