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    Re: Angular Distance Between Stars
    From: Frank Reed
    Date: 2012 Sep 14, 12:53 -0700

    I wrote previously:
    "you can get the refracted distance for any pair of stars when both altitudes are above 45 degrees to a good approximation by decreasing the true great circle distance by 1/3000 (a tenth of a minute of arc for every 5 degrees of distance)."

    So why know this? The exact calculation is very easy. Why bother with this easy rule considering that it's slightly approximate and only works when both stars are higher than 45 degrees? I can think of two reasons. First, it's just a good quick practical calculation. What could be easier than 0.1' for every five degrees? But for me, the more important reason is that it reveals some rather surprising equivalent cases. And these are cases that very few navigators (or even experts in positional astronomy) would guess would be equivalent and many older textbooks actually get wrong. Consider these three cases: 1) two stars are each 85 degrees high and on opposite azimuths: e.g. one due east, the other due west; 2) Two stars are exactly due east, one above the other, one is 50 degrees high, the other 60 degrees high; 3) Two stars are both 50 degrees high but separated in azimuth by some amount (roughly 15.6 degrees) such that the distance between them is 10 degrees. In the first case, the refraction is quite small but it's additive since the stars are pushed together towards the zenith from opposite direction. In the second case, the individual refraction values are much larger but only the difference counts since they're pushed towards the zenith from the same azimuth. And in the third case, both stars are elevated by refraction by exactly the same amount but the distance is shorter since the azimuth lines converge towards the zenith. Those are all very different situations, but nonetheless, the refraction in distance is the same for all three. It's 0.2 minutes of arc since the distance between them is ten degrees. Any time you see ANY pair of stars separated by about one "fist at arm's length" (about ten degrees) and the stars are relatively high (above 45 degrees), without any other information about their orientation or exact altitudes, you know that the refracted distance is 0.2' less than the un-refracted distance.

    Or consider the Great Square of Pegasus. It's above 45 degrees in mid-northern latitudes this year from about 10:30pm, crosses the meridian about 12:45am and falls back below 45 degrees after 3:00am. During that whole time, how does refraction affect its shape and the angles among its stars? Since this is just as special case of the above rule, all angles are reduced by about 0.1' for every 5 degrees. The sides of the Great Square are all about 15 degrees in length so they're each reduced by just about 0.3'. And of course when you reduce the size of a shape by the same proportionate amount in all directions, the shape is not changed at all. So during that long stretch of time from 10:30pm to 3:00am, refraction has slightly reduced the size of the Great Square by a nearly fixed amount, and its shape is not affected at all.


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