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    Re: USCG Student Sample Problems #1, #2 and #3
    From: Antoine Couëtte
    Date: 2013 Apr 25, 13:38 -0700

    RE : your post http://fer3.com/arc/m2.aspx/USCG-Student-Sample-Problems-1-2-3-Rudzinski-apr-2013-g23664

    Dear Greg,

    Thanks for your reply, and please find here-after a few additional comments.



    Your updated results are quite interesting since :

    - The first result which you published for the MOON showed i = -28.9’ / AZ = 335.6° (USNO) while I had found i = -29.1’ / AZ = 335.6°, i.e. a 0’2 difference then, and
    - Your updated results now show I = - 29.3’ / AZ = 335.6° (USNO) while I am still finding this time I = -29.1’ / AZ = 335.6°. Same 0.2’ difference, but … with the opposite sign.

    I will not elaborate any longer on our results differences (+/- 0.2’) since it would be (again) definitely splitting hair. I am aware that hair splitting for a number of our NavList Colleagues start once we start investigating below one arc minute … :-)



    For this UT = 20:05:55 Observation, I first crosschecked the MOON Celestial Coordinates I computed vs. the ones you have published. They all match within 0.1’.

    Therefore the 0.4’ minute difference between our results (which you just recently reduced down to 0’3 now) is certainly worth investigating further.

    Let me first write again your data, with the Index Correction already taken care of:

    Auxiliary Latitude 10°S / Auxiliary Longitude 152°31’3
    Refraction: -0.5’ , Dip -9.7’
    Ho = 62°26’6 and LL Hs = 61°56’0 (the Index Correction has already been performed).

    You found: i = +20.5 T / AZ = 336°9 while my results show: i = +20.1 / AZ = 336°9

    I have attempted to understand why our results show such a significant difference.

    I immediately suspected that I first should investigate our mutual determinations of the Moon Parallax and/or Augmented Semi-Diameter.

    For the Augmented Semi-Diameter, there is a simple formula which is accurate to +/- 0.05’:

    AUGMENTED SD = GEOCENTRIC SD * (1 + sin HP * sin H topocentric).

    With GEOCENTRIC SD = 14.8’, and with HP = 54’4 and H topocentric = 62°, we get SD Augmented = 15’0, which is exactly the value you have used.

    Let us now take a closer look at the Parallax.

    FIRST METHOD with the Nautical Almanac Moon Altitude Correction Tables (Re: the US N.A. for the Year 1982).

    Looking up through the Altitude Corrections Tables, it is not possible to "isolate" the effect of the Parallax itself. Nonetheless the sums of “Parallax + Aug. SD + Refraction” corrections are given in the Tables. From your data, we can easily compute that the Sextant observed altitude with no Index Error and with refraction would be 61°46’3 (LL). From this we derive a “Parallax + Aug. SD + Refraction” equal to 40.0’. If we remove the Refraction correction, we see that the “Parallax + Aug. SD” correction is equal to 40.5'.

    If we use the French Ephémérides Nautiques (Year 1982 Edition) which carry different tables (and these are not so “smart” as the US ones when it comes to using them), we get for the “Parallax + Aug. SD” Correction a value of 40’6.

    Starting with “Parallax + Aug. SD” Correction = + 40.5’ and since we know that Aug. SD = + 15.0’, we conclude that the Parallax Correction = + 25’5.

    Since you have used a Parallax Correction equal to + 25’8, I think that we now have a good explanation to the most significant part of our results difference.


    Over the past 30+ years, I have in-depth tackled the Moon Parallax computations 6 or 7 times and I have worked (hard sometimes) in order to accurately (to +/- 0’1) compute the Moon Parallax Correction from the Earth Ellipsoid. Full accuracy results definitely require 3D computations – a must for artificial satellites applications - , and I have eventually implemented such 3D computations and tested them fully OK against professional Astronomers Occultation Programs (Moon shape forced into being a sphere). For our current case of SEP. 01st, 2013 I have found that the “Parallax + Aug. SD” Correction is equal to 40’513, which resides between the US NA (40.5’) and the French NA (40.6’) results.

    In February 2011 (RE/ the thread starting with “http://fer3.com/arc/m2.aspx/ACCURATE-PARALLAX-COMPUTATION-Cou%C3%ABtte-feb-2011-g15696 “) I have published a solid 2D alternate computation method, which for this example yields Aug. SD = 15’035, Par. = 25’527, and “Parallax + Aug. SD” = 40’562. This result is also between the US NA 40.5’ and the French NA 40.6’ values. This 2D result is 0.05’ from the true 3D value (40’513).

    Also, I have observed that the overall accuracy of this 2D algorithm is always better than 0.1’ when compared to fully accurate 3D algorithms.


    I then have come to the following conclusion:

    ACCURATELY COMPUTING (to +/-0.1’) THE MOON PARALLAX AND SEMI-DIAMETER CORRECTIONS UNDER ALL CASES requires taking in account the individual Observer’s Latitudes (which directly influence their various distances from the Earth Center). For this reason, all Nautical Almanac Moon Altitudes Correction Tables show unavoidable errors exceeding 0’3 under extreme cases. Such errors are reduced into +/-0’2 if the underlying Computations are carried out for “the optimum averaged” Latitude of 30° (N or S), for which the Distance of the Sea Surface Navigator to the Earth Center is equal to .9991671 Earth Radius.

    That’s all folks!


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