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I know that describing the trajectory of all planets in the solar system around the Earth is much more complicated than if we take the Sun as the reference point. But besides this, what is the simplest experiment that can prove that the Sun is a more "inertial" point of reference than the Earth? Do we have to look at the stars in the background or something like that?

Qmechanic
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In fact, if our system consists of only earth and sun, we'd never ever know, because we can never measure the difference, since you can always change the frame of reference between earth and sun, and both statements will be true (earth goes around the sun or vice-versa).

That's why, you need a third impartial object, which typically is a star far, far away. The star gives some light, and we can measure the light abberation caused by the earth's movement, and that's how we know.