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Trawling through some interesting topics that have appeared on Navlist
in my absence, I can't resist a comment on this one, about deducing
altitude from lunar altitudes, which has raised its head once again.

Peter Fogg introduced it with a reference, in Navlist 2146, to a
recent article by George Bennett.

That article provides a reasonably clear and simple procedure for
tackling the problem, by interpolating between two trial-values for
watch-error (or, in the case of his example, extrapolating). It has
advantages over the successive-approximation procedure, proposed on
the Starpath site. For anyone planning to try lunar altitudes to
determine longitude, the Bennett method is certainly as good as any,
and better than some.

And yet....  The problems involved in trying to get longitude from
lunar altitude have been glossed over, in my view, and need to be
spelled out properly. In this context, Bennett says no more than this-

"There is no doubt that lunar altitude methods are inferior to the
lunar distance method but the advantage of the former is that we can
use techniques which are familiar to navigators. However, the accuracy
does depend upon the reliability of altitude measurement such as a
clear horizon and reliable refraction and dip corrections"

You bet, it does. Peter Fogg adds- "It tends to be less accurate than
the conventional method of lunar distances..."

We need to think about where that loss of accuracy occurs.

In any altitude observation, of the angle between a body in the sky
and the horizon, nearly all the measurement error is due to
uncertainty in the direction of the horizon, not in locating the
centre, or edge, of the body. The horizon is often unclear or hazy,
but even when it's clear and sharp, it's seldom exactly straight, but
affected by waves and swell, and the observer's height is also
affected by his vessel's heave on that sea. Even in dead-smooth
conditions, refraction, due to unknown air temperature gradients along
the path between observer's eye and horizon, affects the dip, which
may not follow the "book" value. There's no effective way of telling
when such "anomalous" dip is occurring, and discrepancies of the odd
arc-minute or two are common. For normal navigation, it doesn't
matter, shifting a deduced position only by a mile or two. But in
deducing a longitude from the position of the Moon, every such minute
of error can shift the longitude by about 30 arc-minutes. Depending on
the relative azimuths of the other-bodies and the Moon, these altitude
errors can combine in such a way as to give rise to a 60 arc-minute
shift in longitude for each arc-minute of anomalous dip.

Lunar-distance navigation avoids such problems, because the horizon
isn't involved at all; just the angle between two bodies, taken across
an arc slanting across the sky. That measurement depends only on the
sharpness of the observer's eye and the calibration of his instrument.
The only uncertainty in the corrections is due to refraction, and
provided both bodies are well above the horizon, that is small and
very predictable.

The accuracy of a lunar altitude is also very dependent on the angle
of the Moon's path, with respect to the horizontal. There's usually a
recommendation to measure the Moon altitude when it's near to the
prime vertical (due East or West), as then it's climbing or falling
most quickly. Bennett states "As a general guide, Moon observations
should not be taken in the vicinity of the meridian but near the prime
vertical".

That's all very well. The best circumstances are an observation in the
tropics, when the Moon always rises in the East, or sets in the West,
nearly vertically, its altitude changing at roughly 15 degrees per
hour. Statements of the precision of the lunar altitude method are
often based on such an example.

Compare that with a Summer full-moon seen from southern Britain. Next
year, it won't get significantly above 10 degrees altitude at its
highest. In that season, it never gets anywhere near the prime
vertical, and the maximum usable rate-of-rise is only about 4 degrees
per hour, not 15. In those circumstances, the errors are nearly
quadrupled, compared with an observation in the tropics. Anyone
attempting to use lunar altitudes at such times would be seriously
disappointed.

The basic difficulty about any lunar longitude method is in its
imprecision, because of the unfortunate fact that each minute of
observational error gives rise to about 30 minutes of longitude error,
because of the slow motion of the Moon across the sky. Even at its
best, lunar distance was a crude tool, and only viable because
mariners had no alternative. Anything that degrades further that
already low precision makes the thing unworkable, or of little
practical use.

We need to ask ourselves whether the price that has to be paid, in
using lunar altitudes, is worth the gain. That gain is no more than
this; that observers can use their familiar above-the horizon
technique, and don't need to make a slant distance observation between
two bodies.

The originators of the lunar distance method in the 18th century
(Lacaille, Mayer, Maskelyne) knew what they were doing. No doubt they
were aware of other ways of doing the job, but what they settled on
stood the test of time until chronometers became affordable.

Airy's comments, as quoted by Frank, are very much to the point, when
he says- "But we should guess, in the absence of actual trial, that a
very bad lunar distance would give more trustworthy results than a
very good lunar altitude".

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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