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    Re: Equal altitude sights at local apparent noon
    From: Jan van Puffelen
    Date: 1997 Feb 11, 16:57 EST

    At 18:15 10-02-97 -0600, Pickering, Keith (c) wrote:
    >I must disagree with Jan's negative assesment of this technique for
    >longitude measurement.  This was, indeed, the standard means to measure
    >longitude at sea after the invention of the chronometer.
    I hate to disagree but longitude was preferrably measured by a "longitude
    shot" whereby the spherical triangle was solved with a given latitude,
    yielding the Right Ascension of the measured object (this was given in time,
    now we would use the LHA measured in degrees). Given GMT (from the
    chronometer) the time difference was divided into 15 to give the longitude
    (nowadays the difference between GHA and LHA yields the longitude). The
    object in question was preferrably as much due east or west as possible.
    >The change in
    >the Sun's declination over a period of ~30 minutes is so small that it
    >can be ignored for navigational purposes.  The change in the Sun's
    >altitude is small, but easily measurable with a sextant, except for
    >unusual circumstances such as polar navigation.
    To give an example I have calculated a situation on 50d 00'N 0d 00'W/E and
    the sun having a GHA of 0d 00.0' and declination of 0d 00' N/S exactly at
    12h00'00" GMT. The change in declination is then around 1.0' per hour.
    Time            Altitude
    11h 40' 00"     39d 48.7'
    12h 00' 00"     40d 00.0'
    12h 20' 00"     39d 49.4'
    As we see there is a 0.7' difference between the altitude 20 minutes before
    noon and 20 minutes after noon. If we would plot the altitude, we would see
    that the highest altitude would NOT be reached at 12h00'00" exactly but later.
    This is not the only problem. The ship moves between the first and the
    second observation. No matter in what direction the ship moves the
    observation will have been influenced. Other variables such as dip and
    refraction could have changed in the meantime as well. This example has
    shown that even in 40 minutes difference the declination can be noticeably
    changed. The time of 20 minutes before noon is not sufficient however. BTW
    this situation is even worse for the moon which can have a change in
    declination up to 17 times as much. The situation is better for the stars
    which have no change in declination at all, but unfortunately, the time that
    both the star and the horizon can be seen is limited to the time of morning
    or evening twilight.
    >The reason for doing
    >multiple sights is twofold:  first, to find LAN, and second, to get
    >enough observations to reduce random error to acceptable levels.  The
    >kind of curve-fitting that Malcom describes will work in almost all
    >circumstances, with persistance and patience.
    As I have demonstrated above, curve fitting would yield the wrong LAN
    (because of the difference in declination), even with the ship stationary.
    Obviously, the ship would have moved on half an hour a further 3' or so
    (assuming a speed of 6 knots), thereby distorting the plotted curve even
    more with this very difference.
    Leeway and local (tidal) currents will even further distort the picture.
    Yes, I usually make several observations. I enter the time/altitude pairs as
    X/Y pairs in the lineair regression function of a programmable calculator.
    If the correllation coefficient is satisfactory (which means that all
    observations are more or less on a straight line) I use the average X and Y
    as the actual observation. This is, as you say, to reduce small random
    errors. If the correllation coefficient is insufficient I reenter the same
    time/altitude pairs and check the error against the regression line. Usually
    I have made a mistake and if, by throwing out the wrong pair (this is a
    standard function of the calculator) with the highest error, the remaining
    pairs usually yield a good correllation coefficient and I use the averages
    again. If not, I follow the old fashioned method using plotting paper.
    The double altitude method was indeed a formal method in the 19th century,
    but this method was never used to determine LAN, only the latitude.
    >Keith Pickering
    Jan van Puffelen <puffelej@XXX.XXX>
    52d 24.5'N
     4d 55.0'E

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