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    Re: longitude around noon (a twist)
    From: George Huxtable
    Date: 2008 Jun 4, 15:06 +0100

    Frank Reed wrote-
    
     I chatted in my email a bit about longitude around noon
    | and asked him this:
    | "Which leads to a question: is there an established name in the
    literature,
    | or even in your own jargon, for a fix resulting from a series of ten or
    | twelve sights taken over a relatively short period of time? I've been
    | calling it a "rapid-fire fix". Do you know another name?"
    |
    | His reply:
    | "I don't know of a special name.  You're correct, of course, if you can
    get
    | a bunch of sights on either side of noon, you can get good enough geometry
    | to get a 2-D position.  It works with the LOPs, too, in that they provide
    a
    | good spread of azimuth around then.  There is a slight catch, however, and
    | that is, the higher the Sun is in the sky (and therefore the more rapid
    the
    | altitude and azimuth change near noon) the more you have to worry about
    the
    | curvature of the LOPs.  In some near-degenerate cases (sun within several
    | degrees of the zenith), the usual straight-line plotting -- or math that
    | assumes straight-line LOPs -- may not provide the right fix."
    
    =================
    
    There appears to be some misunderstanding in that dialogue. How does that
    difficulty arise? I think it is quite illusory.
    
    Indeed, the higher the Sun is at noon, the simpler the picture gets, the
    easier it becomes to determine the moment of noon, and the less is the
    influence of any North-South velocity. It's because the whole process
    DEPENDS on that curvature, and the sharper the curvature, the easier it
    gets.
    
    As long as the Sun is within a few degrees of the vertical at noon, you can
    plot the whole event on a Mercator projection, in plane geometry. The Sun is
    travelling around a line of latitude at 900 knots or a bit less, the ship is
    travelling (say) South at 10 knots along a line of longitude, and the zenith
    distance in minutes is simply the distance between them in miles.
    
    Indeed, simplest case of all is if the so-called "degenerate" case where the
    ship crosses the Equator at the same moment that the Sun, travelling round
    the equator at 900 knots, passes that same spot. From the ship, it's a
    zenith noon. Until that moment the ship and the Sun have been approaching at
    a speed of 900 knots and 10 knots, which combine (quadratically) to 900.06,
    so the ship speed is having no effect. Sun altitude increases steadily at 15
    degrees per hour, until that moment. Then, instantly, after they pass, they
    separate at the same rate, so the altitude falls, again at 15 degrees per
    hour. The observer's difficult task is to about-face, from East to West, at
    that moment. But otherwise, determining the moment of noon becomes a doddle.
    
    That's because the plot of altitude against time has developed a sharp
    corner at noon. Its curvature is infinite. And the maximum occurs AT noon,
    whatever the ship speed was.
    
    For noons where the Sun is close to, but not at the vertical, the curve
    becomes less sharp near the top, and ship speed starts to need correcting
    for, but these high Suns are just the sort of observation for which Frank's
    procedure of longitude around noon is, indeed, completely appropriate.
    
    Frank continued-
    
    | That's a good point about sights very close to the zenith. I had mentioned
    | previously on the list that there may be a special case when the Sun is
    | close to the zenith. I still haven't thought through whether it really
    | screws up the graphical technique or merely requires more stringent rules
    | for its application.
    
    Neither. Quite the reverse, in fact.
    
    George.
    
    contact George Huxtable at george@huxtable.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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