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Fred Hebard recommended Bob Young to visit my earlier posting on the
accuracy of trying to measure longitude from an around-noon Sun
observertion.

That actually comprised two postings in the thread " Lat. and Lon at LAN"
on 8th Jan 2004, the second correcting a big numerical error that Fred had
uncovered in the first. Sorry about that.

Because of that error, perhaps it's best if I post an amended (and somewhat
expanded) version of that earlier message. Note that it reflects the
discussion that had taken place up to that date, 8th Jan, and not more
recent postings, which I have been reading but not analysing closely.

TIMED NOON SIGHTS FOR POSITION.

There are two problems in determining the longitude..

1. Accuracy in determining the time of maximum altitude.

Doug said, in a posting sent on on 7 Jan via Dick Savage-
>If the sight taker timed the noon sight within a few secounds of LAN(the
>Sun appears to >hang for 20-30 sec at LAN but still moves in that time
>frame to greater than 180* or less >than 360*)the sight taker has the
>vessel's longitude.

But the Sun doesn't simply "hang for 20-30 seconds", it hangs near its
unchanging maximum altitude for MUCH longer than that. Take, as an example.
a winter-solstice Sun observation from my own latitude of 51deg North.
Local Apparent Noon (LAN) is at 11h 57m 53sec GMT, when the Sun altitude is
at 15deg 33.7'. The Sun doesn't drop by 0.5' from that value until it is  5
minutes (in time) away from LAN, before or after.

How little a change of altitude can Doug reliably detect before he can say,
with confidence, that it's past the maximum and has started to descend
again? Perhaps, as an experienced observer, he might claim to detect that
0.5' change, but it would require a
crystal-sharp horizon to do so, I suspect. Perhaps he can do even better,
but it takes about two minutes before or after LAN for the Sun to drop by a
mere 0.1', and nobody is going to claim to improve on that.

So there's no way for an observer to time the noon sight within a few
seconds, as Doug claims. If the time of LAN could be determined within 5
minutes, then it would establish longitude within about 75 arc-minutes. Not
a great result. In the unlikely event of a timing precision to 2 minutes, a
longitude determination to 30 arc-minutes would result.

An observer can do much better by timing an altitude sometime before noon,
and timing the same altitude after noon, and splitting the difference
between those times. This is the age-old method of determining LAN by
equal-altitudes. And the more widespread that these two times depart from
Noon, the more accurate such a measurement will be. At noon itself is the
very worst moment to try to determine when LAN occurs. If the Sun appears
only at the moment of noon, and no other, it would be best to rely on
whatever DR information exists, rather than fool yourself by presuming you
can determine longitude at noon.

2. Time difference between moment of maximum altitude and LAN on a moving ship.

This second problem affects Doug's big-ships more than it does the small
vessels that most of us sail. I don't know the speed of Doug's ships, but
let's choose 20 knots as a round number. Let's say that he is steaming
South at 20 knots, toward the Sun, from 51deg N, at the exact LAN of 11 h
57m 53s GMT. A stationary observer with a sextant, on a raft that he
passes, will indeed see the Sun "hanging" with a constant maximum altitude
at that time. On his bridge, however, Doug WON'T see the Sun's altitude as
stationary, he will see it as increasing at 20' per hour, because he is
steaming toward the Sun at 20 knots, or 20' per hour. To him, the Sun won't
appear to "hang" in the sky until somewhat later, when it's real altitude
(to the man in the raft) is falling by 20' per hour, but on Doug's vessel,
just matches the rise that's caused by his own speed.

So his LAN will appear to be too late, because what he takes for local
apparent noon isn't LAN at all, it's just the moment of maximum altitude.

How big will that error be?  Charles H Cotter (in  A History of Nautical
Astronomy) deals with the matter on pages 264 to 266, but gets very
confused.

The moment of maximum altitude will be delayed on LAN by 15.3 (tan lat -
tan dec) * v seconds of time, where v is the Southerly component of the
speed in knots, and lat and dec are positive if North, negative if South.
If lat = +51deg and dec = -23.5deg, this works out at 511 sec, or all of
8.5 minutes late. Unless allowed for, this will give rise to an error (not
an uncertainty this time, but an actual error) of 128' in the longitude!

Note that the moment of maximum altitude is delayed after LAN, if the
vessel has a Southward component of speed, and the Sun is to her South, as
in the case referred to above. If sailing toward the Sun when the Sun is to
her North, the delay is the same way. If the Sun has a direction that's aft
of her beam, so her course is to some extent away from the Sun, then the
maximum altitude is correspondingly before LAN.

=================
Effect of Sun declination changing.

Even in the case of the man in the raft (whose latitude isn't changing), he
will see a displacement between the times of max. altitude and LAN of the
Sun, even though he has no motion toward or away from it, because the Sun
can have a North-South motion toward or away from him. Not at a solstice
(which is why I chose that example) but at the equinoxes, when the Sun's
declination is changing fastest. At the Spring equinox, the Sun is moving
toward him at about 1 arc-minute per hour, so that can be thought  of quite
simply as the Sun having a Northerly speed of 1 knot. Even for the
stationary observer on the raft, then, there can be a time-difference
between the maximum altitude of the Sun and the true LAN, which unless
corrected will give rise to an error in longitude of 6 arc-minutes.

This correction, for the time-difference due to changing declination of the
Sun, is referred to as the Equation of Equal Amplitudes, and can be found
under this heading in older editions of Norie or Raper. It was important
for accurate determination of longitudes of harbours from on-land
observations.

It's one of the factors that makes meridian altitudes of the Moon so
complicated, because its declination can change over 35 x faster than the
Sun's.

Except at the solstices, that North-South motion of the Sun needs to be
taken into account, even for the observer on a moving vessel, and added to
his own speed..

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

For slow-moving vessels, determining LATITUDE by a Sun meridian altitude
won't be badly affected by measuring max-altitude at the wrong moment in
time, away from LAN by (say) a couple of minutes, because altitude hovers
so steady for so long. But for faster vessels (20 knots North or South
affecting the timing by over 8 minutes) the resulting errors in latitude
can become rather significant. Correction methods exist (e.g. Raper, 1864,
par. 798-800). Or a mariner can ignore the maximum altitude reading (and
its time) and instead measure the altitude at the true moment of LAN,
predicted from his chronometer and the equation of time: but to do that his
longitude needs to be known, at least approximately. Did Doug Royer depend
on such a prediction for choosing the appropriate time to measure his Sun
altitude, I wonder?

But that was just for latitude. Determining LONGITUDE from a noon sight is,
as we have seen above, fraught with very serious error.

There's little hope of using such a technique to make an improvement on a
previous DR longitude, unless that DR was very dodgy indeed. So I think
it's dangerous to encourage novice navigators to presume that they can
deduce a longitude from their LAN, when that just ain't so; not with any
accuracy that's worthwhile.

George.


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