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    Nitpicking on Moon Height Corrections
    From: Antoine Couëtte
    Date: 2013 May 29, 06:11 -0700

    RE : http://fer3.com/arc/m2.aspx/USCG-Student-Sample-Problems-1-2-3-Couëtte-apr-2013-g23692


    Hello to all,


    Just for your information and in further reference to my post dated April 25th, 2013, I wish to share a few detailed results on the computation of the Moon Parallax and Augmented Semi-Diameter (MPASD) Corrections for Celestial Navigation purposes.

    There are indeed a few "extreme cases" when - for identical recorded Sextant Heights – MPASD Corrections will significantly vary depending on the Moon Declination and the Observer's Position. In other words and for some very same Sextant Heights (corrected for instrument error with Dip correction equal to zero), the Geocentric Heights (computed as a result of applying Refraction, then Augmented Semi-Diameter and finally Parallax) will significantly vary depending on such geometrical position factors.


    *******

    If one requires the utmost achievable accuracy, then all computations are to be made in 3 dimension space. On an Ellipsoid and under such case 2 dimension space computations are not accurate enough since the Local Observer’s vertical (plumb) line generally does not cross the Earth Center.


    All corrections are computed with the following data. Earth Ellipsoid is WGS84 (R=6,378,137 m and f = 1/298.257), and the Moon is assumed to be a perfect Sphere with R = 1,730,000 m.

    Height of eye is assumed to be zero (i.e. Horizon Dip = 0.000’) and atmospheric conditions are supposed to be standard, i.e. P = 1013.25 mb (or 29.92 “ HG) and T = 10°C (283.15°K , or 50°F).

    Note: I am listing hereafter the Moon Apparent Equatorial and Local Coordinates as well as the MASPD Correction values with a great number of digits only for cross-checking purposes so that whoever will use the very same starting data should obtain the very same end results. Suffice it to state that the Moon Apparent Equatorial Coordinates quoted here are accurate to +/- 4" , which renders them fully fit for Celestial Navigation.

    On Sep 29th 1987, with TT-UT = +55.7s , the Moon Center Coordinates were :
    RA = 18h01m47.04s Dec = S -28°43’23.35” HP = 58.824’ , and
    with GHA Arietis = 005°19’372, then Moon GHA = 094°52’603

    From Position S-24°04’6 W094°52’6, at UT1 = 23h48m29.0s on Sep 29th 1987, Navigator #1 observes the MOON LL at H = 85°00.0’. He gets: Intercept = 0.0 NM with Azimuth = 180.0°

    His detailed 3D Space MPASD Corrections are as follows:
    Refraction = -0.085’
    Augmented Semi-Diameter = 16.306’
    Parallax = +4.994’
    Augmented Semi-Diameter + Parallax = 21.300’
    Refraction + Augmented Semi-Diameter + Parallax = 21.215’
    Geocentric observed height = 85°21.215’

    From Position S-33°22’5 W094°52’6 at the very same time UT1 = 23h48m29.0s on Sep 29th 1987, Navigator #2 observes the MOON LL at H = 85°00.0’. He also gets: Intercept = 0.0 NM with Azimuth = 180.0°

    His detailed 3D Space MPASD Corrections are as follows:
    Refraction = -0.085’
    Augmented Semi-Diameter = 16.306’
    Parallax = +4.665’
    Augmented Semi-Diameter + Parallax = 20.971’
    Refraction + Augmented Semi-Diameter + Parallax = 20.886’
    Geocentric observed height = 85°20.886’

    What is interesting to observe here is that from 2 different positions, two identical Sextant Heights yield Geocentric Heights which may differ by up to an amount which can slightly exceed 0.3’ mainly due to the differences in the Parallax Corrections.

    *******
    2D SPACE COMPUTATIONS RESULTS

    NAUTICAL ALMANAC RESULTS

    The US/UK NA (1983) gives a 2D space Correction equal to 20.9’ (i.e. 15.6’ + 5.3’) which is perfect for Navigator #2 but which differs from 3D result by 0.3’for Navigator #1.

    The French Ephémérides Nautiques (1981) yield a 2D space Correction equal to 21.2’ which is perfect for Navigator #1 but which differs from 3D space result by 0.3’for Navigator #2.

