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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Errors in Cotter's book, updated
From: George Huxtable
Date: 2003 Jan 9, 01:00 +0000

```Fred Hebard said-

>George,
>
>It was good to hear more evidence that you're the curmudgeon I
>imagine you to be.

Fred has presented me with the curmudgeon's medal, and it's a badge I shall
wear with pride, and do my best to live up to.

Fred's table of Sun declination-changes came over OK on his email message

I think we can simplify that information, and just consider a worst-case,
at the equinox. Fred's data shows that at the Spring equinox the Sun is
crossing the equator and moving North at 23.7 minutes each day. This is
very nearly 1 minute in each hour, or 1 knot, to put it most simply.

When the Sun is exactly South of as observer, that is at apparent noon,
then its changes in Local Hour Angle do not affect its altitude, but at the
equinox an increasing altitude of 1 arc-minute per hour remains, due solely
to its change in declination. So at that moment, the Sun is not yet
"hanging" stationary in the sky, but is still climbing slowly. It will not
reach its maximum altitude, when it stops going up and starts going down,
until a bit later. What time will that be? It will be that moment when the
Sun's altitude caused by its change in Hour angle past noon is FALLING at 1
arc-minute per hour, to exactly balance the rise that the steady change in
declination causes. At a latitude of 45 deg., this will happen at about 15
seconds after noon.

If you were to make a plot of Sun altitude against time, then near noon you
would see something looking very like a parabola, the arc that a shell from
a gun follows. If the declination wasn't changing (at the solstice) then
the highest point on that parabola would be exactly at noon, and it would
be symmetrical about noon. But near the equinox, the peak would be about 15
seconds late, and in fact the whole parabola-shape would be shifted later
in time by about 15 seconds.

We have discussed at length on this list how impossible it is to determine
the moment-of-noon AT noon, because the Sun's movement has come to a
near-stop, and a perfectly valid alternative is to take the mid-time
between two moments when the Sun has equal altitudes, going up and going
down, preferably at times spaced well away from noon. It should be clear
now that in finding such a mid-time, you are drawing a horizontal line
across our plotted parabola, and finding its mid-point. And if the whole
parabola has been displaced 15 seconds later, then the mid-point of any
line across it will also be displaced to be 15 seconds late, no matter what
height the line was drawn at, however near to maximum altitude that was.
And that's the point we are trying to settle.

You may think that a displacement of noon from the moment of highest Sun,
by 15 seconds, is rather negligible, and so it may be. But the same effect
occurs when the vessel is travelling towards or away from the Sun. If the
vessel's North-South component of its speed is say 10 knots, that will give
rise to an error in the timimg of noon by 150 seconds at a lat of 45 deg.
This has to be added to the effect of declination change.

However and whenever you measure the moment of apparent noon, near to noon
or far from noon, this error in the timing of noon, due to the combined
North or South velocities of the ship and the Sun, has to be allowed for.
But note that this has no observable impact on the value of the maximum
altitude observed, so a latitude measurement is unaffected.

George Huxtable.

```
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