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Copernicus deserves a mention. Galileo's problems resulted (in part) from him being a proponent of Copernicism after the church had declared it heresy.
Heliocentrism was suggested by Copernicus and Galileo built on that, including developing physics to the point where he couldn't believe otherwise.
The heliocentric models predicted the orbits worse than epicyclic geocentric ones and that is the reason Galileo was told to shut up, the court transcript is like 99% science and then a single subordinate clause saying "it also contradicts the bible".
Galileo insisted on circular orbits which was his downfall, ironically "because circles are perfect and god would furnish the universe perfect": That kind of religious language while also being worse science than what was already established did him in. Kepler, based on Brahe's data, was the first one to get a heliocentric model right and more accurate than the epicyclic ones.
Also earth doesn't revolve around the sun. If anything both revolve around their shared centre of gravity but really it's a matter of your frame of reference. Paraphrasing Archimedes: Give me a fixed point in the universe and I will move all your models.
The sun. Does the earth and moon orbit a gravitational center that they share? Does that center revolve around the sun? How imperceptible is this, considering the shared point is likely inside the earth given the difference between its mass and lua's?
Does the earth and moon orbit a gravitational center that they share?
Yes.
Does that center revolve around the sun?
No, it revolves around its shared centre with the sun. Things get complicated fast in gravity with more than two objects involved.
How imperceptible is this, considering the shared point is likely inside the earth given the difference between its mass and lua’s?
It's how we can detect exoplanets. Well, we can tell there must be these and these planets based on the star's wobble dunno if we can do the same for moons of planets: It would certainly work in principle but our instruments might not be good enough.
Does the earth and moon orbit a gravitational center that they share?
https://www.skymarvels.com/gallery/Vid - Earth-Moon Barycenter.htm