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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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