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    Re: Errors in Cotter's book, updated
    From: George Huxtable
    Date: 2003 Jan 9, 01:00 +0000

    Fred Hebard said-
    >It was good to hear more evidence that you're the curmudgeon I
    >imagine you to be.
    Fred has presented me with the curmudgeon's medal, and it's a badge I shall
    wear with pride, and do my best to live up to.
    Fred's table of Sun declination-changes came over OK on his email message
    as received here.
    I think we can simplify that information, and just consider a worst-case,
    at the equinox. Fred's data shows that at the Spring equinox the Sun is
    crossing the equator and moving North at 23.7 minutes each day. This is
    very nearly 1 minute in each hour, or 1 knot, to put it most simply.
    When the Sun is exactly South of as observer, that is at apparent noon,
    then its changes in Local Hour Angle do not affect its altitude, but at the
    equinox an increasing altitude of 1 arc-minute per hour remains, due solely
    to its change in declination. So at that moment, the Sun is not yet
    "hanging" stationary in the sky, but is still climbing slowly. It will not
    reach its maximum altitude, when it stops going up and starts going down,
    until a bit later. What time will that be? It will be that moment when the
    Sun's altitude caused by its change in Hour angle past noon is FALLING at 1
    arc-minute per hour, to exactly balance the rise that the steady change in
    declination causes. At a latitude of 45 deg., this will happen at about 15
    seconds after noon.
    If you were to make a plot of Sun altitude against time, then near noon you
    would see something looking very like a parabola, the arc that a shell from
    a gun follows. If the declination wasn't changing (at the solstice) then
    the highest point on that parabola would be exactly at noon, and it would
    be symmetrical about noon. But near the equinox, the peak would be about 15
    seconds late, and in fact the whole parabola-shape would be shifted later
    in time by about 15 seconds.
    We have discussed at length on this list how impossible it is to determine
    the moment-of-noon AT noon, because the Sun's movement has come to a
    near-stop, and a perfectly valid alternative is to take the mid-time
    between two moments when the Sun has equal altitudes, going up and going
    down, preferably at times spaced well away from noon. It should be clear
    now that in finding such a mid-time, you are drawing a horizontal line
    across our plotted parabola, and finding its mid-point. And if the whole
    parabola has been displaced 15 seconds later, then the mid-point of any
    line across it will also be displaced to be 15 seconds late, no matter what
    height the line was drawn at, however near to maximum altitude that was.
    And that's the point we are trying to settle.
    You may think that a displacement of noon from the moment of highest Sun,
    by 15 seconds, is rather negligible, and so it may be. But the same effect
    occurs when the vessel is travelling towards or away from the Sun. If the
    vessel's North-South component of its speed is say 10 knots, that will give
    rise to an error in the timimg of noon by 150 seconds at a lat of 45 deg.
    This has to be added to the effect of declination change.
    However and whenever you measure the moment of apparent noon, near to noon
    or far from noon, this error in the timing of noon, due to the combined
    North or South velocities of the ship and the Sun, has to be allowed for.
    But note that this has no observable impact on the value of the maximum
    altitude observed, so a latitude measurement is unaffected.
    George Huxtable.

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