NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Joel Silverberg
Date: 2010 Jun 30, 11:51 -0700
I am glad to see the activity my posting has stirred. The problem is not as simple as first appears. I have located Nunes work ( 1542 edition in Latin, and a 2003 translation into modern Portuguese ... I don't know which one strains my limited linguistic abilities more). It is nearly 150 pages long and very difficult to read. I was also tipped off to a nice overview in a Ladies Diary issue in 1817 showing other solutions, and learned that the problem was studied by Delambre in his History of Ancient Astronomy that same year.
The declination of the sun on the day of shortest twilight is equal to the product of the tangent of half the twilight angle (half of 18 degrees for astronomical twilight) and the sine of the observer's latitude with the caveat that the solar declination on that day will be negative if the observer is in the Northern hemisphere and positive if he/she is in the Southern hemisphere.
sine(declination) = tangent (9 degrees) * sine (Latitude) (declination is N or S as described above)
If my back of the envelope calculations are correct (no promises there) that puts the day of shortest twilight on the vernal equinox (say, March 21) at the equator, on March 13 at 20 deg N latitude, March 6 at 40 deg N, March 1 at 60 deg N and Feb 26 at the North Pole. Corresponding autumnal dates are approximately Sept 23 at the equator, Sept 15 (20 deg S) , Sept 8 (40 deg S) , Sept 3 (60 deg S) , and Aug 31 (90 deg S).
Now why that formula works, where it came from, and how to derive it are considerably more difficult questions. If I ever figure it out, I will share it with the list if people are interested.
One source says that "the duration of twilight will be a minimum in any latitude when the sun crosses the small circle bounding twilight (after setting) on the same azimuth on which he rose on the horizon in the morning." But I have learned to take any such pronouncements with a grain of salt, until I can understand for myself why I should believe them.
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