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And there's an important case here that applies beautifully to semi-diameters and also applies to any pair of stars, whole constellations, lunars (refraction only), and so.

If both objects (or points, in the case of locations along the limb of the Sun or Moon) are greater than 45° in altitude, regardless of the actual angular distance between them, their actual altitudes, and their orientations, then the angular distance between those objects (or points) is contracted by 1 part in 3000, very nearly. So suppose we have a pair of stars whose true angular separation is 50° 00.0'. that's 3000 minutes of arc, which means that refraction decreases the observed distance by 1.0' making it 49° 59.0' (measurable with a decent sextant). The true distance is 50° 00'. The refracted distance is 49° 59' --shorter by one part in 3000. And it doesn't matter how the stars are arranged: same side of the zenith, opposite sides of the zenith, both at the same altitude, or very different altitudes. None of that matters. The refraction depends only on the original distance.

Or suppose we have two points on opposite limbs of the Moon or Sun. Those points are separated by about 30'. That's 1800 seconds of arc. The contraction of that angle caused by refraction is 1800"/3000 which is 0.6". For the case where the Sun or Moon are higher than 45°, this applies to the diameter in any direction across the disk and it applies anywhere in the sky. If the Sun's center is 90° high, standing right at the zenith, then the contraction of the diameter of the Sun is just about 0.6" and obviously, by symmetry, it's the same contraction in all directions. More of a surprise, if the Sun is, let's say, 60° high, the contraction of the Sun's diameter is also 0.6" and also the same in any direction across the Sun's disk.

Why such a simple rule? Refraction in altitude is very nearly equal to 1'·tan(z) where z is zenith distance. For small-ish angles tan(z) is nearly equal to z / 57.3° (assuming z is given in degrees), or over a wider range of angles, tan(z) is relatively close to z / 50°. In addition, close enough to the zenith the tricky parts of spherical trigonometry disappear, and we can use plane triangle relationships without worrying about spherical triangles. Now if we pick any three points on the celestial high enough in altitude for that approximation to the tangent to be adequate, then we have a specific form of contraction that makes similar triangles. Suppose one of the points is the zenith and the other two points are stars, for example one star, S1, due west at 60° altitude and the other star, S2, due north at 70° altitude. We start out with the vacuum case, without refraction. We have a triangle with corners at Z, S1, and S2. Two of the side lengths are just the zenith distances of those two stars: ZS1, ZS2. Now we bring in the atmosphere and refraction. Each of those zenith distances will be reduced in direct proportion to the specific zenith distance. They both move towards the zenith to new positions, call them S1' and S2'. The new triangle with corners at Z (unchanged) and S1' and S2' has been reduced in proportion but the shape is exactly the same. This new triangle is geometrically "similar" to the original triangle. And that guarantees that the angle between the two stars has been reduced by the same proportion as the zenith distances themselves. Thus we can ignore the zenith and zenith distances. All that matters is that the distance between the two stars has been reduced in the same proportion.

The effect of refraction on that part of sky higher than about 45° is a pure contraction of all angular distances, just like looking at a painting of the stars and constellations on a wall and taking a step back. Another interesting property of this pure contraction transformation is that it is center-less. As we saw above, the zenith drops out of the story. Every constellation, every pattern of stars, is simply reduced in size by one part in 3000 (approximately).

Frank Reed

PS: There's a geometric connection to the expansion of the Universe here. Yes, really! But this is just an amusing tangent, pardon the pun, so stop reading here if you're not amused!! Many people, including some science fiction authors, learn the basics of cosmology and the expansion of the Universe -- that the galaxies are pulling away from each; the Universe is expanding, and they come upon the very reasonable hypothesis that we ought to be able to trace the expansion backwards and find the "Center of the Universe", the "origin point" from which the Big Bang exploded. The catch is that this expansion is directly proportional to distances between galaxy pairs on a large enough scale. This is usually quoted in terms of the "Hubble constant" which tells us how rapidly distant galaxies are moving away from us (wait... is the Earth the center of the Universe?!). Although different measurement methods still produce slightly different results, a reasonable value for the Hubble constant is around 70km/s per Megaparsec (real astronomers like parsecs, and a Megaparsec is a million parsecs where each parsec is about 3.26 lightyears or 19 trillion miles). Note the units on that number. It's "distance" per "time unit" per "other distance" so the distance units actually cancel out leaving us a pure number or equivalently a "percentage" per some time unit. In fact, we can take standard values of the Hubble constant, drop a zero from the number, and the result is very nearly the percentage change in a billion years. In other words, 70km/s per Megaparsec is very nearly 7% in a billion years. All large-scale distances in the universe are increasing in direct proportion to distance. All triangles we can draw among galaxies, and all lines separating pairs of galaxies, in the Universe, have relative distances that are growing in proportion to the distances. A billion years from now, they will all be about 7% larger than they are now. All geometric relationships and shapes will be "similar" to their shapes today. Any center we might pick for the expansion fades into irrelevance. Any point can be counted as the center of the expansion, but equivalently no point qualifies as the center of the expansion. A proportional expansion of the Universe is "center-free". Just as the proportional contraction of the star patterns we see high in the sky are "zenith-free". Although we do the latter calculations by zenith distances, they drop out of the story because the math leads to simple proportionality.