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Well, it seems that this response to an apparently simple question from 
Joel Silverberg has become a somewhat inglorious eoisode in the history of 
Navlist.

I started off the rot by writing solstices, when I meant to say equinoxes. 
Then Marcel seems to have made a double error, which happened to cancel 
out, and got the right answer for the wrong reasons.

=======================

The story so far.

The situation so far is that we seem to agree that there is a maximum in 
the length of twilight at each solstice, and a minimum near each equinox.

But note that now I say near each equinox, not at each equinox, because my 
view has now changed significantly. Rather belatedly, I realise that the 
question of shortest twilight, that Joel posed, is not so trivial after 
all. I now think that all told, we (or at least, I) have been jumping to 
confusions.

It's true that there's a maximum twilight length around each solstice, 
which is defined simply by symmetry. (Here, I'm neglecting effects caused 
by the eccenticity of the Earth's orbit, which perhaps shouldn't be 
neglected). But those maxima are very different in twilight length, between 
Summer and Winter solstice, as Marcel's graph shows. And that asymmetry, 
between Summer and Winter, affects the symmetry of the minimum, around the 
equinoxes, as is also shown in Marcel's plot.

My initial response, that the matter was a trivial one, was based on my 
understanding that the rate-of-fall of the Sun as it crosses the horizon is 
greatest at the equinox: as indeed it is. But Joel's question was about the 
time taken for its altitude to fall from 0º to -18º, which is not 
symmetrical about the horizon line. If we had been asked about the time for 
the altitude to fall from +1º to -1º, or even from +18º to -18º, we could 
confidently say that those intervals would minimise at the equinox. Not so, 
however, between 0º and -18º

If you look carefully at Marcel's plot, which I recopy here, you will see 
that the dates of the mimima are indeed shifted from the dates of the 
equinoxes. Which makes Joel's question a lot more entertaining. Our answers 
have fallen short.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 

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