Is there a proof that Earth moves?
How do we know that Earth and other planets go around the Sun, and not the opposite?
This is a very good question. For some time after it was accepted that planets (including Earth) circle the Sun, there was no direct proof that it is Earth that moves, and not the Sun.
Ancient astronomer Aristarchus thought that Earth goes around the Sun since he discovered that the Sun is much bigger than Earth. Copernicus (who is usually credited with the idea that planets go around the Sun) thought that Solar system centered on the Sun is more logical and beautiful, but had no definite proof. Kepler discovered that laws governing of orbits become much more simple if the Sun is in their center. Newton has shown that this is caused by the universal law of gravity. If gravity works, Earth and other planets have to go around the Sun, because it is much heavier.
There was no straightforward proof for the motion of Earth until 1725 when James Bradley discovered stellar aberration. This is (apparent) yearly change in positions of all stars in the sky due to Earth's own motion. Aberration arises due to adding up of the speed of light coming from the star and Earth's own speed. This is a very complex phenomenon and its description requires some math.
Another, much simpler, consequence of Earth's motion is stellar parallax. If Earth changes its position relative to the stars, then the stars should appear to change position in the course of the year.
A common experiment illustrating parallax is just looking at a close object (a finger, a pencil etc) with one eye at the time. When you switch from one to the other eye, the object will appear to move against the background. Closer the object is to your eyes, more pronounced the effect is.
Parallax should not be confused with aberration: parallax arises from the change of Earth's position and depends on the distance to the star, while aberration is caused by Earth's great speed and does not depend on how far the star is.
Parallax of a star was first measured by Bessel in 1838. It was not measured before because this change of star's apparent position is very small (the stars are very far from us). This was a very important discovery because Aristotle himself mentioned the lack of observable stellar parallax as the proof that the Earth is not moving (he didn't have a telescope and didn't know that the stars are so distant).
A third discovery proving Earth's motion was that of Doppler effect. Wavelength of the light that we receive from objects moving relative to us becomes a little shorter (i.e. bluer) when we approach the source and becomes longer (i.e. redder) when we move away from the source. When Earth moves toward a star, the star will appear slightly bluer (only high-tech instruments can measure this) while it will appear redder when Earth is on the other side of the orbit and moves in the opposite direction. This effect proves that Earth has a velocity relative to the stars, similar to aberration.
So, aberration (slight change in stellar positions due to Earth's speed), parallax (slight change in stellar positions due to Earth's changing position) and Doppler effect (slight change in color of stars due to Earth's speed) all prove that Earth is moving around the Sun, and not the other way round.
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