Polaris
Polaris is awkward because it is so close to the North Pole. It is about 1/2 degree from the pole, which meant that the time of meridian passage was hard to measure. Polaris moved so slowly in both altitude and azimuth that the fits to its maximum altitude gave large error bars as to the time of passage. This uncertainty pointed out the need for an alternative method to get the dec of Polaris.
I chose to let the scope autotrack on Polaris for an entire night, getting 30,000+ measurements of its instantaneous altitude. With my coarse encoder resolution, many of these points were duplicate. Despite that, when I plotted up the path of Polaris for the entire night, I was rewarded with a beautiful partial circle of data points.
When I fit a circle to this data, I found a radius of 0.66 degrees, the distance of Polaris from the pole at my epoch. This radius is also known as the co-declination, and the true declination was thus 90 - 0.66 = 89.34 degrees. The present day catalog value of Polaris' dec, precessed to 2020, is is 89.352 degrees.
To determine the ra of Polaris, I decided to use Tycho's method of determining the delta ra between Polaris and some other primary star, and then adding that delta ra to the ra of the primary star. This is intrinsically problematical in that most of these triangles are 'skinny'. Nonetheless, quantity can produce a quality all its own, so, by using 30 nearby stars, I was able to estimate the ra of Polaris as 44.92 degrees with a standard deviation of 1.2 degrees even though the triangles were very skinny.
This number looks bad at first. The true ra of Polaris in 2020 was 44.26 degrees, a difference of 0.66 degrees (40 arc minutes). I was trying to get accuracies of a couple of arc minutes. What went wrong?
The answer comes from the coordinate systems involved. An error of 0.66 degrees for a star on the celestial equator (dec = 0) produces an error of 0.66 degrees in apparent position. If you pointed a telescope at this position, the star would be 0.66 degrees from the center of your field of view (FOV). The same 0.66 degree ra error for a star at dec = 60 degrees would produce a pointing error less than one degree, specifically 0.66*cos(60 deg) or 0.33 degrees. The same 0.66 degree ra error for a star at dec = 89.34 degrees would produce a pointing error significantly less than one degree, specifically 0.66*cos(89.34 deg) or 0.007 degrees, which is less than an arc minute. So my position for Polaris does actually meet my goal, even though it looks terrible.
This coordinate effect is important in any evaluation of a stellar catalog; mine, Tycho's, or Gaia's. The relevant error is the pointing error on-the-sky, which can be much smaller than the apparent error in the coordinate itself. This on-the-sky error tells you whether the star will appear in your FOV or on your detector.
