theoretical astronomical dating of ancient archaeological sites

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They say that many archeological sites around the world are nearly perfectly alligned with certain stars and solar coordinates, most varying from absolute by only a few arc minutes. Due to the constant polar shift of Earth the north Star does a bit of a 'wobble' over time and I was wondering...hypothetically...what if these archaeological sites were dead on at the time of construction(?). Thus a calculation determining when the archaeological sites would/could have lined up with their intended targets may prove useful in dating these sites.

Any help is appreciated! TY all in advance!