How Galileo Proved those Spots were on the Sun

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In an attempt to rescue the "perfectability of the heavens," which the Catholic Church teachings required, Galileo's enemy Christoph Scheiner claimed that sunspots were simply undiscovered planets of the Sun occasionally crossing in front of its disk. (Note that Scheiner was inadvertently also acknowledging that the Sun could indeed have planets, such as the Earth, encircling it).

Galileo responded to Scheiner by arguing that sunspots change their shapes and that they are often seen to originate on the solar disk and perish there. Thus they could not be solar planets. They must reside on the Sun, and therefore the Sun is not perfect.

But even these rational arguments did not silence Scheiner, nor Galileo's other critics. However, another of his arguments was so convincing that even Scheiner eventually acknowledged the spots:

Galileo noticed that, while a spot took about 14 days to cross from one side of the Sun to the other, its rate of motion was by no means uniform. That is, the spot's motion always appeared much slower when near the edge of the Sun than when near the center. This Galileo recognized as an effect of foreshortening. That is, a spot coming around the limb is actually traveling towards you. But your eyes can't see the 3-D effect, and the spot likes as if it were moving slowing across the disk. However, when the spot is in the middle 1/3 or so of the Sun, all its motion is across the disk. So it appears to be moving quickly.

Foreshortening would result if, and only if, the spot were on or very near the surface of the Sun. A planet orbiting the Sun would not appear to change its speed when moving across the Sun's disk.

Let's describe this geometrically, as Galileo did: Look at the figure, where the circle represents a view of the Sun from above and "you" represents you, an observer on Earth.

The points A, B, and C are at equal distances apart on the surface of the Sun. Assuming the Sun rotates at a constant rate, the letters represent positions of a sunspot at equal intervals of time. From your perspective, the spot moving from A to B only goes a little ways, so appears to move slowly. On the other hand, the spot moving from B to C covers a lot of ground, and appears to move rapidly.

If the spot were a planet revolving some distance away from the Sun, it's speed would not appear much different as the planet went past the limb and then across the center of the Sun's image.

### Watch the sun rotate

It requires only a simple calculation, performed by Galileo, to express these results mathematically. Can you prove Galileo's theory mathematically?

### Help -- I need a hint!

The only refutation of Galileo's conclusion was if there were indeed planets revolving around the Sun, they were moving irregularly, and always fastest when they crossed the center of the Sun's disk. No sort of reason could be imagined for this behavior, and even if a reason had been found, it would have again indicated that the mysterious planets were moving out of harmony with the uniformity to which medieval astronomy and the Church so strongly clung.