Much of the following is due to a fascinating essay at The Renaissance Mathematicus entitled Galileo’s great bluff and part of the reason why Kuhn is wrong.
An abiding puzzle for many is: why were the physicists and astronomers opposed to Galileo's Copernicanism? Most of his supporters were "Renaissance men," i.e., artists and men of letters who took a dilettante's interest in science. There were some physicists who famously refused to look through the newfangled telescope. This while the Jesuits were making significant telescopic investigations of their own; and were in fact teaching the Copernican calculation methods at the Roman College.
Much of the confusion, imho, comes from looking at history backwards instead of forwards. Backwards, meaning looking from the present into the past, and so looking with foreknowledge of what came after and often with the unexamined assumptions of the Late Modern Ages. But one achieves a better understanding by looking at history from its own past, and not allowing knowledge of next year to affect the view of this year.
A major difference between now and then is that astronomy was accounted a branch of mathematics, not of physics. That is why Galileo held a chair in mathematics at Padua, and why Tycho and Kepler successively held the title of Imperial Mathematician. The purpose of astronomy was to "save the appearances." That is, to accurately predict sunrise, sunset, the positions of the wandering stars, phases of the moon, the eclipses, and so forth -- all for the practical use of astrologers, cartographers, navigators, and churchmen (date of Easter stuff). This was regarded as very distinct from the wooly-headed world of the physicists, which had no practical significance. One of the markers of the Scientific Revolution was the notion that science should be put to the service of engineering and industry.
This also explains Bellarmine's instruction to Galileo, the teaching of Copernican calculations by the Jesuits, and Osiander's preface to Copernicus' book. No one had a problem with a mathematical system. The question was whether it was physically real. There was the revolution in astronomy: the notion that a set of equations that worked well meant that the world actually ran that way. It seemed to the Aristotelians to be rank Pythagorean number mysticism!
Beside, the Ptolemaic calculations worked pretty well already. They were sometimes off, but that was at least as much due to the quality of the data as to the system itself. The data they were using - the old Toledan tables and so forth - had become corrupted over the years by copyist errors. (Another modern unexamined assumption - the nature of the manuscript culture versus the printed book culture.) So the Ptolemaic system was not adding epicycle upon epicycle to correct errors ad hoc, nor teetering on the brink of collapse.
There was a problem in that Ptolemaic astronomy conflicted with Aristotelian physics. There was no room in Aristotle's physical explanation of motion for all those epicycles, let alone for the equants and descants. They would play royal hob with the crystalline spheres, for one thing. Everyone lived with the contradiction the way we live with the contradiction between Relativity and Quantum Theory. (Each provides an estimate of the Cosmological Constant -- but they differ by orders of magnitude.)
Another thing to keep in mind is that there were more than two contenders in the contest for System of the World - and the Copernican was neither obviously better, nor did it win in the end. There were two systems from ancient times:
1. Ptolemaic system: geocentric, stationary earth. Worked out in great detail by Ptolemy in his Almagest [title from an Arabic translation]. It was surprisingly accurate, as noted. Relativity theory tells us that the choice of frame of reference is arbitrary, so it is no less true mathematically than any other frame. As mentioned, the astronomical data was corrupted over the centuries, as errors crept into the data through the process of copying and translating astronomical manuscripts.
2. Heracleidian system. geoheliocentric, stationary earth. The two inner planets, Mercury and Venus, circle the Sun, while the Sun along with the other planets, Moon, Mars, Saturn and Jupiter, circle the Earth. This is actually physically true except for the stationary part! Draw the orbit of the earth. Now draw the orbits of Moon, Mars, etc. They all go around the Earth. Now apply the principle of relativity and designate Earth as the center of the reference frame. The Sun goes around the Earth. Problem: only a few fragments of Heracleidus Ponticus survive. It's possible that the system attributed to him by the Late Classical/Early Medievals (e.g., Macrobius) were reading too much into him. It was also called the Egyptian system, possibly because others had added to Heracleidus.
BTW: Why did Heracleides, Aristotle, Archimedes, Ptolemy and other very smart people put Earth in the center of the reference frame?
Ans.: They were making all their observations from the Earth.
