NavList:

William Hawes, you wrote:
"This time I'll quote directly from no less an authority than the Admiralty Manual of Navigation Volume 2"

It's an authoritative work, yes, but the person who wrote that section may not understand why it's done. Let's not forget that the "manual" of navigation is just that, and its various sections were written by mere mortals.

In general, a celestial object's azimuth, Azm, is calculated from its true declination, Dec, the observer's latitude, Lat, and the (corrected) altitude of the body, H, from this well-known equation:
cos(Azm) = [sin(Dec) - sin(Lat)*sin(H)]/[cos(Lat)*cos(H)],
or any equation equivalent to it by trig identities.

Normally this depends on the corrected altitude so the relationship is rather involved and requires a complete solution of the usual problem of clearing a celestial sight. But there's a way to make this much shorter that allows the results to be tabulated very easily. If we take care to make our observations such that H is zero, then the equation reduces to:
coz(Azm) = sin(Dec)/cos(Lat).
Note that this is identical to
coz(Azm) = cos(pd)/cos(Lat), where pd is the polar distance: pd=90°-Dec. This form can be more useful for picturing relationships since, when you're at the equator, it reduces to Azm = pd, which makes good sense and is easy to remember.

This latter equation (in either form) when H=0 can be tabulated very easily. Columns for Lat, rows for Dec, Azm is read out directly. Easy. That's the REAL reason that amplitudes are supposed to be taken when the corrected altitude H is zero --so that we can use simple tables to look up the azimuth. Notice however that the advice for the observed height of the Sun is somewhat incomplete. The refraction at the horizon is around 34 minutes of arc. That's close enough to twice the Sun's SD that we can just call it that. If we observe the Sun when its lower limb is one semi-diameter above the horizon than its corrected altitude is nearly zero. BUT, and this is critical, that's the correct rule AFTER correcting for dip. That's one of the main reasons you'll find different suggested rules for this. If you're on a large vessel and your height of eye off the waves is 105 feet, then you need to observe the Sun's LL when it's 28' above the sea horizon. Then the total correction is -10' for dip, -34' for refraction, and +16 for SD, which nets out to a corrected altitude of 0° 00' exactly. The difference with respect to azimuth isn't usually measurable except for high latitudes. Rather than fussing over the exact altitude, any altitude for the lower limb between 1 SD and 2 SD (16' and 32') will yield good results for typical heights of eye at sea, given the expectation that azimuths can't be measured to better than a quarter of a degree or so, at best.

But notice that all of this is really a relic of an earlier era when computation was expensive. If you can measure the Sun's altitude, even roughly with a homemade "cross staff" (an index card at arm's length will do just fine), then you can work up the full equation and generate an accurate azimuth when the Sun is just a bit higher in the sky, say 2.5 degrees high, and the refraction is much less variable.

-FER
PS: Of course, I could be wrong. Mere mortal and all that... :)

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