NavList:

Antoine, you wrote:
"If one requires the utmost achievable accuracy, then all computations are to be made in 3 dimension space. On an Ellipsoid and under such case 2 dimension space computations are not accurate enough [...]"

This is generally called the "oblateness" correction (I recall we had some discussion a couple of years ago over the fact that this English terminology is a bit confusing for non-English speakers, so to be specific, "oblateness" here refers to the correction for the ellipsoidal form of the Earth and not the refractional flattening of the Moon when it is low in the sky).

You describe this as "Moon Parallax and Augmented Semi-Diameter (MPASD) Corrections". This is indeed a modification of the Moon's parallax correction since the observer is not standing on a sphere, but the change in the augmented SD is completely negligible. The standard calculation of the augmentation requires no modification.

It's relatively easy to see the scale of these changes in order of magnitude. The oblateness of the Earth places an observer on a surface that differs from the normal spherical shape of the Earth by about 15 miles. That shifts the angular location of the Moon by an angle equal to (15 miles)/(238,000 miles). That's a small angle: a ratio of about 1/16000. We convert it to minutes of arc by multiplying by 3438 (which, as a reminder, is just 60*180/pi). The result is 0.2 minutes of arc. This is (approximately) the maximum error in the Moon's position due to oblateness. The change in the Moon's augmented SD is far smaller. It's the mean SD multiplied by that tiny ratio, or 16'/16000 which, of course, is 0.001 minutes of arc. That's negligible and roughly 30 times smaller than the undulations in the Moon's profile due its own non-spherical shape (in the Moon's case, primarily due to impact basins and other "geology").

You also wrote:
" On an Ellipsoid and under such case 2 dimension space computations are not accurate enough since the Local Observer’s vertical (plumb) line generally does not cross the Earth Center."

Well, this can be over-sold. The fact that the plumb line does not pass through the Earth's center has no real significance. The coordinates account for this. As you have very correctly described, the real issue here is the way that the "3d" shape of the Earth influences the Moon's observed parallax.

You wrote:
"The US/UK NA (1983) gives a 2D space Correction equal to 20.9’"

It may be worth mentioning that the instructions in the NA also note that there is a small correction for oblateness which is necessarily ignored in the calculation of the Moon's standard correction tables. In the section in the Nautical Almanac Explanation with "methods and formulae for direct computation", there is actually a little procedure outlined for calculating those extra couple of tenths of a minute of arc for the Moon's altitude correction. This is hardly a practical concern, but it's nice that they included it. I've never bothered with that correction or the calculation included in the NA, but it's a nice, short formula. How close is it? How do you results compare with that nice, short formula?

-FER
PS: I notice in your account of your calculation you have included info on the value of "delta-T" saying "with TT-UT = +55.7s". I reiterate that this is obfuscation. It is ABSURD to add this information. For any observations within the past fifty years, delta-T is as much as known quantity as the tilt of the Earth's axis. It is not additional information. It is irrelevant. The exact value of delta-T only becomes a concern when we go back further in history. Its value as recently as 300 years ago can only be estimated.

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