All the books say that the Earth revolves around the sun, but how do you prove it? In case you ever time-travel to the middle ages, we have a (relatively) quick way for you to demonstrate heliocentrism to the King.
Today, we have many advantages over ancient astronomers. We have top-notch telescopes that can detect and record a far wider spectrum than mere visible light. We can easily travel to the far reaches of the globe in order to observe the sky. And of course there is the International Space Station.
(Pictured Above: Something Copernicus didn't have.)
Our greatest advantage is the fact that we already know what we are looking for. There's no need to synthesize a vast amount of discordant data into a harmonious theory. We only need to look for ways to confirm a theory that has been presented to us.
Part One: Observation
Most of us have held onto a signpost, or the bar of a merry-go-round, or the hand of a parent, and swung around in a circle. What did we see when we looked outward? We saw one-half of the world around us. We saw that half of the world slide away, as we swung sideways. Then, as we completed our rotation, the original half of the world was restored to us. Since the earth is doing the exact same thing that we did as children, all we need to do is show that its 'view' of the rotation is the same as ours was.
We must simply go outside every night for a year. At first the stars will look like a jumbled mass, but as we get more familiar with them, certain patterns become clear. Over a few months, the patterns will slide sideways, disappearing over the edge of the horizon. At six months, the original patterns will be completely out of view. Then, over the next few months, they will turn back into view, and at the one year mark they will return to their original configuration.
So far, so good. And so far, so obvious. There was a reason why people didn't catch on the idea of heliocentricity for thousands of years.
When we spin in a circle, we can feel the wind on our face, and the effort it takes to hold ourselves to the path of our orbit. We do not feel such things as the earth orbits. Doesn't this present a compelling case that the universe, including the sun, is turning around us, and not the other way around? Since we can't step into space to check its stability, how can we prove that the earth is rotating and the universe isn't?
Part Two: Documentation
This part of the problem requires a more detailed look at the stars. We need to recognize shifts in position of individual stars in relation to each other, not just the movement of entire constellations.
Although looking up at the night sky can give one the impression of looking at lights set into a flat black background, space is three-dimensional. Some stars are far closer to the earth than others, and as the earth rotates around the sun, the distance between them causes us to see them change position in relation to each other.
This is most easily demonstrated visually:
As the earth rotates around the sun, our view of the stars changes. When observing from the earth in the uppermost position, an astronomer would see the red star slightly to the right of the orange star. Someone observing from the earth in the lowest position would see the red star to the left of the orange star. A person observing from the earth in the center of the figure would see only the orange star.
These observations are very different from the ones we would make in a geocentric universe:
A person on a stationary earth would see the orange and red stars only in a single position as they moved around the earth.
By keeping a careful record of the stars at night, including their proximity to each other, we are able to see apparent shifts in star position over time. These shifts in star position, paradoxically enough, prove that universe around us is (relatively) stationary, and we are the ones moving. And all it took was a year out in the freezing cold, carefully documenting star positions, to prove something that we already knew was true.