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Take any pair of stars with both altitudes above 45°. Refraction compresses the angular distance between them. The correction is beautifully simple: it's one part in 3000, which is equivalent to 0.1' per 5° of angular separation, and this works for all distances, no matter their relative altitudes, no matter how they are oriented, and no worries about temperature and pressure. Atmospheric refraction decreases the apparent angles between the stars. The corrected angles are slightly larger than what we observe. This is an approximation, and the result can be wrong by as much as 0.1' but it's fast and easy and really that's an excellent approximation for our needs.

How do you get the correct distances to compare against? Use any source to find the GHA (or SHA) and Dec for the stars you're using. Be sure to use data for the current month. The year doesn't matter much since the distances change very slowly over decades (except for a few stars, like Arcturus). But the month matters due to stellar aberration. Then calculate the great circle distance using the standard cosine formula.

Let's try one. Anyone in mid-northern latitudes at any longitude who steps outside tonight around local midnight (assuming daylight time/summer time in effect and assuming you're not too far from the standard meridian for your zone) will find Vega and Arcturus both high in the sky. They will both be above 45°.

The SHA and Dec for Arcturus May 20, 2019 are 145.8627°, 19.0841°. For Vega they are 80.5985°, 38.8002°. Running the cosine formula:
  cos D = sin Dec1 sin Dec2 + cos Dec1 cos Dec2 cos dSHA
yields a distance of 59.1327° or 59°08.0'. Note that if we had used data from 2018, there would have been no difference, but since Arcturus is unusual and has high proper motion, almanac data from ten years ago would not be acceptable. Also note that the angle changes by at least +/-0.2' cyclically during the year due to stellar aberration so it's important to use data near the current date.

I measure this distance with my sextant at any old time around or after midnight so long as both stars are above 45° altitude. Over the course of maybe half an hour, I observe the angular separation six times and average the results. Suppose my observed angular distance (index error always zero) is 59°06.9'. The refraction correction is 0.1' for every 5° of angular separation. This angular distance is nearly 60°, so the correction is 1.2'. That yields a corrected distance of 59°08.1'. That would be 0.1' greater than expected, as calculated above, but this is right at the limit of expectations for sextant observation so we can safely ignore it.

Frank Reed