NavList:

George H, you wrote:
"Whatever the phase of the Moon, if we observe it near the moment of
meridian passage, its shadow-pattern will always be symmetrical about a
horizontal line through its centre. In that case, there's always a
sharp line round 180 deg of its edge, from its upper limb to its lower
limb, on one side or the other, with a fuzzy edge on the other side. In
that case, it doesn't matter which limb is used, because there will
always be a crisp edge or cusp extending to both upper and lower
limbs."

That's not quite right. Consider the case of the First Quarter Moon
around 6pm local time on December 21. As its crossing the meridian,
nearly on the celestial equator, the line of cusps will be tilted by
about 23 degrees with respect to the meridian. An observer in
mid-northern latitudes would see the Moon on the merdian "tilted"
towards the southwest horizon, where the Sun set an hour or more
earlier. The navigator would then have to use the Lower Limb (also for
a substantial period before meridian passage).

The condition to find the correct limb is easy to describe and use
visually, and not too tough to calculate.

Visually, you look at the line of cusps (or near Full Moon the line
running more or less from the Moon's north to south pole which an
experienced observer should know from the appearance of the bright and
dark patches on the Moon's face --see PS). If the line of cusps is
vertical, neither limb is prefered. If the line is tilted so that the
upper cusp is to the right of the lower cusp (and the Sun is to the
right), then the Lower Limb should be used. Tilted the other way (or
with the Sun to the left), the Upper Limb is used.

To calculate this, draw the Moon-Zenith-Sun triangle leaving enough
room to draw a little picture of the Moon including the line of cusps.
That line will always be exactly perpendicular to the Moon-Sun arc.
Calling LD the angle between the Sun and Moon, and h1 and h2 the
altitudes of the Moon and Sun respectively, you can calculate a
quantity which is the "corner cosine" at the Moon in the triangle:
   A = [sin(h2) - sin(h1)*cos(LD)] / [cos(h1)*sin(LD)].
This follows directly and simply from the cosine formula of spherical
trigonometry (and swapping altitudes, h1 and h2, for zenith distances,
z1 and z2. The quantity A is the cosine of the angle between the
Moon-Sun arc and the vertical arc passing through the Moon. If A is
less than zero, the navigator should shoot the Lower Limb. If A is
greater than zero, shoot the Upper Limb.

By the way, I'm not suggesting that anyone should go through any of
this math in practical navigation. Although it's not particularly
difficult, the visual test is sufficient. But it would certainly be
useful information in sight planning software, for example. Also notice
that the altitudes and the angle LD (the center-to-center lunar
distance) can be measured directly, as long as the Sun is still in the
sky, or calculated based on the estimated position and the local time.
Great accuracy is not required.

Jim Van Zandt's very slick vector approach to the problem should yield
the same results (though he works in terms of the azimuth difference
between the two bodies instead of the great circle angle between them).
Say, Jim, just of out "math culture" curiosity, where, or rather, in
what course of study, did you learn to do angular positional
calculations that way? I'm so accustomed to the spherical trig approach
that I never think of using vectors for these problems, though of
course they are geometrically equivalent.

-FER
PS: At Full Moon, the great circle arc from the Sun to the Moon may be
many degrees away from the ecliptic --even 90 degrees. In that case,
the line of cusps is nowhere near the Moon's N-S line. But as George
has already mentioned, the difference is insignificant for cases so
close to Full Moon.


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