Can anyone explain why from the XVII century standpoint, heliocentrism required stars to be huge, much bigger than Sun? Which observation or measurement hinted at that?
Yes there was lack of observable star parallax. But XVII measurement precision was so, so much worse than the one required to detect parallax (15" was best precision achieved at the time vs <1" required to tentatively detect parallax and ~0.3" required to be sure about it), that stars could "be" 50x closer than they really are, and 50x smaller, and the world would still "fit" the observations if one assumes that parallax is just tad small enough to escape detection. Which means, nearly every star being much smaller than Sun. Such a small star is impossible, but since source of the stellar energy wasn't known anyway, that couldn't be determined.
"The most devastating argument against the Copernican universe was the star size problem. When we look at a star in the sky, it appears to have a small, fixed width. Knowing this width and the distance to the star, simple geometry reveals how big the star is (right). In geocentric models of the universe, the stars lie just beyond the planets, implying that star sizes are comparable to that of the sun (below). But Copernicus's heliocentric theory demands that the stars be extremely far away. This in turn implies that they should be absurdly large—hundreds of times bigger than the sun (bottom). Copernicans could not explain away the anomalous data without appeals to divine intervention. In reality, the stars are far away, but their apparent width is an illusion, an artifact of the way light behaves as it enters a pupil or telescope—behavior that scientists would not understand for another 200 years."
Ah so the Airy disk was confused for the star size? Then true; but that is sick, one had to just observe same star with the same telescope, once with full aperture and once with the aperture diaphragmed to half diameter, to observe that the star has become 2x larger (and 4x dimmer).
Also let's see, so a typical telescope aperture of the era have been 25-30mm (bigger will make focal ratio too big and chromatism totally unacceptable), which means Airy disk 5" big, or 1/40000 of the distance. Assuming stars being 1/15 parsec away to make their parallax just barely unobservable, it means yeah, stars 25x the size of Sun, which is whole lot and isn't a reasonable assumption.
But measuring Airy disk size is easy by using a hair put in the focal plane to see how quickly it crosses (will be 1/2 vs 1/4 of a second)... Easy experiment to check that Airy disk isn't a "star".
Diffraction of light was only understood in the mid 19th century. They simply had no idea what to look for. In Galilei's time, making optical instruments was mostly based on trial and error.
OK one easier way to confirm that Airy disk isn't a "star". Speed of Moon movement vs stars was well known and is around 0.5'' per second, and Airy disk was 4-5'' wide. So, when a star is occulted by Moon, it should have disappeared slowly as the Moon "slides" over it, during ~10 seconds. In fact, it disappears instantly, because Airy disk isn't a star...
They wouldn't have needed to know what to look for if they had noticed that the "width" they ascribed to the stars changed with every measurement. Every single telescope size would have led them to a different figure, and if they had cared about pursuing quantitative inconsistencies they would have not only realized that they weren't measuring the width of the star, but also that the determining factor was the telescope's aperture.
Came to here to say thank you for this comment, this made me understand what everyone was talking about. I know some photography so your comment made my mind "click" and understand. And I also agree with the comments the same level as this - they didn't understand these things back then. I couldn't at first understand why they'd believe the stars must have been huge.
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).
Nature [1] cites Graney [2] that Galileo may have confused stars for their Airy disks. Nature's phrasing makes it seem that we're not entirely sure of Galileo's exact confusion.
> Graney suggested in 2008 that Galileo's observations of stars were actually diffraction patterns called Airy disks
Yes there was lack of observable star parallax. But XVII measurement precision was so, so much worse than the one required to detect parallax (15" was best precision achieved at the time vs <1" required to tentatively detect parallax and ~0.3" required to be sure about it), that stars could "be" 50x closer than they really are, and 50x smaller, and the world would still "fit" the observations if one assumes that parallax is just tad small enough to escape detection. Which means, nearly every star being much smaller than Sun. Such a small star is impossible, but since source of the stellar energy wasn't known anyway, that couldn't be determined.