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    Re: Lunar Distances v. Meridian Angles
    From: George Huxtable
    Date: 2002 Feb 27, 17:43 +0000

    Chuck Griffiths asked the following sensible question-
    >OK, I know George H. promised he'd cover using moon altitudes in place of lunar
    >distances in his next part but I'm going to go ahead and jump the gun by asking
    >a question. Now that George has helped me start to understand lunar distances I
    >can't help but consider an alternative approach to finding GMT. Why can't we
    >observe the altitude of the moon and one other body and, using our assumed
    >latitude, solve for  the meridian angle of both bodies. The difference between
    >the two angles should change by the rate at which the moon moves through
    >the sky
    >faster than another body. If that's true, can't we find the meridian angle
    >between the two bodies for the even hours, say on either side of what time we
    >think it is, and use the same inverse linear interpolation approach to find the
    >time of our sight?
    >Of course, I can think of a couple issues with this approach worth discussion.
    >First, this only works when the altitude of the moon and the other body change
    >reasonably with time, i.e., we can't do it when either body is close to being a
    >meridian sight. Second, we need both altitudes simultaneously. I think this
    >could be solved by alternately observing one body then the other several times
    >and graphing the sights so that we could derive an averaged simultaneous
    >altitude from the graph.
    >Lastly, why bother when the other methods thus far described work? It seems to
    >me that if this is a workable solution it provides a method of checking time
    >using techniques that are already in most navigator's "bag of tricks". That is,
    >we get to correct for refraction, horizontal parallax, augmentation, etc. using
    >the tabular methods we use for other sights.
    >Chuck Griffiths
    George Huxtable responds-
    First, I would like to endorse Dan Allen's welcome to Bruce Stark and his
    lunar expertise. We appear to be gathering a useful little band of lunar
    enthusiasts on NAV-L.
    The next part of my stuff about lunars is indeed "on the stocks", but not
    yet ready to launch on the ether. But that's no reason for Chuck, or anyone
    else, to hold fire on any questions that come to mind. And Chuck's question
    is a very fair one. It has been asked before, however; starting in 1674.
    Francis Chichester, the famous single-handed circumnavigator, proposed such
    a method in 1966, and a spate of publications followed, on similar lines.
    These were answered in an authoritative article by David Sadler, then
    director of HM Nautical Almanac Office, in the RIN's "Journal of
    Navigation", 31, 2 May 1978, page 244, entitled "Lunar Methods for
    'Longitude Without Time' ".
    From my point of view, the drawback of Sadler's article is that it is
    illustrated by a diagram of such devilish cunning and complexity that I am
    quite unable to make head or tail of it. If anyone on this list manages to
    penetrate its mysteries, I would be grateful if an explanation was posted
    Bruce Stark made a number of pertinent points in reply, and I would like to
    expand on his statement-
    >First: Consider what you know about the reliable accuracy of altitudes taken
    >from the sea horizon, especially from a small boat. You will be combining the
    >errors of several such altitudes.
    It's important to bear in mind that in any measurement that uses the Moon's
    motion to provide time and hence longitude, accuracy in determining the
    Moon's position is all. This follows from the fact that each minute-of-arc
    error results in an error in the final position of the vessel of 30 minutes
    of arc (which near the equator corresponds to 30 miles) or sometimes more.
    It is FAR more demanding that the normal run of astronavigation.
    The main virtue of a lunar is that the all-important measurement in which
    so much accuracy is required, the angle-in-the-sky between the Moon and the
    Sun (or other body) does not involve the horizon AT ALL. True, the
    altitudes of Sun and Moon do have to be measured up from the horizon as an
    auxiliary measurement, but this is only to get a correction to a
    correction, and a rough value for those altitudes will be perfectly
    Why is the accuracy so degraded whenever the horizon is involved?
    First, if there's any haziness in the air, the first thing to become
    indistict is the line of the horizon.
    Second, even if the horizon is really sharp, it isn't exactly a
    well-defined straight line (except in millpond conditions), especially from
    a small craft. The horizon-line is made up from the peaks of overlapping
    waves and swell, and the vessel, too, is riding on those waves. The
    observer does what he can by timing his shots when he judges his vessel to
    be on the top of its "heave", but it is inevitably a compromise.
    Third, even if the horizon is both sharp and straight, its angle  can be
    affected by anomalous refraction, which causes the dip to vary from its
    predicted value. We have discussed this matter at some length before, on
    this list, so unless anyone asks further I will say no more about it than
    Air layers at different temperatures near the horizon can cause the sun's
    image to be distorted as it rises and sets, can in extreme cases cause
    mirage effects when a distant vessel is observed as floating well above the
    horizon, sometimes even inverted. Where none of these objects is there to
    give a clue to the odd behaviour of light on its path from skimming the
    horizon to the observer's eye, anomalous dip may nevertheless be present,
    quite unsuspected and undetectable. An error in dip of 1 minute may be
    quite usual, and 2 or 3 minute errors can also occur occasionally. There is
    no way for the observer to correct for it. (Special instruments to measure
    the dip-of-the-moment have been devised but are very uncommon).
    These errors may present no real problems in normal astronavigation. After
    all, what significance has an error of 2 or 3 miles in an astro position?
    However, to the lunar observer, where any such errors are multiplied 30
    times or more in calculating his longitude, they are intolerable. This was
    well-known to eighteenth-century navigators, who accepted the practical and
    arithmetical difficulties of measuring lunar distance up in the sky, rather
    than altitudes up from the horizon, to cling on to all the precision that
    they possibly could.
    George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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