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    Re: Grenadine Lunar Distances
    From: Arthur Pearson
    Date: 2003 Feb 1, 13:09 -0500

    My experience is that the altitudes do make a difference, and I suspect
    the way I handled them may have introduced some error. So I welcome any
    critique of my method which was as follows:
    I took altitudes of sun and moon before and after the lunar series (what
    is listed below are the raw data).  I created a spreadsheet to make the
    calculations. The spreadsheet requires altitudes (Hs) of sun and moon
    that are simultaneous to the selected lunar. The limb of either body can
    be upper or lower, and the reduction assumes they both were taken from
    the same height of eye (HoE) for the dip correction. The peculiarities
    of my altitudes required some manipulations in advance:
    * My moon altitudes were upper limb before but lower limb after. Using
    CelestNav, I found Ho for the LOWER limb observation, then by trial and
    error I found an Hs UPPER limb that produced the same Ho.  That gave me
    Hs for the UPPER limb both before and after. I interpolated linearly
    between them to get an Hs simultaneous to the selected Ds.
    * My second sun altitude came down on a small island so I had to reduce
    it with dip short. I used 15.2' for 8 foot HoE with shoreline 0.3nm
    away. As above, I found Ho for the altitude, then worked back to an
    equivalent Hs that when reduced with the standard dip for 8 foot HoE
    gave the same Ho. That gave me comparable altitudes before and after,
    and I interpolated between them to get Hs sun simultaneous to selected
    * For the record, the simultaneous altitudes I got for my selected Ds
    For GMT = 19:42:04
    Hs Sun = 27� 23.1' LL
    Hs Moon = 65� 29.3' UL
    HoE = 8 feet
    The problems I suspect could affect the results are:
    * Linear interpolation assumes a linear change in altitude which isn't
    strictly the case.  I took about 32 minutes between before and after
    altitudes. The moon was approaching its meridian passage. I am not sure
    how this might bias the results or by how much. I suspect (hope) that it
    is not significant.
    * The manipulations I went through before interpolating may have errors
    in logic or execution. Again, I suspect (hope) that they are not
    I am delighted to hear any comments, critiques or suggestions.
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of William Allen
    Sent: Saturday, February 01, 2003 1:37 AM
    Subject: Re: Grenadine Lunar Distances
    Thanks for sharing your data with us.
    Let me ask a question, though.  I see you have taken a series of Ds
    sights over a period of about 17 minutes.  You chose one as your best
    using graphs.  But then you took the heights of the moon and the sun
    about 10 and 13 minutes after the time of your selected Ds?
    I would have thought that you would have needed to get the moon and sun
    heights at a time that was as close as possible to your Ds observation?
    Or take the height of the moon and sun at the beginning of your Ds run
    and then again at the end and average the two?
    Maybe this doesn't make that big a difference, I don't know, but I would
    be curious what you and others think about that.
    Bill Allen
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of Arthur Pearson
    Sent: Friday, January 31, 2003 8:48 AM
    Subject: Grenadine Lunar Distances
    Ladies and Gentlemen:
    In early January I had the opportunity to take two sets of lunar
    distance observations while on a 40' bareboat charter in the Grenadines
    in the southeastern Caribbean.  Having only taken lunars on land
    previously, it was a great fun wrestling with the practical
    considerations of taking lunars more or less at sea.  This posting will
    share the data for those who would like to work with them and offer some
    comments on the practicalities.
    My first observations composed a complete series intended to find
    latitude and longitude by the "1800 method".  I took a local apparent
    noon sight (LAN) for latitude.  About 3 hours later, I took sun and moon
    altitudes before and after a series of 7 lunar distances.  The first sun
    sight was used to determine local apparent time (LAT) which was
    converted to local mean time (LMT).  I graphed the lunars and selected
    the fourth for solution.  Interpolating the altitudes to the time of the
    selected lunar, I cleared the lunar and solved for GMT per lunar. The
    difference between GMT and LMT was converted to longitude per lunar.
