NavList:

Antoine, you wrote:
"By which effect - ( unknown to me ) - do the classic methods of the lunars era seem to be able to overcome the parallactic retardation effect as regards the final UT determination accuracy?Any crystal clear explanation from anybody ?"

I hate to say it, but you are caught in a trap. It's a trap that captured George Huxtable eighteen long years ago and left him confused for two more years. He eventually realized that this phenomenon which he had annointed with the scholarly-sounding name parallactic retardation is of no concern at all (*). It is not a problem (except in exotic cases). This is a trap, a rabbit hole. 

Both you and Paul Hirose miss out by investing your energies in minute calculational details, but with all that effort you lose the forest for the trees. Rather than obsessing over trivial details, like exact values of delta-T, and and working out numbers to the nearest hundredth of a minute of arc, why not try some genuinely simple cases? The simplest lunars analysis case of all is always the vertically-aligned scenario. Clearing a lunar when the Moon and the other body are on the same azimuth (or on the same vertical circle but opposite azimuths) is nothing more than subtracting ordinary altitude corrections. Here's one for you to ponder:

On some date (it doesn't matter when), almanac data informs us that the geocentric lunar distance at 16:00:00 UT is 29°00.0'. An hour later at 1700, the distance is 29°30.0'. During the hour, the almanac also informs us that the Moon's HP is 57.0'. These are just nice round numbers to make the example easy.

We are somewhere at sea close to the equator on this date, and we see the Moon and a star above it both nearly due west not long after sunset. They are aligned vertically on the same azimuth (within some negligible difference). I shoot the altitudes of the Moon and star and (after adjusting the Moon's altitude for SD and both altitudes for dip), I have 30°03' for the altitude of the Moon's center and 60°01' for the altitude of the star. These altitudes have been observed with an understanding that they don't need to be all that accurate. I also observe the exact lunar distance, and (after adding in the Moon's exact, augmented SD) I find that the center-to-center lunar distance is 30°00.0' exactly (just for this example). 

With this observational data and the HP from the almanac, we can directly clear the lunar. The star and Moon are aligned vertically, so clearing the distance is just a matter of combining the standard altitude corrections (refraction for the star, refraction - parallax in altitude for the Moon). No spherical trig required! What do you get? You should find 29° plus something like a dozen minutes of arc. Your exact result is not important, but it should be something like that. Meanwhile from the almanac data, we know that the geocentric (cleared) distance is increasing at a rate (inverse rate) of exactly 120 seconds per minute of arc. So if your cleared distance is 29°12', then the number of seconds elapsed since 16:00:00 is 12'·120sec / m.o.a. or 1440 seconds which would imply a time of exactly 16:24:00. This should all be quite straight-forward, right? Now work the calculation again with an error of 1' in the observed lunar distance. So what happens? ... Nothing. That error passes through the lunar digestive system from beginning to end unchanged. One minute of arc error in the observed distance yields one minute of arc error in the cleared distance (corresponding to exactly 120 seconds of time in this scenario). This slippery idea of "parallactic retardation" doesn't enter into it at all.

To experiment further, I suggest two more simple scenarios: 1) drop the altitude of the Moon. What would happen if you did not measure the altitude of the Moon? How would you replace it with calculations? 2) Move the star to the other side of the sky so that the observed lunar distance is 90°+a few minutes but still aligned so that the Moon and star are on the same vertical circle which makes clearing trivial. Does the Moon's altitude matter when the LD is nearly 90°? 

For anyone else reading along, I cannot emphasize enough that this parallactic retardation is not a real issue. It is not some new nuance in those devilishly-tricky lunars. Lunars are, in fact, easy. Don't dive down the rabbit hole chasing pseudo-problems!

Frank Reed

* Purely historical note: I should add that it was Jan Kalivoda who eventually convinced George Huxtable that he was wrong. Apparently this took some long exchange of private emails back in early 2004 (go digging in the archives if you're curious), and after he was convinced, George really wanted the whole thing to go away. He regretted bringing up the idea and realized thereafter that the topocentric distances are not important. I caught up with Jan Kalivoda a few months ago by email. After years of deep interest in nautical astronomy and lunars especially, he eventually gave it all up over a decade ago and moved on to linguistics. He has become an expert in classical and medieval Latin texts, if I understood correctly.