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    Re: Exercise Lunar Distance with Mercury
    From: Antoine Cou�tte
    Date: 2009 Sep 20, 09:48 -0700

    La Bruy�re, Sep 20, 2009
    
    
    Dear Jeremy,
    
    
    Thank you for interesting examples to work ! First time I ever heard of a Moon-Mercury Lunar.
    
    Please find here-after my solution and comments.
    
    This is a typical case when we can work on averaged values both for Sextant Distances anf for UT values.
    
    Therefore, I have worked all results from the following values :
    
    - Mean Sextant Distance = 41�05'58 (41 Degrees 05 Minutes 58/100 of a Minute, 
    further down shortened to the following format : 41D05.58M). This value has 
    been corrected for your Index error of 0.8' (remove, i.e. substract the error 
    , i.e. remove 0.8' ) or your Index correction of -0.8' (add the correction, 
    i.e. add -0.8'). We can readily observe on your individual sextant reported 
    readings is that values(possibly) there is a "inaccurate" result somewhere 
    for either Observation 2, 3 or 4 since observed values should all increase as 
    times increase, and
    - Mean UT tagged to the value herabove : 16h47m15.0s
    
    I have therefore (re)computed heights since they are required, even to a 
    rather good accuracy (probably better than what is usually mentioned on 
    Navlist) if you want to make accurate Distance clearing and take in account 
    Body Shape distortion due to differential refraction (probably not 
    significant in this current case since Moon is fairly high in the sky, but 
    this may become a quite significant effect when observed bodies are (much) 
    lower).
    
    For all subsequent altitudes computations I have assumed that your height of 
    eye is 106 ft (rings a bell ???? . this seems a value you often use) and that 
    both Temperature and Pressure are standard.
    
    NOTE 1 : I am hereby publishing a number of digits which sometimes do not have 
    any physical meaning (such as heights to 0.001 '), or Celestial Bodies 
    Coordinates to better to 1 arc second while I have computed then to an 
    accuracy of 3 arc second only here.
    
    
    SINCE YOU have NOT "TIME TAGGED" YOUR (GPS?) POSITION, I HAVE MADE TWO ASSUMPTIONS :
    
    CASE ONE : FIRST ASSUMPTION
    First assumption (the most probable one I think ... ??? ) : Your indicated 
    (GPS ?) position is for 17:00 UT, in which case your 16h47m15.0s position is 
    S3629.7E2624.5 , then at which time, you get  sextant observed altitudes of 
    14�48'559 (14D48.559M) for Mercury and 55�06'118 (55D06.118M) for the Moon 
    lower limb.
    
    CASE TWO : SECOND ASSUMPTION
    Second assumption (less likely case ??? ) is that your indicated (GPS ?) 
    position for 16h47m15.0s is exactly the one you are mentioning, i.e. 
    S3631.1E02620.8, then at which time, you get  sextant observed altitudes of 
    14�51'328 (14D51.328M) for Mercury and 55�08'565 (55D08.565M) for the Moon 
    lower limb.
    
    NOTE 2 : Shortly simplified, it can be stated that a Lunar can be seen as a 
    set of four data : choose any two and the other two result of your choice, 
    sometimes in a unique way, sometimes not. The four data considered hereafter 
    are as follows:
    
    (1)	UT of Sextant Distance,
    (2)	Sextant Distance,
    (3)	Position at UT Sextant distance, and
    (4)	Heights observed
    I will not extensively cover all cases hereunder, but only the few one interesting for immediate use.
    
    NOTE 3 : All computations results presented hereafter have been corrected for 
    all known effects of parallax, Earth oblateness, augmented semi-diameter, 
    refraction and body shape distortion due to differential refraction, horizon 
    depression, and in case of Planets for phase correction as follows : I am 
    assuming that you "swing-kiss" the center of gravity of the Planet light over 
    the Moon Limb.
    
    NOW, HERE ARE MY RESULTS
    
    Since they have not been published in this thread so far, the following data pertain to Mercury :
    
    - Distance to Earth is .907240981 UA, giving an Equatorial parallax close to 9.69 ",
    - Visual magnitude is 0.33 ,
    - Semi-diameter is 3.707 arcseconds, and
    - Corrections for phase angle are -1.605 arcseconds for RA and +0.851 
    arcseconds for Declination, both reckoned from body center of gravity towards 
    center of light.
    
    CASE ONE : Your assumed position at 16h47m15s0 is assumed to be S3629.7E02624.5
    
    CASE ONE Subcase ONE
    
    If we assume that Both (1) and (3) just defined immediately above show no 
    error at all, then the Observed Sextant distance is equal to 41�05'127 
    (41D05.127M), in which case we can compare it with your sextant observed 
    value corrected for Index error of 41�05'58 (41D05.58M). You had an average 
    observation value error of 0.45 arc minute, which is already quite good. 
    Observed position is S3629.7E02624.5 since (3) is forced into the 
    computations.
    
    CASE ONE Subcase TWO
    
    If we then assume that both (2) and (3) show no error at all, then you get a 
    UT error equal to 1m32.8s, which gives an observation UT of 16h48m47,8s and 
    obviously an observed position of S3629.7E02624.5 since (3) is still forced 
    into the computations.
    
