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Dear Ed, Sean, and Frank,

• Re:
  • 1 - First-MoonStar-Lunar-EdPopko-dec-2017-g41011
  • 2 - First-MoonStar-Lunar-SeanC-dec-2017-g41056, and :
  • 3 - MoonStar-Lunar-EdPopko-dec-2017-g41062 , and

Décidément ... your First Moon Lunar, Ed, is raising a lot of interest.

This time I have a question for you, Frank, directly related to solving this Lunar.

Nonetheless, both Ed and Sean may react, since in setting up my upcoming twofold question to Frank (see *7* further down) , I am quoting "your" own numbers.

Fast Readers can proceed directly to *4* here-under, because I am doing a bit of "History" so that Lunarian Crazy Readers can get deeper into this subject.

*1*

Quick recap (Ref1) : This Lunar was shot from N28°41.2' W082°18.3' on Dec 24th, 2017 at 0h05m04s GMT . We have assumed that Height of Eye = 0', Temperature = 50°F and Sea Level Pressure = 29.92" Hg. Ed and Sean possibly worked in a slightly "Weather" environment, but this should be almost negligible for meaningful results comparison.

*2*

This Lunar was independently solved :

• First by Ed through "Using Reed's algorithm in [his] HP calculator" (Ref 1). Then :
• By Sean through "  using Frank's fomulae " for Sean (Ref2)  with his own derived starting data slightly different from Ed's. And finally :
• Again by Ed through his same method " I input your updates and again used my calculator program based on Frank's workshop algorithm" .

*3*

From what I have been able to understand -and please Frank correct me if I am wrong here - it looks that that Frank's Algorithm requires the following starting data :

• Moon Center and Aldebaran Geocentric heights, corrected for refraction [and corrected for semi-diameter since we deal with Moon Center] , and named "Ho" . And :
• Moon Center and Aldebaran Topocentric heights (i.e. as observed from the Earth Surface), not corrected for refraction [but corrected for semi-diameter since we deal with Moon Center], and named "Ho+Corr" . And:
• Distance from Aldebaran to Moon Center (or better : to the "Moon Center Best Determination") as seen from the Earth Surface and affected by refraction, i.e. as measured through a "perfect and ideal" sextant able to "pick up" the exact refracted Moon Center. Such Sextant distance is named "LDpc". Finally :

Processing all these data through Frank's Algorithm yields a "Cleared Lunar Distance" named "LDc", which is the geocentric angular distance between the Moon Center and Aldebaran, thus directly comparable to Nautical Almanac derived data.

*4*

Post Ref 2 by Sean definitely triggered my immediate attention since it looks that through a fully classical method - as I understand Frank's Algorithm is - and from almost the very same starting data than Ed's, Sean was able to get a different Cleared Distance than Ed. And interestingly enough, when reworking this Lunar with the same data as Sean, Ed did not get the exact same Cleared Distance as Sean.

This gave me the idea of solving this Lunar through the Classical Lunar Formulae I have been using from time to time in the past 6 years or so. These are the Chevalier de Borda's Formulae I referred to in a recent post. At least I believe them to be so, and if I am wrong, please Frank be so kind as to correct me here and please be so kind as to let me know who published them first. Let's then call these ones "B Formulae" . As you can see in the enclosure, these are "straight brute force" Formulae.

Let's also call Frank's Algoritm - as used by both Ed and Sean - the "F Formulae" .

I am not familiar at all with "F Formulae".

I decided to compare the results of "B Formulae" with "F Formulae" and I decided to check them against results very close from Frank's On Line Calculator results.

*5*

And ... lo and behold ... I just discovered this afternoon that "B Formulae" and "F Formulae" require exactly the same starting data in order to yield the Cleared Lunar Distances. Are they [closely] related ? Let us see :

Ref 1 Lunar computations by Ed

• Moon Ho+corr = 37°07.3'   Aldebaran Ho+Corr = 36°12.0'     LDpc = 95°38.7'   Moon Ho = 37°50.1'      Aldebaran Ho = 36°10.7'   
• "F Formulae" Ed's LDc = 95°04.5'    "B Formulae" results LDc = 95°04.709'    Difference "F Formulae" - "B Formulae" = -0.209 '

Ref 2 Lunar computations by Sean

• Moon Ho+corr = 37°07.8'   Aldebaran Ho+Corr = 36°12.1'     LDpc = 95°38.6'   Moon Ho = 37°50.1'     Aldebaran Ho = 36°10.7'    
• "F Formulae" Ed's LDc = 95°04.8'    "B Formulae" results LDc = 95°05.097'    Difference "F Formulae" - "B Formulae" = -0.297 '

Ref 3 Lunar computations by Ed from the very same data as Sean's

• Moon Ho+corr = 37°07.8'   Aldebaran Ho+Corr = 36°12.1'     LDpc = 95°38.6'   Moon Ho = 37°50.1'  Aldebaran Ho = 36°10.7'       
• "F Formulae" Ed's LDc = 95°04.9'    "B Formulae" results LDc = 95°05.097'    Difference "F Formulae" - "B Formulae" = -0.197 '
• Difference between Ed and Sean working from the same algoritm and onto the same starting data : 0.1'  .

*6*

A bit surprised at these results, I decided to check how "B Formulae" would perform against my own "Big" Lunar Software - let's call it "C Formulae". Here are the relevant data derived through "C Formulae" as follows :

• Moon Ho+corr = 37°07.416'   Aldebaran Ho+Corr = 36°12.001'     LDpc = 95°38.576'   Moon Ho = 37°50.048'  Aldebaran Ho = 36°10.679' . And :
  • C Formulae LDc = 95°04.746' . Note : Frank, I am think that your On Line Calculator results are extremely close (to better than 1") from these "C Formulae" results.
• I then ran "B Formulae" with the data immediately here-above :  "B Formulae" results LDc = 95°04.741'    Difference "C Formulae" - "B Formulae" = +0.005 '

I also found that Frank's On Line Computer LDc = 95°04.7' (probably very close from 95°04.75' as suggested here-above). 

*7*

Hence my two-fold question to you Frank :

• Since Ed and Sean do not get the very same LDc from identical data (Ref 2 and 3), what is the exact LDc yielded by your own Algorithm in this specific case when processing such Ref2 or Ref3 data? And :
• How do the so-called "Borda's Formulae" in the enclosed document relate to your own Algorithm, and are they safe for use under any and all Configurations ?
  • ​​​​​​​I have obtained my "Borda's" results through electronic "brute force" computation.
    • Maybe this could explain some differences between your upcoming results should your Algorithm require some "manual computation" steps as it seems to be the case.

Thank you very much, Frank, for your Kind Attention and for your reply.

Meanwhile .. Happy New year again to all !

Antoine Couëtte