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    Re: Rocky Mountain Lunar Distance
    From: Arthur Pearson
    Date: 2002 Dec 16, 21:57 -0500

    As usual, George has good comments and questions. Here are my responses:
    1) The bubble horizon I have is the practice model sold by Celestaire.
    It is a simple plastic tube with no magnification. There is a reference
    line in the view, and the trick is to bring the body down to the
    reference line while bringing the reference line into the center of a
    bubble. I wouldn't doubt that mounted on a tripod or locked in place
    somehow, the instrument has accuracy within 2' per David's comment.
    However, held in the hand I find it is a real challenge to bring all the
    elements together at the same instant. I followed the instructions
    Celestaire provides once and found an installed error of -2.3', but you
    need a true horizon to find the error, so it is not something that can
    be checked when you really need it. That said, it is inexpensive and
    completely satisfactory for a quick experiment like this. If one wants a
    greater accuracy, I would think a reflecting horizon would be the best
    alternative, but I have never tried one.
    2) I measured the sun's altitude once about 2 minutes after the last
    lunar. I did not measure the moon's altitude. All my sights and times
    are recorded below if anyone wants to work with them.
    3) I never checked the wrist watch against accurate zone time, so I
    can't assess the accuracy of the time sight. One of the virtues of this
    method is the watch only had to keep accurate time over the period
    during which one takes the lunars and the time sight, about 10 minutes
    in this case. I think you could do this with a stop watch if you wanted
    to (I hope I will hear it if I am wrong in this).
    4) George is quite right about my casual selection of which sights to
    favor. When I take a simple average of the longitudes derived from the 4
    sights, I get 105� 14'W which is about 45' too far east.
    5) George's expression for LHA moon is more exact (LHA Moon =LHA Sun +
    GHA Moon - GHA Sun adding or taking off 360� where necessary). It is
    worth noting that while this was the easiest way for me to understand
    the calculation of LHA moon, the examples provided by Patterson to Lewis
    in his "Astronomical Notebook" take a different path to the same result.
    I found it easier to construct calculations by Patterson's method.
    Recall that by assuming longitude we have established a correction
    between local time (LT) and Greenwich Time (GT), and by our time sight,
    we find a correction from watch time (WT) to LT.  With WT of the lunar,
    we correct to estimated GT and take GHA and Dec of moon and sun from the
    almanac (in my case, I interpolated within the hour for GHA, and within
    the 12 hour period for Dec). LHA of either body is simply (GHA ~
    Longitude). I puzzled quite a while before accepting that this arrives
    at the same value as George's expression. The elegance of the method is
    in the linkage of assumed longitude and assumed GT. Preston provides a
    wonderful word picture of this in Note 23 on page 190 (page # refers to
    the page of the journal, not the length of the article!).
    6) On review, I can't find a longitude calculation by Thompson in the
    Gottfred article. I got the impression that he worked out longitude in
    the field because Gottfred transcribes field notes in which Thompson
    calculates altitudes, applies reverse parallax and refraction, and
    clears the distance. That is the hard part! It is easy to get to
    longitude from there (look in the tables, interpolate for GT, find the
    time difference from LT, convert to degrees of longitude). But
    Gottfred's language on this point is ambiguous:
    "Thompson also computed longitudes from his knowledge of Greenwich and
    Local Apparent Times, set his watches to local apparent time by
    observing the sun or other stars, and computed the magnetic variation at
    his locale.
    To demonstrate how these values were determined, I will use a
    hypothetical case (since Thompson leaves us no calculations) using the
    data from November 3, 1810."
    Did he or didn't he? I can't tell.
