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Paul Hirose wrote: "Refraction also decreases Moon semidiameter"

And Antoine Couëtte, you replied: "Just for fun ... I wanted to check own results on that one."

You can validate this in your head, without any fancy code. This follows from an easy rule that provides an excellent estimate by using a property of refraction at higher altitudes above the horizon. For any angular distance, regardless of scale, regardless of orientation, so long as both points are above roughly 45° in altitude, the effect of refraction is a simple compression, a contraction, a reduction in angular size, by about one part in 3000.

If you have two stars separated by 50° (both at altitudes above 45°), then the refraction of the angular distance between them amounts to 1'. That is, the "true" distance is reduced by refraction by one minute of arc, from 50° to 49°59' (50° is, of course, exactly 3000 minutes of arc, so an estimate of one part in 3000 is equal to one minute). 

Similarly if we have two stars separate by 50 minutes of arc, then the refraction of the distance between them would be one second of arc. Again that's one part in 3000. And in the case of the Moon's semi-diameter, since that's typically about 15 minutes of arc or 900 seconds of arc, the contraction of the semi-diameter is just about a third of a second of arc (900/3000).

Note that in this altitude range, above roughly 45°, this nice easy rule applies to angular separations in any orientation. In the case of the Moon it applies to the contraction of both the horizontal diameter and the vertical diameter of the Moon. The entire sky above 45° is simply reduced in size by direct proportion: one part in 3000. It's "squeezed" towards the zenith. And because this compression is proportional, you shouldn't even think of the zenith as a key point. Pick any arbitrary point in the sky higher than 45°. Refraction slightly compresses all nearby star patterns towards that point. The whole top "half" of the sky, including the Moon if it's up there, is "squished" just a bit.

When you step outside this evening and see the Big Dipper high overhead (assuming you're in a high enough northern latitude such that its stars are above 45° high), remember this: it has been slightly compressed. The angular length of the Big Dipper is about 25°. One part in 3000 of that angle is half a minute of arc. You could measure that. 

Suppose you haul out your telescope, and you decide to look at the Whirlpool Galaxy, M51, tonight. That's just off the end of the handle of the Big Dipper. It's about 10' in angular diameter. You decide to view it with a magnification of 60x. But oh, no... The magnification caused by refraction in your telescope is slightly diminished by the refraction due to the atmosphere. Fortunately, it's not much! Your 60x of magnification is reduced by one part in 3000 to 59.98x by the atmosphere itself. Maybe they should add that to the fine print in the warning label (*).

Frank Reed

* Warning label about telescope magnification? It's true. The IAU (International Astronomical Union) in circular 4E712-1-APR-2023 has insisted that all observatories add a warning label on new telescopes stating that "objects in telescope are more distant than they appear".