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    Rocky Mountain Lunar Distance
    From: Arthur Pearson
    Date: 2002 Dec 14, 15:39 -0500

    Inspired by recent threads on this list, and by additional reading about
    Lewis and Clark and David Thompson, I decided to test the early 1800�s
    methodology for determining position on a recent ski tour in the
    Colorado Rockies.  With the help of various postings to this list and
    some off-list information from George Huxtable, I constructed a
    spreadsheet to make the calculations, loaded it on my palm pilot, and
    headed off toward the continental divide with a sextant, a cheap bubble
    horizon and common wristwatch set roughly to zone time.  Apart from
    using modern almanac data and a spreadsheet to speed the calculations, I
    followed exactly the procedures used by the early explorers. With some
    good luck, I got some good results.  What is more remarkable is how
    forgiving the procedure is toward flawed assumptions and even sloppy sun
    sights that lead up to the lunar distance observation.  In short, it is
    a wonderfully robust procedure for the rough conditions of its early
    practitioners, and not a bad procedure for today if you can make the
    calculations conveniently.
    
    The procedure I used is essentially the �old timers� method we discussed
    in �Use of Sun Sights for Local time, and Lunars for Longitude�
    (http://www.i-DEADLINK-com/lists/navigation/0210/0069.html) wherein longitude
    is obtained by: 1) noon sight for latitude; 2) afternoon or morning sun
    sight to determine local time (LT); 3) lunar distance to determine
    Greenwich time (GT); 4) longitude taken as the difference between LT and
    GT. The procedure also relies on the insights of �Calculated Altitudes
    for Lunars� (http://www.i-DEADLINK-com/lists/navigation/0210/0172.html) where
    we agreed that with the proper procedure, calculated altitudes for the
    sun and moon could be used to clear the lunar distance with sufficient
    accuracy to get a good longitude on the first iteration.
    
    The best documentation I have found on the use of these methods concerns
    Lewis and Clark (1803-1806) and David Thompson (1790-1812).  Richard
    Preston�s excellent article on Lewis and Clark has been mentioned on
    this list before and is available at
    http://www.aps-pub.com/proceedings/jun00/Preston.pdf.  He provides an
    outline of their procedures and a full discussion of the instructions
    provided by astronomer Robert Patterson in the �Astronomical Notebook�
    which Lewis carried on his journey.  George Huxtable kindly supplied an
    email transcript of the �Astronomical Notebook� which provided detailed
    examples of the procedures including the proper method for calculating
    altitudes.
    
    David Thompson explored western Canada and his navigational procedures
    are documented by J. Gottfred at
    http://www.northwestjournal.ca/dtnav.html.  Thompson�s use of calculated
    altitudes is elaborately reconstructed by Gottfried who provides a
    comprehensive set of diagrams and trigonometric formulas in explanation
    of the technique.  It is interesting to note that Thompson worked his
    own sights to a full solution in the field. L&C only recorded their
    observations and then turned them over to Ferdinand Hassler, a
    mathematician at West Point who spent 10 years working them before
    giving up in frustration. Preston suggests that his inability to work
    the data may have been in part due to the controversy and confusion
    surrounding the calculated altitude method.
    
    My spreadsheet for calculation follows Preston�s description of the
    procedure. The details and any errors are my own:
    1. Assume a longitude. This determines a corresponding estimate of the
    GT of Local Apparent Noon (LAN) and an estimated correction from LT to
    GT.
    2. Take a noon sight at LAN. From estimated GT of LAN, estimate
    declination sun and calculate latitude.
    3. Take an afternoon sun sight and note watch time (WT).
    4. Assume WT is roughly equal LT, determine estimated GT of afternoon
    sun sight.
    5. With latitude and altitude in hand, estimate declination sun,
    calculate LHA sun and convert to time to determine actual LT of sight.
    6. Determine correction to WT to arrive at LT.
    7. Take a lunar distance and note WT.
    8. Correct WT to LT using correction determined in step 6. Correct LT to
    estimated GT of lunar distance.
    9. Take latitude, estimated declination of sun, convert time of lunar
    since LAN to LHA of the sun, and calculate altitude of sun.
    10. Take latitude, estimated declination of moon, find LHA moon as (LHA
    sun +/- difference between GHA sun and GHA moon), and calculate altitude
    of moon.
    11. Clear the lunar distance.
    12. Determine actual GT.
    13. Convert difference between GT and LT to longitude.
    The description above glosses over some complications of modern almanac
    data and the equation of time, but the framework is complete.
    
