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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@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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