A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
From: Frank Reed
Date: 2018 Aug 1, 10:59 -0700
Oh, if you want a formula, it's not too difficult to derive yourself. Consider the spherical triangle with corners at the Sun, Moon, and Zenith. The angle, ZMS at the Moon in that triangle is the same as the inclination of the line through the Moon's horns. Draw a picture with the Moon about two days old (around 26° away from the Sun) to convince yourself of that. Now you can calculate it, right? It's nice to know that you can calculate the orientation of thee Moon's horns (and it's vital for simulation software), but there's really no need to do so for the issue at hand. Instead, you can just look at the Moon in the sky.
The cosine of that same angle, ZMS, incidentally, is the "corner cosine" for clearing lunars by a series method --it tells you the fraction of the Moon's altitude that acts along the lunar arc. If the angle is zero, with the horns horizontal and the bright edge of the Moon pointing toward the zenith, the altitude correction is applied 100% as a negative correction to the lunar distance. And that also implies then that you can make a rough estimate of the cleared lunar distance just by looking at the orientation of the Moon's horns (not connected directly with the issue we're talking about with longitude by lunar altitude, but it's the same math). This also, as we have discussed previously, explains a seeming oddity that Ed Popko noticed: in a majority cases, both in textooks and in the real world, the cleared center-to-center lunar distance is less than the observed. That's because in convenient cases the Moon is usually observed for a lunar distance with the illuminated limb "up".