    CASE ONE Subcase THREE (a)
    
    If we assume that all (2) and (3) show no error at all, then we get a UT error 
    of 0m53.2 s, a UT Time of 16h48m08s2 and an observed position of 
    S3632.0E02610.8 .
    
    CASE ONE Subcase THREE (b)
    
    We can notice that the following data also fulfill (2) and (3) if considered 
    as exact : UT error of 0m54.1s with a 16h48m09.1s UT position at 
    S1935.0E02805.3
    
    CASE ONE Subcase FOUR
    
    An interesting one here : what if we assume that both (1) and (2) show no 
    error ? I have not seen anything published for this case.
    
    Well, we can start with a simple case : a grazing Moon occultation with a 
    pinpoint body (star, planet or asteroid) is fairly simple to visualize, 
    likewise if and when we consider a pinpoint light source with Moon.
    Its becomes a funnier story when the second body has a non zero apparent 
    diameter : Mister Sun to name it !!! Any ideas here ???
    
    *******
    
    CASE TWO : Your assumed position at 16h47m15s0 is assumed to be S3631.0E02620.8
    
    CASE TWO Subcase ONE
    
    
    If we assume that Both (1) and (3) just defined immediately above show no 
    error at all, then the Observed Sextant distance is equal to 41�05'179 
    (41D05.179M), in which case we can compare it with your sextant observed 
    value corrected for Index error of 41�05'58 (41D05.58M). You had an average 
    observation value error of 0.40 arc minute, which is good. Observed position 
    is S3631.0E02620.8 since (3) is forced into the computations.
    
    CASE TWO Subcase TWO 
    
    If we then assume that both (2) and (3) show no error at all, then you get a 
    UT error equal to 1m22.1s, which gives an observation UT of 16h48m37,1s and 
    obviously an observed position of S3631.0E02620.8 since (3) is still forced 
    into the computations.
    
    CASE TWO Subcase THREE (a)
    
    If we assume that all (2) and (3) show no error at all, then we get a UT error 
    of 0m47.0s, a UT Time of 16h48m02s0 and an observed position of 
    S3633.0E02608.7 .
    
    CASE TWO Subcase THREE (b)
    
    We can notice that the following data also fulfill (2) and (3) if considered 
    as exact : UT error of 0m47.9s with a 16h48m02.9s position of S1933.3E02804.0 
    
    
    CASE TWO Subcase FOUR
    
    Already covered in CASE ONE Subcase FOUR hereabove.
    
    *******
    
    NOTE 4 ( Last Note) : Planetary data computations are based on the following theories :
    VSOP82 and NGT : These data handed over to me by my late good friend Dr Pierre 
    Bretagnon of the Bureau des Longitudes in Paris. Dr Pierre Bretagnon "merged" 
    both VSOP82 and NGT theories into the VSOP87 theory. So I am essentially 
    getting the same results as VSOP 87. As stated earlier, Moon and Mercury 
    apparent coordinates have been computed to an accuracy of 3 arcseconds here. 
    Mercury phase correction on both Right Ascension and Declination are computed 
    to an accuracy better than 1/100 th of an arcsecond.
    
    For the Moon I am using the ELP 2000-85 theory from M. Jean and Mrs Michelle 
    Chapront from the CNRS in Paris and who are working in Bureau des Longitudes, 
    a copy of ELP-2000 85 was handed over to me by Mrs Michelle Chapront.
    
    For Precession I am using Dr Pierre Bretagnon's Precession values to their 
    full accuracy as derived from NGT. Their current precision better than 1/1000 
    arcsecond and will degrade to about 1/10 arc second after 1000 years 
    before/after 2000.0 .
    
    For Nutation I am using here J. Wahr's Theory (Wahr, J. 1981) to its full 
    accuracy. Its current accuracy is  5/100 arc second and it is degrading to 
    about 1 arc second after 1000 years before/after 2000.0 . Although not 
    ourstandingly accurate by to-day standards, this Theory had the great merit 
    of being the first one used by the IAU which incorporated a elastic Earth 
    model.
    
    For Stars Apparent computation positions, I am using the FK6 positions when 
    available, and the Hipparcos Position by default of FK6 published values. It 
    enables to compute stars apparent positions - including gravitational light 
    deflection - to 1/1000 arc second (when compared with documents using the J. 
    Wahr's Theory such as the AA for a few years starting with the 1984 volume).
    
    For the TT-UT values I am using the information given in :
    
    eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html
    
    which I consider being the best single comprehensive source both for recent 
    and ancient historical times (Thanks to M. Morrison and al)
    In a few words, the accuracies I am using are definitely above the required 
    standards for CelNav with a conventional Sextant.
    
    *******
    
    Comments on the hereabove (all or any part) are welcome. I may not have the 
    immediate time to reply since I am in "standby" and might fly again (to the 
    USA ??) to-night or to-morrow morning.
    
    ******
    
    Till then, Thank again Jeremy and
    
    Best Regards to you all
    
    Enjoy !!!!!!!
    
    Antoine M. "Kermit" COUETTE
    N4641.753W00054.875
    Hopefully, no typos hereabove.
    
    
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