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How would I go around answering these questions (e.g., for $X=10,000$ -- IOW, historical observations are not available and the time is too long ago to rely on a fixed calendar):

  1. How many days ago was the spring equinox X years ago?
  2. What phase (how many days from conjunction) was the moon on that day?

Here is my "preliminary research":

Methodology

I suppose I could solve the 3 body problem (Sun, Earth, Moon) numerically backwards in time, but I suspect that the numerical errors and gravitation perturbations from other planets will accumulate rather quickly (how quickly?)

I could do arithmetic with the durations of lunation and year, but it appears that they are not as stable as I wish they were (are they?)

Software

Online calculators (NOAA, Stellafane) do not go that far back. Presumably they rely on a version of Almagest but, again, it is not clear if these formulas are reliable for 3-5-10k year calculations.

Question:

How would I answer 1 & 2 above?

sds
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    "but it appears that they are not as stable as I wish they were (are they?)" No, they are not and this would be difficult. Astronomers, of course, can do this but the are familiar with formulas we are not. If we knew the formulas the math would be straightforward (not necessarily easy) but straight forward. But with out the formulas we don't have a rat's fart in a Michelin star restaurant chance of doing it.... tl;dr.... ask an astronomer. Not a mathematician. – fleablood Apr 15 '21 at 19:33

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