Pythagorean note: Some ancients like Aristarchus of Samothrace were said to have proposed a true heliocentric system. But there is no direct evidence for this. No relevant writing of his survives. Archimedes relates that he has the earth going around the sun (which is silly, right?) but no statement that he has the other planets doing so. The Pythagoreans held that the sun was a big bronze mirror reflecting an unseen and unseeable Central Fire. Later, some held the sun to be the Central Fire. The Central Fire went in the center because i) fire was nobler than earth and ii) the center was a nobler position. Ergo... There are many names for this, but science, observational astronomy, and mathematics are not among them. There do not seem to be any astronomical tables developing this system the way Ptolemy developed the stationary geocentric system.
And then there were five more models in Early Modern Times.
3. Copernican system . heliocentric, rotating earth, circular orbits. This system had more epicycles than Ptolemy, and not notably simpler. It produced less accurate results in some cases, partly because it relied on the same inaccurate tables. It predicts that Venus would have phases. Copernicus knew of Aristarchus, but had to develop the mathematical system himself. He needed the epicycles because he was trying to use circles for the orbits.
Tycho Brahe, the Imperial Astronomer then began a program of creating new tables based on meticulous modern observations. [Still no telescopes, remember]. Nicholas Reymers Bär, known as "Ursus," visited and allegedly copied Tycho's work and published first, leading to a major feud with Kepler caught in the middle. Kepler went to work for Tycho.
4. Tychonic system [1587 et seq.] geoheliocentric, stationary earth. Everything revolves around sun, which revolves around Earth/Moon. Modeled on Heracleidian system. It predicts that Venus would have phases.
5. Ursine system [1587 et seq.] geoheliocentric, rotating earth. Created by Nicholas Reymers Bär ("Ursus.") Everything revolves around sun, which revolves around Earth/Moon, and the earth rotates. That is, same as Tycho's except for rotation. It predicts that Venus would have phases.
6. Gilbertine system  geocentric, rotating earth. Created by William Gilbert of magnetism fame. This was very popular, and when pure geocentrism was abandoned led many to adopt Ursus rather than Tycho. For some reason, astronomers seemed more open to a rotating Earth than a revolving Earth. The lack of stellar parallax seemed to falsify the latter.
1601. Tycho dies; Kepler gets his job.
7. Keplerian system . heliocentric, rotating earth, elliptical orbits. Created by Johannes Kepler. Predicts the phases of Venus. Eliminates epicycles. Generally Kool.
1610. Galileo Galilei observed the phases of Venus, thus supporting 2, 3, 4, 5, or 7. Pure geocentrism -- #1 and #6 -- was O-U-T out. This was possibly Galileo's sole original contribution to the debate. Otherwise, he merely supported pure #3 and ignored the correct model #7.
1632. Galileo Galilei published the Dialogue of the Two Chief World Systems. A very dishonest book. It presented #1 and #3 as the only two options, even though #1 had been pretty much knocked out of the contest and no astronomers were championing a pure geocentric system by then. In the book, he made no mention of the Tychonic, Ursine, or Keplerian systems. And his alleged killer proof of the Earth's rotation -- that the tides were sloshing due to the spin -- was utterly bogus. Not only did people know it had something to do with the Moon -- Heck, Aquinas knew that much! -- but it also contradicted his own argument against the Objection of the Winds.
1651. Giovanni Riccioli published the Almagestum novum, the frontispiece to which shows the Copernican and Tychonic systems weighing in against each other while Ptolemy lies prostrate on the ground, though crying "I will rise again!" This was the book Galileo should have written. But because Galileo had poisoned the well with his personal quarrel, Father Riccioli was constrained to support a stationary centered earth. But look carefully at the scales on the frontispiece. (It was Fr. Riccioli who named the first craters on the Moon -- after Copernicus, Tycho, Galileo, Gassendi, Clavius, etc. He clearly saw all of them as contributing to astronomical knowledge, whichever model they had supported.
But remember: Bellarmine has simply required that the heliocentric system no be taught as fact until empirical evidence could prove it so, and so far the parallax objection still falsified the theory. (Nor does there seem to be a record of any experiment like Gugliemini's ca. 1790 for eastward deflection of falling bodies.)
The Scientific Process
The Heracleidian model was the first to go, in 1587, superseded by Tycho and Ursus.
Ptolemy and Gilbert were tossed after 1610 and the discovery of phases in Venus. Copernicus, Tycho and Ursus survive. A new contender, Kepler, has entered the fight.
