NavList:

Bob Goethe, you wrote:
"I have struggled with this whole concept.  It seems like the moon's orbit, as one of the class called "the three-body problem" may be chaotic:  fundamentally unpredictable.  But then I read other things that suggest we can predict the moon's orbit with enormous accuracy centuries into the future."

Chaos is about the long-term, and the long-term is relative. Compare with the weather. Your local weather forecast can give you a fairly decent forecast for the weather for tomorrow afternoon, 16 Oct 2022. But what would the forecast be for tomorrow afternoon one year from today, 16 Oct 2023?? We could make some general climatological statements about the weather a year from today, but nothing more. For a period of "some days" in the weather, we can predict the future (and it's now done by "numerical integration", more below). Months ahead? Almost nothing.

The motions of the moons and planets in the Solar System are chaotic in the very long-term, but we can also predict their coordinates far into the future. Here our "few days" of weather is a "few centuries" of celestial mechanics. Beyond that? The planet Mercury may well be ejected from the Solar System in a few hundred million years. We can assess the statistical probability to a significant extent. But will it happen in a million centuries or a million and one? That's hidden behind chaotic dynamics. The time horizon is much longer because celestial mechanics is simpler than meteorology in terms of fundamental laws and because the system itself is radically simpler. We can work out a detailed ephemeris of the planet Mercury centuries into the future and its chaotic catastrophe millions of years into the future by accounting for fewer than a hundred massive objects in the Solar System.

"I lack the math background to understand the phrases "analytical integrations" and "numerical integrations".  I look forward to the discussion here."

Unfortunately, you'll encounter a certain amount of browbeating on this topic. Be aware.

Analytical solutions are equations that you can write down that appear to allow us to calculate the future from a few lines of math. They have a "Euclidean" feel to them, as if all of physical science could be written down as a fixed and eternal set of rules that can be derived by mathematical "analysis" from fundamental laws. The traditional equations for so-called "Keplerian elllipses" are like this. They are short, clean, sufficiently intricate to invite a certain "religious devotion" and they depend on only a handful of parameters for each orbit. They're a bit like magic. Of course even from the time of Newton, experts in celestial mechanics, like Edumund Halley, for example, understood that the Keplerian ellipses were not the whole story. The orbits of comets could be perturned by Jupiter and other planets. Working out the changes in orbits caused by these higher order gravitational "perturbations" occupied the great mathematicians of the 18th and 19th centuries (here's a readable and minimally mathematical, albeit antique and dusty, introduction to the topic by George Biddell Airy from the 19th century). Until the development of modern computing techniques in the 20th century, analytical solutions were really the only option.

Numerical integrations are a lot like dead reckoning, but due to the nature of the system, radically more successful --there are no winds or currents in space (well, almost none)! You start out with assumed coordinates and motions for major objects like the Sun and Earth. Then you drop in some new object, like maybe a satellite passing near the Earth , moving with some specific velocity (example: Lucy spacecraft zooming low over Australia in less than 24 hours on its way to the Trojan asteroids). Next you calculate the gravitational acceleration of the satellite due to the Sun and the Earth. Those accelerations will slightly adjust the velocity. Then you move ahead one second in time (or one minute, or one day, whatever works in the problem at hand). You calculate where the satellite ends up after one second. Then repeat. Its position is a little different so the gravitation from the Earth and Sun are a little different. With the new gravitational accelerations, you calculate new velocities and then new positions. Over and over again. The techniques are highly optimized and involve mountains of detail, but essentially that's what's going on. It's similar to determining a boat's position by dead reckoning. That's "numerical integration". 

Numerical integration has exploded in recent decades, and it's even how weather is predicted today. The idea was present over a century ago, but it was almost unimaginable science fiction. How could we ever have detailed data on the entire atmosphere and the oceans, too, and how could we numerically integrate the complex and highly inter-connected equations that describe the thermal and hydrodynamic properties of the atmosphere? And yet we do now. When you hear about different models predicting different hurricane tracks, these are runs of different numerical integrations. 

You concluded:
"I have observed reduced accuracy in my own moon sights...but when I did, I always assumed that the moon's illuminated limb didn't QUITE wrap around to the exact bottom for a lower limb observation, or to the exact top for an upper limb."

First, you should test that hypothesis. Do you really get convincingly worse results under those circumstances? Second, the illuminated limb of the Moon should not really be in doubt. It's really rather rare that the proper limb will be uncertain. Only for perhaps two days on either side of Full Moon should there be any question at all. Even then, there are easy tricks you can learn to make sure you're using the right limb. Honestly, though, it's probably not a major issue.

Frank Reed