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    Re: Errors in Cotter's book, updated
    From: Fred Hebard
    Date: 2003 Jan 8, 11:46 -0500

    George,
    
    It was good to hear more evidence that you're the curmudgeon I
    imagine you to be.
    
    
    Regarding the following:
    
    >Page 264. Cotter says, about finding the moment of noon by equal Sun
    >altitudes-
    >
    >"By taking the equal-altitude sights shortly before and after noon the
    >necessity for applying a correction for the change in the Sun's declination
    >in the interval is obviated, since any such change will be trifling."
    >
    >I disagree with Cotter's analysis here. It seems to me that the correction
    >necessary for a changing declination does not reduce as the interval chosen
    >gets closer to noon.
    
    The rate of change of the declination would not be changing very much
    no matter what the interval, as long as it's less than 12 hours or
    so.  But the amount by which the declination changes would decrease
    as the interval were shortened.
    
    Once in a while, one can take advantage of sexigesimals and simple
    circular models to get an idea of what's going on with celestial
    phenomena.
    
    As a first approximation, the declination can be modeled by a cosine
    curve with an argument of 1 day = 1 degree = first day of spring,
    with a maximum value of approximately 23 degrees at zero days and a
    value of zero degrees at 90 days.  At equinox, the first derivative
    at 90 days would be (23/360)*sine90 or 23 arcminutes per day.  The
    second derivative would be zero (cos90=0).  At solstice, the first
    derivative is zero but the second derivative would be maximal.  I
    don't know how to calculate the values of these derivatives from the
    formula without more head scratching than I have time and patience
    for, being but a biologist, but here are some data from the USNO
    site, equivalent to Nautical Almanac data:
    
    Equinox
    
    hour       date   declination   rate of change    rate of change of change
                                   =first derivative      =second derivative
      gmt       U.S.       o    '          '                       '
    --------------------------------------------------------------------------
    Equinox
    12:00:00  3/19/03   S 0  36.5
                                         23.7
        "      3/20/03   S 0  12.8                                0
                                         23.7
        "      3/21/03   N 0  10.9                                0
                                         23.7
        "      3/22/03   N 0  34.6
    
    Solstice
    12:00:00  6/19/03   N23  25.3
                                          0.7
        "      6/20/03   N23  26.0                                0.3
                                          0.4
        "      6/21/03   N23  26.4                                0.3
                                          0.1
        "      6/22/03   N23  26.3                                0.6
                                          0.7
        "      6/23/03   N23  25.6
    
    In between
    12:00:00  5/8/03    N17   3.2
                                         16.2
        "      5/9/03    N17  19.4                                0.4
                                         15.8
        "      5/10/03   N17  35.2                                0.2
                                         15.6
        "      5/11/03   N17  50.8                                0.3
                                         15.3
        "      5/12/03   N17  66.1
    
    
    The data are not precise enough to get much of a handle on the second
    derivative, which is probably pretty steady in May, but you can see
    that it's jumping around quite a bit in June and zero in March.  The
    simple model could give you precise values, which wouldn't be off too
    much in accuracy for getting a handle on the second derivative at the
    precision required.
    
    Fred
    --
    --------------------------------------------------------------------------
    Frederick V. Hebard, PhD                      Email: mailto:Fred{at}acf.org
    Staff Pathologist, Meadowview Research Farms  Web: http://www.acf.org
    American Chestnut Foundation                  Phone: (276) 944-4631
    14005 Glenbrook Ave.                          Fax: (276) 944-0934
    Meadowview, VA 24361
    
    
    

       
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