    Having GMT and exact position from a handheld GPS, we have a gauge to
    judge my accuracy.  The data and my solutions are as follows:
    Date: Jan. 7, 2003
    Index Correction: -1.1'
    LAN Sight:
    Height of Eye: 10 feet
    Hs LAN: 54� 37.3'
    Latitude per LAN: 12� 49.0'N
    Altitudes Before Lunar:
    Height of Eye: 8 feet
          GMT        Hs
    Sun   19:18.00   32� 15.0' Lower limb
    Moon  19:21:21   63� 24.0' Upper limb
    GMT        Ds
    19:33:27   59� 33.2'
    19:35:08   59� 34.0'
    19:40:22   59� 34.8'
    19:42:04   59� 35.4' (after graphing all 7, I used this one)
    19:45:17   59� 36.6'
    19:48:07   59� 36.0'
    19:50:24   59� 37.5'
    Altitudes After Lunar:
          GMT        Hs
    Sun   19:52.39   25� 27.2' Lower limb, dip short at 0.3 nautical miles
    Moon  19:55:15   66� 18.6' Lower limb
    "DR" Latitude of Lunar: 12� 37.2'N (see explanation below)
    LMT of Lunar per 1st sun sight: 15:36:22 (=LMT sun sight + time elapsed
    GMT per Lunar:  19:44:55
    Longitude per lunar: 62� 8.2'W
    Time Error of Lunar:     2m 51sec fast
    Distance Error of Lunar: 1.3' too long
    Longitude Error:         47' too far west
    Total Error in position: 46 nm including 1 mile error in latitude
    Position per GPS:
    12� 38.2'N
    61� 21.2'W
    This LAN sight was taken on a broad reach in 20-25 knots in six foot
    seas, common conditions between the islands of the Grenadines at this
    time of year.  The error was 1 nm compared to GPS. In the course of the
    afternoon we took a varied course around and between islands to our
    anchorage behind the reef of the Tobago Cays. A strict DR was not
    practical, so for the "DR" latitude for the lunar, I simply applied the
    1nm error in my LAN sight to the latitude per GPS of my lunar
    The lunar was taken in the very rolly anchorage behind the reef. Not
    open ocean, but definitely a lot of motion.  The moon happened to be
    nearing its highest point, providing several challenges.  It was very
    difficult to get the sun over to the moon, and the high altitude made
    for a very awkward posture when I did. This sight was a prolonged
    struggle; I tried several postures but found nothing comfortable.  The
    first moon altitude used the upper limb, the second one used the lower
    limb because that looked best at the time (a mistake; I latter
    calculated that the moon had not yet reached the meridian by the second
    lunar). This made interpolating between the moon altitudes cumbersome
    and may have introduced error by measuring an incomplete limb.  Finally,
    the second sun sight came down against one of the Cays, so I had to
    correct for dip short for HOE 8ft, distance of 0.3 nm, which I calculate
    to be 15.2'.
    The high moon also brought George's parallactic retardation into play; I
    calculated the hourly change in apparent distance as only 14.5' which is
    quite slow.  All in all, I am pleased with how well it came out given
    the difficulties.  I also solved this lunar using the "Lewis & Clark"
    method by which I calculated the altitudes based on latitude per LAN, an
    assumed longitude, and the LHA of the bodies based on the afternoon sun
    sight.  This method came in only 31' too far west, an improvement of
    16'. As I used the same distance observation, the difference can only be
    attributed to using different altitudes to clear the distance
    (calculated vs. observed). Perhaps I made errors in the unusual
    corrections that were needed for the observed altitudes. In any case, it
    is another demonstration of the robustness and efficiency of the "Lewis
    & Clark" methodology.
    The second set of observations was a series of lunar distances alone. I
    cleared and reduced to GMT based on calculated altitudes that were in
    turn based on known GMT and position.  There is no point to this process
    other than improving one's lunar technique.  This set was taken hard on
    the wind in 25 knots and 6 foot seas just south of Bequia. My data and
    results are as follows:
    Date: Jan. 10, 2003
    Index correction: -1.1'
    GMT        Ds
    18:29:25   92� 28.4'
    18:32:31   92� 29.0'
    18:36:06   92� 30.8'
    18:38:47   92� 31.1' (after graphing, I used this one)
    18:43:44   92� 33.8'
    18:46:13   92� 33.4'
    Position per GPS at time of selected distance:
    12� 56.9' N
    61� 16.7' W
    GMT per Lunar: 18:42:59
    Time error of lunar: 5m 12sec fast
    Distance error of lunar: 2.3' too long
    With a much lower moon (Hs~17�) and thus a much faster rate of change in
    apparent distance (21' per hour), this was a better time to take a
    distance than the high moon of the first set.  Although a longer
    distance, the posture was much more comfortable.  I was braced solidly
    in the companion way and could pivot comfortably from the waste up.
    However, the motion of the boat had the sun and moon dancing
    boisterously around in the scope.  I felt I got a good look at a couple
    of the contacts but struggled with the others.  The graph of the
    observations is more consistent than I would have guessed although I
    ended up pretty far off.  I would guess that had I worked the longitude
    as I did with the first set, I'd be off by over 80 miles. Practice,
    practice, practice... I suppose I'll just have to spend more time in the
    Had I been depending on these lunars at the end of a transatlantic
    passage to the Grenadines, the practical consequence would be a landfall
    7 to 15 hours later than anticipated. Late landfall might mean after
    dark when one was planning for daylight.  More unsettling would be
    arriving early in the dark had the error gone the other way.  Clearly a
    broad margin of error and extreme caution are required when depending on
    these techniques.  On the other hand, I would love to know the
    satisfaction of leaving the Canaries, crossing an ocean, and arriving
    within a few hours of ETA using only sun, moon and a sextant.
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