    Lastly, I did have a great time on the ski trip but prefer to stay
    closer to sea level in slightly warmer surroundings. In early January,
    my sextant and I are headed for a long anticipated cruise through the
    Grenadines, and as the moon will be "in distance", I hope to make the
    best of the opportunity. This time I'll leave the bubble horizon at
    NOON SIGHT (IC = -3.0')
    Hs              Approx ZT 11:55:00
    27� 29.0'   11:55:00
    LUNAR DISTANCES (IC = -3.0')
    Ds          Watch Time
    58� 9.4'    14:30:40
    58� 9.4     14:31:55
    58� 12.6'   14:34:18
    58� 13.9'   14:36:23
    SUN SIGHT FOR TIME (IC = -3.0')
    Hs          Watch Time
    16� 21.2'   14:39:44
    -----Original Message-----
    From: Navigation Mailing List
    Sent: Monday, December 16, 2002 2:32 PM
    Subject: Re: Rocky Mountain Lunar Distance
    Well done, Arthur Pearson!
    Instead of sitting in an armchair arguing it out (I plead guilty to
    you have gone out and tried it. And given a good coherent account of it
    the rest of us to follow. Thank you, Arthur.
    It's left me with a few assorted questions and comments.
    1.I've never used a "bubble horizon" with a sextant, and wonder what
    performance can be expected from it on land. Is Arthur's experience of
    bubble horizon being "a real challenge" a common one, and does an error
    7 minutes strike other users as the sort of accuracy (or inaccuracy) one
    has to expect? For measuring such on-land altitudes, would an aircraft
    bubble-sextant have done better?
    2. If the afternoon Sun altitude had been measured halfway through the
    set-of-four lunar distances, then this could have been treated as an
    observed altitude (which it was) rather than an altitude to be
    Did Arthur also observe a Moon altitude at or near that time? I
    that he was demonstrating the art of calculated altitudes, rather than
    mmeasured ones (and has done it well). It would be interesting to see
    3. On returning to civilisation, did Arthur check his wristwatch against
    Zone time, to retrospectively confirm the time-accuracy of his afternoon
    sun observation? Just for cross-checking. It depends, of course, on how
    reliable was the timekeeping of his wristwatch, over that interval.
    4. I hope Arthur doesn't mind too much if I quibble somewhat about the
    he treated his four lunar-distance observations. His rejection of the
    two (because although more than a minute apart they gave identical
    would not find favour in a science lab. The lunar distance changes only
    about half-a-degree in an hour, or in a minute of time by about 0.5
    arc-minutes (maybe somewhat less if parallactic retardation is
    significant). Even if the error of each observation was as little as
    arc-minutes, it would be quite unsurprising to find two observations, 1
    minute of time apart, giving exactly the same lunar distance. Arthur
    should, I suggest, plot out all four observations against time, and draw
    best-line between them by eye, with a slope that he can work out in
    advance. That line will probably steer a path roughly midway between
    first two observations. It would be easier for us to check on this if
    the raw data was provided.
    If this plot was made, and all four observations taken account of in
    way, my guess is that the resulting longitude would come within 45
    or so of the true value, which wouldn't be a bad value for a lunar,
    especially at first attempt. Captain Cook would have been pleased with
    5. Arthur quotes the expression for LHA Moon as-
    (LHA Sun +/- difference between GHA Sun and GHA Moon)
    but the ambiguities in this could cause trouble and it should be
    more exactly as-
    LHA Moon =LHA Sun + GHA Moon - GHA Sun (adding or taking off 360� where
    GHA values have been provided the nautical almanac since 1952. Earlier
    almanacs gave Right Ascensions (RA) instead, which were in terms of time
    rather than angle, and measured Eastwards rather than Westwards, from a
    different base. Because of that, LHA Moon was  derived from-
    LHA Moon = LHA Sun + 15*( RA Sun - RA Moon) where the 15 is to convert
    in hours to degrees, and the subtraction is the other way round.
    6. Arthur says-
    >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
    >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.
    Well, I have read Jeff Gottfred's account, and it that he quotes no
    longitudes obtained by Thompson. Thompson may indeed have calculated
    longitudes in the field, but I can't find that in Gottfred's paper. I
    have missed something, though.
    I agree with Arthur's conclusions about insensitivity to errors, but
    out that he has demonstrated this just for one particular configuration
    Moon and Sun, and it will not be true to quite the same extent with
    differing geometries.
    I do hope that Arthur enjoyed the skiing part of his trip, and that he
    take his observing gear with him next time too, though it can't be much
    lugging a sextant about on skis.
    George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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