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If you are receiving this posting for the second time, please accept my
apologies. I'm struggling, to some extent, with the differences between my
new PC and my old Mac, and suffering from some finger-trouble. I thought I
had sent the following message to Nav-l around midnight last night, but
haven't had it reflected back to me by Nav-l, so here it is once again.

===============================

On 7 June, discussing the "Frank Reed proposal for determining latitude and
longitude around noon, and his method for "Northing correction to noon
latitude", I wrote-
 >
 >Nevertheless,
 >we seem to agree that choosing a different zero-point for the corrections
 >will not shift the timing of the corrected peak, which depends on the slope
 >of the corrections, but not their amount.
 >
 >Then I went on to-
 >>"However, it looks to me as if an error in
 >>that initial  presumption of noon would give rise to an error in the deduced
 >>maximum  altitude, and so in the latitude. Perhaps Frank will comment."
 >
 >Frank did, as follows-
 >
 >"Nope. No  error. See above."

And I replied-

 >However, I urge Frank to rethink his flippant dismissal of the point that I
 >have made. What's needed, to calculate latitude simply, is the Sun's
 >altitude AT MERIDIAN PASSAGE, and not at any other time. To obtain that,
 >Frank tells us to take the altitude from the peak value of the corrected
 >Sun-altitude curve, at his "folding" point, which will be at meridian
 >passage. But that's not the observed altitude, it's the corrected altitude,
 >at meridian passage. The correction that's been made to observed altitude,
 >at that moment, depends on how far it is away in time from the zero-point
 >of his corrections, and that zero-moment was chosen quite arbitrarily. Only
 >if the zero-point of the corrections happened to be at the moment of
 >meridian passage, would the peak of the corrected-altitude curve correspond
 >to the observed altitude at that moment.
 >
 >So I suggest that Frank's proposed method should be somewhat modified. Yes,
 >certainly, use the corrected-altitude curve to determine, from its
 >symmetry, the moment of meridian passage. But then, read off, corresponding
 >to that moment of meridian passage, the UNCORRECTED value of altitude,
 >which will NOT in general be its peak value.
 >
I later reminded him-

 >I haven't seen any response from Frank to this suggestion, and wonder if he
 >thinks that it is perhaps wrong, or negligible.

to which he replied-

"At this point, I think it's a minor detail, but I haven't devoted much
time to it (so maybe I'll change my mind later!). What little time I have
spent on this 'lat/lon at noon' business has been focused on refining the
small stuff and trying to reach a happy balance between economy of
paperwork and economy of paper (that is, I don't think there's advantage to
this method if it requires lots of tricky calculations, and at the same
time, there's not much advantage if it requires purchasing and carrying a
volume of tables)."

Which is fair enough.

=======================

Now, I have though a bit more deeply about that problem of determining
latitude, and have come to the conclusion that Frank and I are both right.
To determine the moment of noon,  (using the Frank Reed method) first
correct the series of measured altitudes, taking some arbitrary moment,
near noon, as the starting point at which the correction is zero. It has to
be that way, because at that stage you don't know the longitude, so can't
predict just when local noon will be. Then the centre of symmetry of the
resulting corrected curve gives the true moment of meridian passage.

Then you can EITHER read-off from the curve the UNCORRECTED altitude at
that moment of meridian passage,  which will not in general be at its peak
value, and which will allow the latitude AT NOON to be deduced. That was
what I suggested. OR, as Frank suggested, you can read off the peak value
of the CORRECTED altitudes, which will occur around the moment of meridian
passage, and this will give the latitude, not at noon, but at the arbitrary
moment which was chosen at which the correction was zero.

Looking back, that's what Frank was explaining, and I was rather slow to
cotton-on.

========================

On a related topic,  I wrote-

"There's another phenomenon that a navigator might not expect when sailing
in gusty/squally conditions in a North or South direction near noon. If his
vessel is speeding up in the puffs, then slowing in the lulls, then so will
his rate of change of Sun altitude due to those changes in speed. That
changing slope has to be added to the expected parabolic change in Sun
altitude caused by the Sun's transit through the meridian. And the end
result is a wavering of the curve of altitude with time, about its peak. In
an extreme case, the Sun could show more than one maximum altitude.

I don't suggest that that's going to be a cause of serious error. The
effects on the resultant wavering parabola can be averaged out, rather
well, by the proposed folding of the graph, particularly if the series of
measurements extends well away from noon"

And Frank responded-

"As long as you're considering more intricate scenarios like this one, it's
a good chance to point out the advantage of the simple altitude adjustment
technique that I've already outlined. Imagine a scenario where you're
sailing along with a nice breeze making 7 knots from the beginning of your
sight series twenty minutes before noon until right around noon when,
suddenly, the wind dies. Though you might need to think through a few
little details, by adjusting each sight individually, this case can be
handled easily. "

Well, the precise way to handle the altitude adjustment, which would
correct for such speed changes properly, would be to read the log at a time
close (but not demandingly close) to the moment of each altitude
observation; then take the differences in log reading, before and after
that arbitrary zero-moment. Those log differences, in miles, should be
multiplied by an adjustment factor amounting to cos (course angle), because
it's only the North-South component that is required. And that's the
correction to be made (in arc-minutes) to each observed altitude. I've
neglected any allowance for ocean current here.

It's actually the Northing in miles, for each observation, before and after
the arbitrary zero-point, that is required for a proper  correction.
Multiplying the estimated momentary speed by the time elapsed is an
inaccurate alternative. Nevertheless, it isn't going to make a big
difference to the overall result; I accept that.

===================================

As for my alternative suggestion,that instead of correcting each observed
altitude, one should apply a calculated difference between the times of
maximum altitude and meridian passage, based on estimated average Northing
speed . That correction amounts to
15.3(tan lat - tan dec) seconds of time, or 3.8(tan lat - tan dec) minutes
of longitude, for each knot of Northing. If the complexity of calculating
that multiplier (once for the day) is regarded as challenging, a compact
table relating it to lat and dec ( or perhaps, even simpler, to lat and
date), giving the multiplier to say 1%, could be readily provided.

George.
===============================================================
Contact George at george@huxtable.u-net.com ,or by phone +44 1865 820222,
or from within UK 01865 820222.
Or by post- George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13
5HX, UK.

   

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