At the time, heliocentrism explained most phenomena with fewer assumptions than did geocentrism:
1. Why do some planets undergo retrograde motion, while others don't?
2. Why does retrograde motion only occur when a planet is opposite the Sun?
3. Why is there retrograde motion at all?
4. Why is it that we always observe smaller objects orbiting larger objects? (the Moon orbits the Earth, Jupiter's moons orbit Jupiter, implying that the Earth probably also orbits the Sun)
5. What are the orbital distances of the planets? This is completely undermined in the geocentric system, but very tightly constrained in the heliocentric system! In a geocentric world, there's no reason why it should be possible to even create a reasonably fitting heliocentric model. The opposite is not true.
Different oservations of apparent angular diameters were wildly inconsistent at the time of Galileo. It's natural that he would give less credence to an argument based on such shaky observations, and more credence to the many types of observations that favored heliocentrism.
Beyond these observations, there was a more general consideration. The geocentric model had been based on a view of the Universe that held the Earth to be fundamentally different from the heavens. Galileo's observations (of mountains and craters on the Moon and of sunspots, for example) showed that the heavens were actually very much like the Earth. The heavens were not a separate realm of perfect forms. They were made up of real worlds with their own landscapes and imperfections. Putting the Earth at the center of that Universe seemed absurd, whereas it hadn't before.
> Procyon has the same diameter and brightness as Saturn.
How could he possibly that have determined? I don't think that Brahe could measure this he hadn't even a telescope. The apparent diameter you see from stars is diffraction limited.
Tycho was pre-telescope... so he had no idea that Saturn has a bigger visual diameter than Procyon... But, he could at least figure it out indirectly - by noticing that all stars twinkle a lot, while planets do not (or only a bit, on the worst nights).
- Procyon has the same diameter and brightness as Saturn.
- If Procyon is much farther than (say) 100 times Saturn’s distance, simple geometry proves its actual size would dwarf the sun.
- All the stars would dwarf the Sun, which would then be the only pea in a universe of melons, which is absurd.
- But if Procyon were any closer, there would be visible parallax from the Earth's revolution.
- There is no visible parallax.
- Lack of parallax plus the apparent size of the stars therefore requires a stationary Earth. QED.
The problem with heliocentrism is that it requires a lot of additional hypothesis that are not needed by other (better performing !) models.