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):
- How many days ago was the spring equinox X years ago?
- 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?