    The US Navy site ( http://aa.usno.navy.mil/data/docs/celnavtable.php )yields the following Correction of for Navigator #1:

    Celestial Navigation Data for 1987 Sep 29 at 23:48:29 UT

    For Assumed Position: Latitude S 24 04.6
    Longitude W 94 52.6

    Almanac Data | Altitude Corrections
    Object GHA Dec Hc Zn | Refr SD PA Sum
    o ' o ' o ' o | ' ' ' '
    SUN 179 32.6 S 2 30.0 + 5 53.3 269.9 | -8.8 16.0 0.1 7.3
    MOON 94 52.7 S28 43.4 +85 21.2 180.0 | -0.1 16.3 4.8 21.0


    , which gives an accurate value (85°21’2) for Hc corresponding to a Sextant Height equal 85°00.2’ , hence a 0.2’ difference with our 3D space results. And

    The US Navy site yields the following Correction of for Navigator #2:

    Celestial Navigation Data for 1987 Sep 29 at 23:48:29 UT

    For Assumed Position: Latitude S 33 22.5
    Longitude W 94 52.6

    Almanac Data | Altitude Corrections
    Object GHA Dec Hc Zn | Refr SD PA Sum
    o ' o ' o ' o | ' ' ' '
    SUN 179 32.6 S 2 30.0 + 5 49.7 270.8 | -8.9 16.0 0.1 7.2
    MOON 94 52.7 S28 43.4 +85 20.9 360.0 | -0.1 16.3 4.9 21.1


    , which again gives an accurate value (85°20.9’) for Hc, and corresponding to a Sextant Height equal to 84)59.8’, hence a 0.2’ difference (in the opposite direction, but still towards the Moon) with our 3D space results.

    We can therefore conclude that the algorithm used by the USNO server is most probably NOT a 3D space algorithm, but most probably a 2D space algorithm (of the kind listed hereafter).

    A 2D space direct computation taking in account the Observer’s Latitude (as it influences Distance from the Earth Center) such as the one described in “ http://fer3.com/arc/m2.aspx/ACCURATE-PARALLAX-COMPUTATION-Couëtte-feb-2011-g15696 ” gives the following Corrections:

    Navigator #1
    Refraction = -0.085’
    Augmented Semi-Diameter = 16.306’
    Parallax = +4.847’
    Augmented Semi-Diameter + Parallax = 21.154’
    Refraction + Augmented Semi-Diameter + Parallax = 21.069’
    Geocentric observed height = 85°21.069’ which differs by 0’15 from 3D space result.
    and,

    Navigator #2
    Refraction = -0.085’
    Augmented Semi-Diameter = 16.306’
    Parallax = +4.845’
    Augmented Semi-Diameter + Parallax = 21.151’
    Refraction + Augmented Semi-Diameter + Parallax = 21.067’
    Geocentric observed height = 85°21.067’which differs by 0’2 from 3D space result.

    And finally a “brute force” 2D space straight (no loop) computation performed for the average Latitude of 30° (N or S) yields the following result for both Navigator #1 Navigator #2:

    Navigators #1 and #2
    Refraction = -0.085’
    Augmented Semi-Diameter = 16.302’
    Parallax = +4.846’
    Augmented Semi-Diameter + Parallax = 21.148’
    Refraction + Augmented Semi-Diameter + Parallax = 21.063’
    Geocentric observed height = 85°21.063’which also differs by up to 0’2 from 3D space result.

    OVERALL CONCLUSIONS

    Although it has very little (if any) consequences in Classical Celnav, under extreme cases – i.e. with MOON extreme declinations and Sextant heights close to 90° - the Moon Height Corrections do not only depend upon our well known traditional parameters (parallax, height and atmospheric conditions), but also to a limited nonetheless significant extent upon the Observer’s position and upon the Moon Declination, with an overall cumulated amplitude which can exceed 0.3’ for these latter 2 variables.

    The earlier Nautical Almanacs correction tables (US/UK NA 1983, and French Ephémérides Nautiques 1982) show errors reaching up to 0.3’ under such “extreme” cases.

    The more recent USNO data server as well as 2D space computations restrict such errors to less than 0.2’, an amount which is about the best any 2D space computation can achieve.

    3D space computations are necessary in order to achieve the highest accuracy, even for the Moon if claiming 0.1’ accuracy for its Heights corrections. 3D space computations are also a must for bodies with much higher parallax (such as satellites). For satellites, accurate height reductions also requires a lot of additional refinements, including an accurate method to reduce the refraction induced parallax for close bodies (see André DANJON’S excellent “ASTRONOMIE GENERALE” Chapter IX § 79).

    I am fully aware that these results are more a curiosity than anything else for our daily Celnav routines. I would still rank them into the “nice to know” list.

    Last but not least, I would be grateful should any of you could bring an independent confirmation to the 3D space results I have indicated here (Paul? or any other NavList Member?).

    Thanks again for your very Patient and Kind Understanding

    Antoine M. “Kermit” Couëtte

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