David C, you wrote:
"Now to more serious things - your comment had me puzzled. How can the azimuth jump to due east or west in a millisecond? I am guessing that it is something to do with a discontinuity when the lat =0°. I have attached the relevant page from Davis. Note thatat lat=0, dec =0 lha=0 the azimuth field is empty. Note also that as the LHA slowly changes the jump to east or west only occurs when dec = lat. This is confirmed by looking at the pages for lat=1° and the jump to east or wst occurs when dec=1°."
Yes. And doesn't that make excellent sense to you? Declination is Latitude. The declination of a celestial body is the latitude of the subStar point for that body --the latitude where the object is at the zenith. Similarly, GHA is Longitude. The GHA of a celestial body is the longitude where that object is at the zenith. And what happens to azimuth at the zenith? It's a coordinate singularity, and from a purely mathematical sense, azimuth is undefined at the zenith. In the real world, no object is ever exactly, perfectly at the zenith. Instead its azimuth rapidly swings from east to west as it passes very close the zenith.
On any day of any year, pick a star with some declination. I'm using a star here since their declinations are almost exactly constant for any single day. If you visit the latitude that is equal to that declination to the nearest mile, then at some point during the day the star will be within one minute of arc of your zenith. That star will rise from some point on the eastern horizon (not typically due east) and as it gets close to the zenith its azimuth will settle in to almost exactly due east. Then as it passes some fraction of a minute of arc from the zenith, its azimuth will swing rapidly around to very nearly due west. If you happen to observe it during that rapid flip, and if at that time you find it on an azimuth close to north or close to south, then you know that you're observing right in the middle of the flip. It's "meridian passage" but extremely close to the zenith. This is not something that you likely see with a sextant since it happens so fast. But you could see it in calculated data, and that's what I was pointing out in the data (image) that I posted yesterday. At 13:30:00 the Sun's azimuth was within a few degrees of due north --it was in the middle of the flip.
This is very similar to what you would see if you were skiing cross-country near the south pole. Imagine yourself aiming for some little flag on the snow and ice that represents the true pole. As you ski along, your course would be due south on approach. South... south... south... and then as you zip past the flag, still travelling on a straight line, you would suddenly be heading north... north... north... There's a sudden flip in the coordinate direction even though you're travelling in a straight line. If you could collect data on your course's true azimuth, fine-grained in time, maybe once every tenth of a second, as you ski towards and away from the pole, you would be able to detect the quick flip from southbound to northbound, and in between you would see a few points connecting the two asymptotic limits since, of course, you would never, in the real world, ski perfectly across the exact pole. The azimuth data, expressed as compass directions, might read once a second over 30 seconds as follows:
N, N, N, N, N, N, N, N, N, N, N, N, NxW, NW, W, SW, SxW, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S.
That instant where you are travelling due west was also the instant when you were closest to the pole. Similarly an observer "under the Sun" watching it from a latitude nearly identical to the Sun's declination would see a string of azimuths over the course of a dozen seconds as follows:
E, E, E, E, ExN, NE, N, NW, WxN, W, W, W, W, ...
"Does this mean that mariners sailing along the equator should avoid a compass check on the equinox?"
No, but they should avoid a compass check when the Sun is high in altitude, and that's true in general. Modern navigators are erroneously taught that they should not measure high sextant altitudes, nothing above 75° or even 65°. There's no reason for that. I think it's mostly a result of poor skill and worse advice on how to swing the arc. Navigators in the 19th century routinely measured the Sun's altitude at noon on days when it was near the zenith and I've never seen any complaints about the process. But the modern (and historical) advice to avoid measuring a celestial object's azimuth when it's above 20° or 30° max is reasonable. It's quite difficult to judge azimuth when the Sun is high.