Kepler worked for Tycho and succeeded to Tycho's job in 1601 when Tycho died. Much of Kepler's work was started under Tycho's direction, using Tycho's more accurate observations in place of the error-filled tables Copernicus had used. (All of this before the telescope.) His Astronomia Nova was delayed over copyright hassles regarding use of Tycho's data.
The Copernican system faded away in the face of the superiority of Kepler’s elliptical system. By the early 1630s only three systems were left:
b) Tychonic systems without diurnal rotation.
c) Tychonic systems with diurnal rotation. [Ursine]
"At this point in the contest the Tychonic system with diurnal rotation was well ahead on points. This is the point at which Galileo published his Dialogo in which he presents a contest between the Copernican and Ptolemaic systems blithely ignoring the fact that both were effectively already out of the running. His book proved popular amongst literati who were not astronomers who enjoyed his very obvious polemic writing skills but contrary to popular opinion it didn’t play a significant role in the contemporary scientific discussion."
But how to choose between Kepler and Ursus/Tycho? All three predicted the phases of Venus; all three gave accurate predictions of the planets. The math was simpler for Kepler and two of his three "laws" were generally accepted. The "equal area" law seemed more weakly grounded. What was needed was an overarching physical theory that would make sense of one or another model. Mathematical simplicity was not a good reason for preferring one model over another. That is rank number mysticism. Tychonic/Ursine models remained popular into the 1660s. Newton's universal gravitation  finally put paid to them, as there was no similar physical theory that would account for geo-heliocentric systems.
Wikipedia states: "Within Kepler's religious view of the cosmos, the Sun (a symbol of God the Father) was the source of motive force in the solar system. As a physical basis, Kepler drew by analogy on William Gilbert's theory of the magnetic soul of the Earth from De Magnete (1600) and on his own work on optics. Kepler supposed that the motive power radiated by the Sun weakens with distance, causing faster or slower motion as planets move closer or farther from it."
Not a bad guess. There is a nascent notion here of universal gravitation, although religion and analogy seems to have played a primary role in developing it.
Newton finally came up with a theory that was relatively simple and from which the Keplerian system popped out as a consequence. This did not answer the empirical falsification of the earth's revolution -- no stellar parallax -- but the system was so elegant that everyone assumed that the stars must be much farther away than the ancients, Arabs, and medievals had estimated, and therefore the parallax must be very very small. They were right; but they assumed it. They didn't prove it. Science does not mean guessing right. So legitimate doubts continued. The Newton theory/Kepler model sure seemed right: concise, coherent, and the math worked out right. Seemed a shame to have that Popper-falsification thingie going on. This was not actually settled until the late 1700s. Stellar aberration convinced the scientists, but the effect was small, intermittent, and needed special equipment. Not very empirical for anyone lacking the equipment. That the earth was an oblate spheroid supported the rotation of the earth, assuming there was no other reason why the earth might be oblate, but again: not something that most people could experience and verify for themselves. Guglielmini's experiment dropping weights down the center of the Tower of Bologna detected the eastward deflection predicted by rotation. (Previous efforts had evidently run afoul of windage or other effects; that's why G dropped the weights inside the Tower.) In 1806, Calandrelli claimed to have found parallax in α-Lyrae; but ironically this may have been a false alarm. Bessel's discovery of stellar parallax in 1830 was undoubted. Foucault's pendulum provided further proof of Earth's rotation. Like the falling weights, it was direct and empirical, just as the telescopic observation of parallax was something actually seen.
Meanwhile, back in the world of physical theory, The Renaissance Mathematicus tells us....
"Even following the publication of Newton’s Principia we still don’t have a single, dominating. unchallenged model. Both the Cartesians and the Leibnizians challenged Newton’s model because it was not mechanical (with its action-at-a-distance concept of gravity). The Cartesians fought a rearguard action supporting Descartes’ vortex system against Newton’s gravity until deep into the 18th century. Leibniz challenged Newton’s assumption of absolute time and space, proposing a relative system instead; a challenge that would with time lead to the refutation of Newton’s system..."
It turned out Newton was wrong about both the nature of gravity and the absoluteness of space and time. See Einstein for details.
+ + +
For a discussion of Galileo's place in the scheme of things, history of science-wise, see The Renaissance Mathematicus: Pulling Out the Stopper.