    My test of the method took place Dec. 8, 2002 near Breckenridge,
    Colorado. I skied with friends up a trail that threaded toward the
    continental divide to a high point at about 10,500 feet (3,200 meters).
    While the sun was burning through the high cirrus clouds, the moon was
    obscured and the prospects looked bleak. We reached a meadow just before
    noon and I took the LAN site.  I find my bubble horizon to be a real
    challenge, and in the excitement of the moment and the sharing of the
    "view" with my mountaineer pals, my sight was 7' off compared to the
    latitude on the topographic map. Not a good start, but not an impediment
    to a reasonable longitude as we shall see.
    
    We continued on working east and about a 1/2 mile south toward the
    divide before emerging into a clearing just over two hours later. The
    clouds cleared and the quarter moon emerged into a bright, blue sky.
    Pressed to begin the return trip, I took only four distances with a
    standard scope (Ds=~58�), switched to the bubble horizon and got just
    one altitude of the falling sun (Hs =~16�).  I can't judge the accuracy
    of the afternoon sun as there was no authoritative local time to compare
    it to.  As we shall see, any inaccuracy in this sight has a limited
    effect on the determination of longitude.
    
    I worked out my results for the location of the lunar distance
    observations, a spot known as Bakers Tank which is shown at
    
    N 39� 26.5'
    W 105� 59.8'
    
    on the USGS topo maps. As stated, my noon latitude sight was 7' off, and
    thus my observed latitude, adjusted for DR between lunch and Bakers
    Tank, was 7' too far north. This flawed latitude became an input into
    both the sun sight for LT, and the clearing of the lunar. I retained the
    error in my subsequent calculations to assess the accuracy of the entire
    sequence.
    
    My first two lunars were suspect as I got identical angles measured more
    than a minute apart. Solved individually, they yielded longitudes 80'
    and 90' too far east (very poor compared to Preston's calculation of
    L&C's results). The latter two distances seemed more appropriate in
    their distribution. Graphing the sights and plotting slope of calculated
    Da for the approximate hour favored the last two. They produced
    longitudes within 16' and 3' of truth respectively. Accepting the flawed
    latitude and averaging the two latter distances, we get a fix of:
    
    N 39� 33.5'
    W 105� 53.0'
    
    My luck, skill or inconsistency is less interesting than an analysis of
    the tolerance of this procedure to flawed assumptions and early errors.
    As my calculations are done in a spreadsheet, I have the luxury of
    easily recalculating the results to test their sensitivity to changes in
    the inputs.
    
    As usual, the lunar itself must be of the highest precision. A 1' error
    in the measured distance shifts the derived longitude of Bakers Tank
    about 30' (greater distance shifts to the west, lesser to the east).
    There is no escape from the rigors of lunar precision.
    
    However, one need not start with an accurate estimate of longitude. My
    initial calculation assumed W 110�, about 185 miles west of the truth.
    After the first iteration, I shifted my assumed longitude to W 106�.
    The second iteration longitude shifted a mere 0.1' west. Swinging my
    assumed longitude from W 110� to W 100� (a 460 mile shift) shifts the
    result a whopping 0.2' west. The procedure is remarkable tolerant of
    error in the initial assumption of longitude.
    
    Error in the LAN sight has a greater but still modest impact. A 1' error
    in here shifts the derived longitude about 1.1'. Had I gotten my LAN
    exactly correct (correcting my 7' error) my derived longitude would have
    been about 8' further east. This is a greater error in longitude, but a
    lesser error in distance (we trade a 7 mile error in latitude for a 6
    mile error in longitude). Better to get the latitude right, but all is
    not lost if you don't.
    
    Error in the sun sight for time has a bit more leverage on longitude. A
    1' error here shifts the derived longitude about 1.3'. I have no way of
    knowing how well I did on this front.
    
    In summary, the potential contribution to error in longitude was roughly
    as follows for Bakers Tank on Dec. 8, 2002:
    1' error in lunar distance     = 30' error in longitude.
    1' error in sun sight for time = 1.3' error in longitude.
    1' error in LAN sight          = 1.1' error in longitude.
    5� error in assumed longitude  = 0.1' error in longitude.
    
    I think this test is a strong affirmation that the old time method for
    determining longitude, including the use of an assumed longitude and
    calculated altitudes based thereupon, is remarkably robust and utterly
    suitable for the use it was given by early explorers. The longitude
    assumption has remarkably little impact on the results even for the most
    disoriented explorer. Greater skill and a decent reflecting horizon
    would leave little excuse for the sun sights to be outside of a minute
    or two of accuracy. This eliminates any significant error in the
    calculations that clear the lunar distance. As is the case with other
    methods, the precision of the lunar is the all important determinant of
    accuracy.
    
    I would be most interested in any articles or books that document the
    use of these procedures by sea captains of the era as thoroughly as
    Preston and Gottfred document their use in the heart of the continent.
    
    
    

       
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