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    Re: Speed of Sunset
    From: David Pike
    Date: 2016 Sep 30, 14:10 -0700

    Lu you write:  If it's for both solstices, then I'm at a loss for an explanation and leave it to the more erudite members of this list (whoops, sorry for using "list," the more erudite members of this GROUP) to explain...

    Sorry for the late post.  I’ve been at the Lincolnshire Showground for the last three days confusing ten year olds about finding the size of the Earth with a GPS receiver and a trundle wheel.  I’ve never really thought about the duration of sunset, but I do remember being taught long ago that the duration of twilight varies with both latitude of the observer and the declination of the Sun.  It strikes me that the nautical twilight zone is like one big stretched out sunset, approximately 12 degrees of arc compared to 32 minutes of arc (if you neglect the effects of refraction).  That’s a 24x stretch, so anything true for the duration of twilight, is probably also be true for the duration of sunset.  The reason, at least for different latitudes, is that whilst the Sun’s altitude is measured with respect to the perpendicular to the observer’s horizon, time is measured by the rate the Sun crosses the Earth’s meridians, so the more slanting one plane is to the other, the longer the duration of twilight (and also sunset).  This is easily shown pictorially as in Figure 1 below, and several navigation books use similar diagrams.  The changing duration is also easily spotted by observing just one daily page of the Nautical Almanac.
    The variation of the duration of twilight with declination is, in my opinion, harder to demonstrate pictorially, and I think many books find the same.  At least one falls back on the well tried phrase “It’s obvious that …..”.
    Fortunately, the variation with declination at say 50N, 0N/S, and 50S is clear from observing three daily pages of the Nautical Almanac around 22Jun, 23Sep, and 22Dec (Figure 2 Below).
    Interestingly, while the duration of twilight differs with changes in declination, it only does so at significant values of the observer’s latitude.  If the observer is at 0N/S the change in duration of twilight with declination is minimal, and the actual duration equates very roughly to the Earth’s rate of rotation of 15 degrees per hour.
    Therefore, the problem becomes how to show why the duration of twilight differs with declination at a significant value of latitude.  60N suggests itself from a trig function point of view, but there’s the possibility of permanent twilight around the summer solstice, so let’s use 45N.

    It will be seen from Figure 3 that whereas the loci of the observer’s horizon and the Sun at dec = 0 degrees are great circles; the locus of the Sun about the meridians at any other value of declination is only a small circle centred on the Earth’s axis of rotation.  Projecting the twilight zone onto the smaller circle doesn’t seem to affect the number of meridians the Sun crosses, but the section of small circle is slanted more into the projection of the twilight band, and this will increase the number meridians that the Sun crosses whilst traversing the twilight zone, so the duration of twilight at latitude 45N (in this case) increases with declination.  By adding to the diagrams, you can see that the same is true for summer and winter.  The same should also be true of the duration of sunset.  DaveP.




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