NavList:

I would like to reformulate this problem in terms of the Sun's declination and local hour angle (LHA). When we say "date" we are really referring to the declination of the Sun and by "length of twilight" we mean the difference between the Sun's LHA at sunrise/sunset and LHA at the start/end of twilight. (Changes in the Sun's R.A. and declination during twilight as well as refraction and dip are ignored). The problem can be stated as: For a given latitude, find the declination of the Sun for which the difference between its LHA at sunrise/sunset and the LHA at the end/start of twilight is a minimum.

Using the cosine rule of spherical trigonometry it can be shown that
0 = sin(d)*sin(L) + cos(d)*cos(L)*cos(LHA1)
cos(90+18) = sin(d)*sin(L) + cos(d)*cos(L)*cos(LHA2)

where
LHA1 is the Sun's LHA at sunrise/sunset
LHA2 is the Sun's LHA at the start/end of twilight
d is the Sun's declination
L is the observer's latitude

We seek to minimize the difference between LHA1 and LHA2 and therein lies the problem. The cosines of the angles are known but we want to minimize the difference between the angles themselves.

The equations above do allow the length of twilight to be plotted and solved for numerically. I have reproduced Marcel's 41 degrees north plot in terms of declination and LHA difference (see attachment). For that latitude the shortest twilight occurs when the Sun's declination is 5.9 degrees south.

It would seem that the discussions here on Navlist provide an answer to Joel's original question. The interest of this problem is presumably in its apparent simplicity, relative mathematical intractibility, and counterintuitive result.

I am reminded of the somwhat similar counterintuitive problem of finding the date of earliest sunrise. This one however has an elegant geometric interpretation involving the analemma (see http://www.analemma.com/Pages/OtherPhenomenon/OtherPhenomenon.html )

Regards,
Robin Stuart

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