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    Re: Longitude via lunar altitudes, simplified
    From: George Huxtable
    Date: 2007 Mar 11, 15:16 -0000

    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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