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Was it during Galileo's era or was it a much earlier time with Greek philosophers when the idea of heliocentrism was rejected because the lack of parallax movement of the stars? The idea of stars being so far away they wouldn't show parallax movement wasn't acceptable without stronger evidence than what was available at the time, given how massive that would make outer space, so the simpler explanation was that the sun moved.

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Earlier, for the most part. There was a fairly well known set of standard objections. A lot of the others were to do with the implications of the earth rotating, like the fact that things seem to fall straight down (not quite, but it was only after Galileo's death that someone managed to demonstrate this clearly, and not for lack of trying), and that you don't get a huge wind generated by the rotation (it does affect the winds, but you need a far more sophisticated framework for understanding motion that only really started to take shape with Newton to put this together). The star parallax thing was further complicated by an quirk of optics, which is that stars don't appear to be perfect point objects, they (to the naked eye and to telescopes) do have some apparent diameter, and that diameter implied that if they were far away enough to not have parallax, not only would the universe need to be much bigger, the stars themselves would need to be huge, many times the size of the solar system.

It also didn't help that Galileo's model was still incorrect and inelegant, due to the insistence on circles and epicycles. It actually needed more cycles to explain than the geocentric version of the model! It was Kepler that actually got things right and elegant by allowing orbits to be ellipses.

(As an aside, there's one exchange of letters between Galileo and a bigwig member of the church, where amusingly, to modern observers, the church guy was more correct about the astronomy and Galileo was more correct on the theology!)

This is a really good overview. However:

> It actually needed more cycles to explain than the geocentric version of the model!

This part is not true. The main advantage of Copernicus's model was that it reduced the number of epicycles needed. In the Ptolemaic system every planet (except the Moon and Sun) required an epicycle with a period of one year, which we now know was needed in order to account for the relative motion of the Earth on its own orbit. In a heliocentric system these epicycles could be eliminated. Copernicus presented his model more in terms of it requiring fewer calculations.

The Ptolemaic system was already not particularly favored and was well and truly killed in Gallileo's time (in part by his observations). The competition was Tycho Brahe's model, in which the sun revolves around the earth but the planets revolve around the sun. They for the most part make identical predictions as far as astronomical observations, but Brahe's was very slightly simpler and didn't cause the other objections.

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