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@ Joel Silverberg

Just in case this subject is still of interest to you.

I still imagine - as I mentioned earlier - that this problem
originates from calculating the Muslim prayer times. In the early days
of this religion general rules were established to help for those
religious needs. Later it was realised that these rules were not
sufficiently correct and required a refinement. Regarding the twilight
between sunset/sunrise and begin/end of night (sun at -17 to -19 deg)
seems to be relevant not only for the two prayers which I mentioned
earlier (evening and night) but also for the morning prayer. These
prayers have apparently to be done in the corresponding time-spans,
preferably at their beginning. I could now imagine that the time-span
for the twilight was first considered to be a fixed time interval. The
original problem may then have arisen from realising that this time
interval is not so constant and thus requires some corrective
adjustment.

If this is really the origin of this problem you may eventually find
more on it in the publications of this Gentlemen:
http://web.uni-frankfurt.de/fb13/ign/ign2/people/kinge.html

===

Now to those of you who still try to imagine the possible results of
some special cases. To answer those the initial question on the
shortest twilight is not sufficiently precise. It actually requires to
distinguish between two cases:

Duration of twilight ...
(1) ... with sun being "somewhere" between horizon (sunset/sunrise)
and -18 deg altitude
(2) ... between sunset/sunrise and sun actually passing -18 deg altitude

The attached graph shows the duration of twilight for a location at 65
deg North. With the first definition for the duration of twilight the
yellow points are valid, for the second definition not. The two
definitions lead thus to two different results. (Note: George may be
happy to see that his first wrong answer (solstice) happen to be
correct in this particular case.)

At the North Pole - and in a corresponding way also at the South Pole
- there are only two values for the full year, one for the
Winter/Spring-time and the other for the Autumn/Winter-time. The
shorter of the two is at the North Pole the Winter/Spring one. During
this time the earth is nearer to the sun and thus moving faster in its
orbit.

Marcel

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