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Tony Oz http://fer3.com/arc/m2.aspx/Recent-occultation-TonyOz-jun-2022-g52768 asked:
Is there any "honest" solution?

Here's a way forward but I'll let others be the judge of whether or not it is an honest solution.

An explanation Bessel's method of occultation reduction can be found in Chapter XV of Smart's Textbook on Spherical Astronomy  https://archive.org/details/textbookonspheri0000smar. It yields an equation for points Earth's surface, specified by geocentric latitude and longitude, where the star be seen on the Moon's limb (section 204, eq. 8). It is equivalent to Kermit's "3 D intersection curve(s) of a sphere and a cylinder." mentioned here http://fer3.com/arc/m2.aspx/Recent-occultation-Cou%C3%ABtte-jun-2022-g52785 however for occultations you should really be using the ellipsoid.

Armed with equations giving the locus of points at both immersion and emersion, numerical solutions can be found for their intersection. With the inputs given in this post http://fer3.com/arc/m2.aspx/Recent-occultation-Walden-jun-2022-g52772 I find 40°46'30"N 73°37'02"W and 82°00'50"N 109°41'51"W with the latter being rejected due to daylight.

As a check, using MICA or similar, one can calculate the Moon's topocentric position and confirm that is equal to the Moon's topocentric semi-diameter.

This problem, although interesting as a challenge, is not very practical. A more reasonable way to proceed is to get latitude by a meridian sight and then use either the time of immersion or emersion to fix longitude. The solution is then much simpler as it only involves just 1 unknown.

Robin Stuart