A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 Feb 14, 17:55 -0800
Yep. Midnight sun is a different case. Another name for this is "meridian passage below the pole".
No, there is no generalization of the noon rule, Lat = Z.D.+Dec, and given the rarity of "below the pole" meridian passage sights, I don't think there's any good reason to seek a generalization.
Of course there is a separate rule that you can write for this case:
Lat = altitude + coDec,
where coDec is the co-declination (=90°-dec). If you draw a little diagram, it's easy to re-derive this on the spot. The altitude of the celestial pole is your latitude. That's a given. So picture some celestial body directly below the pole. The angle from the horizon up is just the altitude (corrected). And the angle from the body to the celestial pole is the "polar distance" which is another name for the coDec. This is a case where ignoring the +/- signs is allowed and also beneficial.