Dear Frank et al,
After a second night sleeping on your nice puzzle I
realised that I DID make a small goof in one of my
assertions and no one has picked me up on it!!
In the formula:
tan(dec) = [-]sin(d)/sqrt(t1^2 - 2.t1.t2.cos(d) + t2^2)
I asserted (correctly) that the argument of the square
root function couldn't be negative. Dave Bell kindly
confirmed this. I (also correctly) asserted that the
argument could be zero.
I then (incorrectly) asserted that when the argument is
zero, the associated declination would be 90 degrees.
My (false) reasoning was that with zero as the value of
the square root, the right-hand side would have a zero
denominator and hence an infinite value overall.
In fact, for the argument of the square root function
to be zero we require...
Either t1 = t2 and d = 0
or t1 = -t2 and d = 180
In both cases the numerator, sin(d), is also zero and
we have nought-over-nought which is well known to give
bad vibrations to mathematicians!
These two sets of requirements correspond to:
Either two places at the same latitude and with
zero separation of longitude (meaning that
the two places are coincident).
or two places of opposite latitude and 180 degrees
of longitude apart (meaning the two places are
diametrically opposite on the Earth's surface).
In both sets of circumstances there is no unique great
circle through the pairs of points and, in consequence,
no single associated solar declination.
The second case is interesting in demonstrating something
fairly obvious but perhaps not widely known:
If, standing in a particular place, you note the
instant of sunset (or sunrise) you can ponder the
thought that someone at the other end of the Earth's
diameter from you is simultaneously experiencing
sunrise (or sunset). This is independent of where
you are or the time of year.
One of the many nice features of this puzzle is that there
is no need to know the time of sunset (or sunrise) and, in
consequence, you can lay any understanding of hour-angles
on one side.
Hey, isn't that just what you wanted to know!
Best wishes
Frank
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