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    Re: GMT from Lunar Photo
    From: Antoine Couëtte
    Date: 2022 Jan 18, 04:44 -0800

    Re : http://fer3.com/arc/m2.aspx/GMT-from-Lunar-Photo-FrankReed-jan-2022-g51845

    Hello to all,

    Further to Frank's initial post - Thanks again Frank - a number of contributions have been published.

    Warmest thanks again here to all the NavList Members who gave me additional information data when requested - including Star ID's - to help me solving this quiz.

    In this entire post and its enclosure I am using the following data derived from the Original Picture by Fred Espenak :

    - A plate scale of 2.691" / px from Peter Monta

    - A Moon Diameter of 728.3 px as an averaged value from Greg Rudzinski's (private communication) 728.6 px and Bill Ritchies' 728 px  (Tibor Miseta's 732 px "rediscovered" too late ... thanks to you too, Tibor :-)  ).

    - A HD 202284 Star Far Limb Lunar of 966.9 px as an averaged value from Greg's (Priv. comm.) 967.2 px, Bill's 967 px and Peter's 966.5 px , giving a Lunar Distance of 43.37'

    From HD 20284 (the 4 o'clok star) with Far Limb Distance = 43.37'  I am getting GMT = 01h11m37.2s

    From HD 202635 (the occulting star) with Near Limb Distance = 0.00'  I am getting GMT =  01h11m34.3s

    From HD 202672 (the 1 o'clock star) with Near Limb Distance = 2.84' from Bill Ritchie with Peter's scaling I am getting GMT = 01h11m30.8s

    From averaging these values and through giving "double weight" to the occulting star, I am eventually getting GMT = 01h11m34s which is my own final solution.

    This end result is quite close from Frank's estimate and "own solution" of "01:11:15 UT +/- 15 seconds simply by playing the time back and forth until it looked like the image".

    *******

    In this NavList thread the values quoted for the Moon [refracted] Augmented Diameter are all different. This can significantly modify the GMT end-results  :

    - Bill's value of " 2 * 16.37’ ", i.e. 32.74' , and subsequently used by Greg (private communication). 

    - Value of 2*16.361' , i.e. 32.72' quoted by Dave Walden (private communication).

    - Value of 32.66' derived from Plate scale (2.691" / px ) and Moon Diameter of 728.3 px

    A difference of .08' in the Augmented Diameter induces a difference of about 0.1' in the HD 20284 (the 4 o'clock star) Far Limb Distance, i.e. a GMT change of 20 s.

    I then decided to investigate this a bit deeper.

    First of all, from the Classical Formula : Augmented / Geocentric Diameter = 1 + sin H * sin HP , with Geocentric Diameter =  32.588' , H = 16° and HP = 59.798' , we get augmented Unrefracted Diameter = 32.744'

    Then, from the F. Espenak's picture, the only usable Diameter is the one between the Horns ends. Any other Diameter is useless because of irradiation.

    It is my understanding that whoever "measured" such Augmented Diameter did measure it between the Horns end.

    Then I decided to compute the actual refracted augmented Diameter through my own LD program.

    It involved the various sketching operations detailed in next page.

    I also took this chance to compute other Refracted Diameters with different tilt angles. All results are summarized in the next page.

    As a conclusion : I feel that using the "plate scale" determined by Peter Monta should yield the most reliable end results. This scale is using all 18 stars scattered on the picture. On the other hand computing a "scaling factor" just from even the "best computed" Augmented Diameter relies only on one part of the plate and cannot be as good.

    Last important note :

    (1) - Provided that we can consider that the Moon is a "perfect" Sphere with radius = 1,738 km - which does have actual limitations to about 3" to 4" earlier addressed in NavList (e.g. Moon mountains) -  I trust that the Refracted and Unrefracted Augmented Diameters published here are correct to the published digits, i.e. very close to 0.001' of arc.

    I believe so for the following reasons :

    (1.1)  - For such Lunar application, the Ephemeris I am computing and using is quite close from "real world" values : absolute errors less than 4" on RA / GHA and less than 0.5" on DEC and HP.

    (1.2) - The errors in relative positions between bodies are one order of magnitude smaller than the errors in their absolute positionsThe Augmented Diameters computed here are obtained through Differences between Far Limb and Near Limb Lunar distances. 

    (1.3) - For the very same reasons this also stands true for the Refraction values I am using. Although there is no "universally accepted" Refraction Theory, at heights above 15° or so, all such Theories predict extremely similar refraction values. Since the Refracted Diameters published here are subject mainly if not only to differential refraction, between various Theories the differences between differential refractions for heights around 16° are extremely small and should not exceed the 0.001' value quoted here-above.

    (1.4) - The "vertical" Diameters are obtained here by a 1D "brute force" use of Refraction (cf VI / 1 on page 2) .

    (1.5) - While using the same Refraction theory, all other "tilted" diameters - including the vertical and horizontal ones - are computed through a totally different chain of calculation : 2D vector and differential geometry on the surface of a sphere.

    (1.6) - All Diameters computed through such 2D algorithms show exactly the expected results, namely:

    (1.6.1) - The Refracted "almost vertical" Diameter (Moon Center to HD 202672) given in VI / 6 (32.651') almost exactly matches its "brute force" counterpart in VI / 1 (32.652'). The 0.001' difference is due to the fact that this "12 o'clock star" is 2° off the vertical.

    (1.6.2) - The Fictitious star HD3* Height (Hs = 15°58.1') "built" to be as close as possible from the Moon Refracted Center Height (Hs=15°57.9') yields a Horizontal Refracted Diameter "KL" exactly equal to any other non refracted diameter (32.750' in VI / 7 ).

    (1.6.3) - All non Refracted Diameters - rightmost columns - whether tilted or not show exactly the same value of 32.750' .

    (1.2) Hence I keep thinking that these Published refracted Diameters results can be used as benchmarks to whoever wishes to check its own Lunar Software. This can be easily done as follows : from any Near/Far Limb distance value, add/subtract the published applicable refracted Diameter and rerun your Lunar Program with a Far/Near limb instead. You should then derive the same GMT value.

    Hope it helps.

    Thanks again - and first to Frank for this interesting quizz - and to all for your support when requested.

    Antoine M. "Kermit" Couëtte

    antoine.m.couette[at]club-internet.fr

    File:
    220117-Influence-of-tilt-angles-on-Moon-refracted-Diameters-.pdf
       
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