What goes into making a diagram like this? I recognize that the math of it may be tedious, if not complex, but what I’m really wondering about are what variables must be known to do the projection. I assume that the list would include the relative positions and velocities of the Earth, Moon, and Sun, as well as the direction and speed of rotation of the Earth. What have I missed?
Your guess was pretty good. You don’t need to “know” the position or velocity of the Sun, since you can use that as your reference location. But the position and velocity of the Earth and the Moon at any single time, along with the masses of the Earth and Sun, are enough to calculate everything about the orbital motions of the bodies. In fact, almost any set of data on those objects will be sufficient, so long as there are enough degrees of freedom in the data. For instance, you could in principle get all of the data you need for the Moon from three dated photographs of the Moon plus one range measurement, or from two photos, two range measurements, and a Doppler measurement, or almost anything that gives you seven numbers total (though of course some sets of measurements will be much more accurate and useful than others). In practice, one uses a set of numbers called the orbital elements, which contain information like the eccentricity of the orbit, how much the orbit is tilted relative to some reference plane, which direction the long axis of the orbit is pointing, etc., since the math is generally easier using these elements.
Of course, once you have the orbital motions, you also need to find where on the Earth the shadow will be. For this, you need the direction and speed of the Earth’s rotation, as you surmised, and you also need to know the orientation of the Earth at some reference time.
As I understand it, the orbit of the Moon is much too complicated to describe with just the usual orbital elements. Those will give a rough approximation of its position which is not good enough for eclipse predictions.
I believe eclipse predictors use numerical simulations of the Moon’s orbit, but I don’t know any details. It’s possible there may be something in Meeus if you want to do it yourself.
Thanks for your answers.
One of the things I’ve wondered about in this regard is how the early explorers were able to consult an almanac of eclipses (and, apocryphally, to coerce the ‘natives’ by threatening to take away the moon, or was it the sun?) if accurate prediction required anything remotely approaching precise knowledge of the variables I mentioned above.
There’s some discussion of the timescale on which such predictions improved towards the end of this old thread. Basically, through the 16th and 17th century the prediction of where solar eclipses would be visible from gradually improved, with Halley then publishing what was essentially a modern eclipse map for the one visible from England in 1715.
Stories about explorers using eclipses to scare natives are surely all versions of Columbus knowing that Regiomontanus had predicted an eclipse for February 29th 1504. But that was a lunar eclipse. It’s always been much easier to reasonably reliably predict the visibility of those.
I think that any orbit can be described with the same elements, but the problem with respect to the Moon is that the elements change fairly rapidly over time, primarily due to the influence of the Sun. So to predict eclipses at any distance into the future, you have to model not only the orbital elements, but how they change.