NavList:

Antoine, you wrote:
"Just one question here : I am not quite familiar with the definition of "refracted" semi-diameters. They imply some kind of "Center". From which such "center" are they reckoned ? Obviously, given the irregular apparent shape of both bodies - and especially the Sun who is quite low with such a distorted limb - refracted semi-diameters will vary according to the directions in which they are measured. This is a bit intricate to compute, but this can be accurately done with to-day computing power."

This is nothing more than "refractional flattening". And despite the rather long description in a later post, it is in fact very easy to calculate and tables for correcting for it were included in many 19th century navigation manuals, though only in the dying days of lunars from about the 1830s. All you do is look up the refraction at the Moon's center and then look up the refraction 16 minutes of arc higher. The difference gives you the refractional flattening for the upper half of the Moon (very nearly equal to half of the total refractional flattening). This maximal amount then varies by cos^2(position angle) as you work your way around the limb. In Thompson's very popular tables, and in others who followed his lead, the recommendation was to estimate the position angle at the time of observation since you hardly need that position angle better than the nearest ten degrees. So imagine this case: you see the Moon rather low in the sky as you're shooting a lunar. It's maybe ten degrees high, so you know you should correct for refractional flattening of the Moon's shape. The Sun is, let's say, halfway up the sky towards your right. The "lunar arc" --that great circle arc passing through the centers of both objects-- touches the Moon at about "two o'clock" (seeing the Moon's face as a traditional analog clock with 12:00 at the top. That corresponds to a position angle of 60 degrees. You note that mentally before your start clearing the sight. During the pre-clearing process, when you're working out the semi-diameter of the Moon, you take the cosine of that position angle, square it, and you get 0.25. So the flattening in the direction of the lunar arc is 25% of the maximal flattening in the vertical direction. That's all there is to it. Many navigators simply ignored this. It only matters when the Moon or Sun is quite low in the sky, and most navigators back then believed that refraction tables were unreliable at such low altitudes anyway. Since lunars were only required occasionally, it was easier to just wait for better geometry.

-FER