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    GMT from Moon and Body Observed Altitudes...??
    From: Ron Jones
    Date: 2019 Feb 16, 15:46 -0800

    GMT from Moon & Body Observed Altitudes … What am I missing?

    Some time ago I started to try and understand the process of determining GMT by measuring the angular distance from the moon to another body.  My current understanding of the normal process  involves the following five steps:

    Step 1  Simultaneously measure, using a sextant, the altitudes of the moon (hsMoon) plus another celestial body (hsBody) and the angular distance between the moon & body (LD).  This requires the coordinated effort of three people with sextants or the taking a series of alternating Moon Body sextant altitudes & LD measurements, then using linear regression to determine LD and altitudes at a common point in time.

    Step2  Correct the LD measurement for the sextant Index Error & the semi diameters of the Moon & Body thus producing an apparent LD.

    Step 3 Using the atmospheric refraction & parallax in altitude associated with the sextant altitudes of  the Moon &  Body, correct the apparent LD to obtain the “cleared” LD.  This is easy if the Moon & Body are vertically aligned (near identical Zn values)   and  somewhat arduous if the Moon & Body have different azimuth (Zn) values.

    Step 4 Using GHA & Dec values from the Nautical Almanac and the spherical law of cosines for great circle distance, calculate the Geocentric Lunar Distance (GLD) from the Moon to the Body at the time of the sextant measurements. This usually accomplished by calculating GLD1 for the whole hour prior to the time of the sextant measurements and GLD2 for the whole hour following the time of the sextant measurements.  Then using  linear interpolation and the time increment past the whole hour of the sextant measurements, calculate the Geocentric Lunar Distance (GLD) for the common point in time of the sextant measurements.

    Spherical law of cosines for great circle distance:
                   D° = acos(sin(Lat1)sin(Lat2)+cos(Lat1)cos(Lat2)cos(Long1- Long2))
                   consider Dec equivalent to LAT and (GHAMoon – GHABody) equivalent to  (Long1- Long2)
                   Then   GLD
    ° = acos(sin(DecMoon)sin(DecBody)+cos(DecMoon)cos(Dec2)cos(GHAMoon – GHABody))

    Step 5. Using the difference between the “cleared” LD & the GLD compute GMT by linear interpolation using the time increment past the whole hour prior to the time of the sextant measurements.

    As an alternative to the above, consider the following, we still take a series of alternating Moon Body sextant altitudes and use linear regression to determine their sextant  altitudes (hs) at a common time (as was done in Step 1 above).  Next we follow the process of converting sextant altitude (hs) to apparent altitude (Ha) by applying sextant index correction and dip.  Then we convert apparent altitude (Ha) to observed aka true altitude (Ho) by applying atmospheric refraction, semi diameter and parallax in altitude.  We can now estimate the Geocentric Lunar Distance by again using the spherical law of cosines for great circle distance.  But now instead of the terrestrial coordinate system we use the celestial sphere coordinate system using our Zenith as the elevated pole and consider the observed altitudes (Ho) as equivalent to Lat and the difference in true azimuth directions of the moon & body (ZnMoon – ZnBody) as equivalent to (Long1- Long2).  Using the Spherical law of cosines for great circle distance we can write:
             True LD
    ° = acos(sin(HoMoon)sin(HoBody)+cos(HoMoon)cos(Ho2)cos(ZnMoon – ZnBody))
    Next
    using the time increment past the whole hour prior to the time of the sextant measurements and linear interpolation between GLD1 & GLD2 we can obtain the associated GMT for the True LD (same as Step 5 above). 

    Using this method we have eliminated the need to measure the angular distance between the moon & body.   

    See attached PDFs  

    So what am I missing? 

    Ron Jones

    File:
    Lunar-Distance-from-Moon-Body-Observed-Altitudes.pdf
    File:
    Moon-Body-Nautical-Almanac-Data.pdf
       
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