NavList:

Thank you Frank for your clarification : April 30th, 2025 is THE date with UT unchanged at 02:58:19

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By the way, an almost similar problem has already been submitted on NavList in early 2022 by Dave Walden. It has been solved in 3D with all required accuracy here and also independently here by its Author himself.

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Back to this current problem :

I differ from your view-point in the sense that the cones "apexes" are the centers of the stars themselves and not the Moon Center. Hence such cones are virtually not different from cylinders since the stars are so far away. Nonetheless this difference in respective view-points does not seem to invalidate subsequent conclusions.

So - fully agree here - the intersection of such cones (2 infinite lines with one of relevance here) with the Earth surface is our solution.

As you point out here, with both cones being so close - both apex stars being only about one half degree apart - and with such cones more or less "grazing" the earth surface rather than being more or less "locally vertical", this "worsens" the situation and we are facing a most unfavorable environment with a problem quite ill-conditioned here. Nobody's fault, just the environment here.

In conclusion - again full agreement here - it can be expected that whatever the method used to solve it, whether 3 D (I can do it again since I have kept all my notes and specific Software), or through by Besselian Elements on the Fundamental plane, or any other method (direct use of Stellarium or equivalent software) there is little doubt that in one specific "direction" on Earth the slightest angular deviation from the Moon limb will generate a quite important displacement on the Earth Surface, which might present [iteration] stability problems.

If I have time, I will work it and publish own 3D solution.

Kermit