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Dear Paul,

Thank you for your commented results over Lunar # 2 (re: http://fer3.com/arc/m2.aspx/Modern-Lunars-Couëtte-sep-2016-g36570) .

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Lunars #2 and #3 (re: http://fer3.com/arc/m2.aspx/Modern-Lunars-Couëtte-sep-2016-g36570) have been devised for the following reasons:

- Since in both cases, the Sun and Moon Azimuts are the same, it is easy to compare the heights between Sun UL - Moon LL and their Apparent Limb-to-Limb Distances yielded by a Lunar Distance Program.

- They also demonstrate the difference bewween Limb-to-Limb Apparent Topocentric closure rates, and Geocentric closure rates. Hence the importance of accurately computing such topocentric closure rates - now permitted by "modern" software (and out of reach the Lunars "classical" methods) - in order to get a good understanding of the reliability of the Lunars observed.

- They also show that at low altitutes, Refraction can either slow down the apparent closure rates - to the extent of even changing its sign in the example which I admit is a limit case where both bodies have the same azimut with one of them extreeeeemely low - or accelerate such closure rates (Re: Willam E. Chauvenet Fomalhaut-Lunar on 30 Aug 1855).

- They also show that - for such local configurations - the primary effect of refraction at low altitude is to (greatly) modify the closure rates since this effect is observed whatever the Refraction model used at low altitudes. This effect seems to me much more important than the differences themselves between [our] various Refraction Models.

- They finally demonstrate that for one same geocentric Closure rate - directed related to our "Classical Methods" with the "Cleared Distances", the locally observed Limb-to-limb closure rates may vary enormously in relative values.

Note: nothing at all here against Classical Methods and their current implementations. Simply showing that compared to Classical Methods, Modern Methods can benefit of much more efficient computing power and finally lead us into a better understanding of some of the Lunars limitations and caveats, most often set aside in the past because there was no easy way to accurately compute the “Local Longitude change / Sextant Distance change ratio” and to understand the importance of this concept 150 years ago.

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Here I am going to look more closely into our numerical results.

Re: LUNAR #2 (re: http://fer3.com/arc/m2.aspx/Modern-Lunars-Couëtte-sep-2016-g36570) 

I have studied carefully differences between our results and I can see that we are using some slightly different data.

Main difference concerns Refraction, your refraction of the Sun LL is -0.1652° vs. -0.166 335 659° for the value I am using. Difference between both is close to 4", well within the known uncertainties for Refraction at such a low altitude (Sun LL at 5.2°)

One thing puzzles me ... the difference between our Azimuts:

For the Moon, you get  85.4227° , while I get: Moon Z = 85.42280° including a Parallax in Azimut equal to dZ = -14.59022” , which - without the parralax in Azumuth - would put the Moon at 85.4268 °

For the Sun, you get 85.4223°, while I get: Sun Z = 85.42227°, including a parallax in Azimut equal to dZ = - 0.02538”

Maybe a difference in Aries GHA ? I am computing it at ARIES GHA = 107.057 931 8 °, but I am ignoring some higher order cross-terms of the "Equation des Equinoxes".

Strange ...

Best Friendly Lunarian Regards  :-)

Kermit