I was once asked “Why do you believe the Earth to be revolving around the Sun, and not the other way around?”
Here’s the answer I gave:
I don’t. Let me explain.
Imagine for a second that the Earth and Sun are the only massive objects in the Universe. If you are an observer on the surface of the Earth, you might feel that this description is less useful than the one centered around exactly where you are: you look at the sky and see the Sun going roughly in circles around you. That is, until you look a little more closely and you notice that the circles move with the seasons. That’s fine, you can take that into account by introducing perturbations of sorts into the trajectory of the Sun and find a reasonably simple equation that seems to account for it. You still have no explanation about what's going on, but at least you are getting closer to a predictive model. You can start wooing the rest of your tribe by predicting the next eclipse for example. Oh wait, we haven't even put the Moon into the universe yet. Hold on, we will in a second.
Let's not bring planets and satellites into the picture yet, but let's add the rest of the universe. Now we have stars in the sky that appear like a sphere with some bright dots on it and that rotates around us. To the naked eye, this is a very stable object that moves in a very simple way. It looks perfect and eternal, except for the occasional supernova. And it looks very much like we are at the center of it. That is, until you look with a big telescope like Bessel did in 1838 and make measurements of star positions at different times of the year precise enough to notice that only some of them move with the seasons, and some others, such as Sirius, wobble over a few decades (another one of Bessel's discoveries). But let's assume you don't have access to a big telescope and that you don't know that.
So let's bring the Moon in now. That is an object that has many striking features but it is yet another object that is just going in circles around us. Because it occasionally passes in front of the Sun, we know that the Moon is closer to us than the Sun. The phases of the Moon show us that it is a sphere that gets lit by the Sun differently as it revolves around the Earth. Good.
But there is another interesting observation that you can make at this point: tides are synchronized with the trajectories of the Moon and Sun: the tides are the strongest when the Moon and Sun are aligned with the Earth, and the weakest when the Moon, Earth and Sun form a right triangle with the Earth at the right angle. That gives us a first hint at an influence at a distance of the heavenly bodies onto Earth-bound objects.
Now's the time to bring the rest of the solar system in. To the naked eye, we have new dots of light that don't behave like the other dots of light: they are moving relative to the background of dots, but all roughly stay on the same circle that the Sun is moving along. But whereas the Sun is moving steadily along that circle, the new dots don't. Two of them wobble back and forth around the position of the Sun, while the others go all the way around but not at a steady pace, even reversing direction sometimes.
Notice that so far we've only used observations that anybody can reproduce with a pair of eyes and optionally some modest travelling around the world and a small telescope.
So how do we interpret that data?
We can try to fit it with a set of functions, which is more or less what epicycles were used for. It does work to a degree, and when the precision of the model is not good enough, we can add more and more epicycles and get closer to the data. That does have a predictive virtue, but all it proves is that the movement of celestial bodies is smooth and regular: you can actually fit any smooth function with a composition of elementary ones. That’s basic mathematics. For example, any sound can be approximated with a series of harmonics, and the more harmonics you add, the closer you get to the original sound. The difference here is that we're talking about the music of the spheres instead of actual sounds.
Nothing in all this of course explains where the epicycles come from.
Now let's shift perspective and imagine that the Sun doesn't move (much) and that everything else but the background of stars -including the Earth- rotates around it. If you get the distances right, the very complicated movements of the planets become ellipses around the Sun, ellipses being about the simplest shapes after a straight line and a circle. You can now get a much closer fit to the actual movement of planets, with a much simpler model. That's what Kepler did in 1605. But you don't have to take his word or any scientist's word for it: it's easy to build a computer simulation of that model that computes the position of each planet in the heliocentric referential and then translates the positions back into geocentric coordinates. You can then compare the results with what you see in the sky.
At this point, we have successfully decomposed what seemed like very complex trajectories into a composition of simple elliptical trajectories: that of the Earth and that of each other planet. Divide and conquer.
We still don't have an explanation of the ellipses though. It took the genius of Newton to come up with that. He single-handedly invented infinitesimal calculus (at the same time as Leibniz) and applied it to expose the laws of motion and then discovered the law of universal gravitation. With those tools, he was able to further reduce Kepler's results and laws to one single law: all bodies in the universe attract each other with a force that is inversely proportional to their distance and proportional to each of their masses.
Not only does this explain all of the data we've talked about so far (including tides), it also unifies the fall of earthly objects with the movement of planets. Furthermore, it accounts for the movements of the Moon and the satellites of Saturn and Jupiter, two "mini-solar-systems" in orbit around the Sun. By the way, imagine if intelligent life were to evolve on a satellite of a giant planet: their inhabitants would have an even harder problem to solve than ours as they would have to untangle the composition of three orbital movements instead of two.
An interesting remark to make at this point is that Newton put the final nail in the coffin for the geocentric view of the world, but he also was the first to point out that his own theory made the heliocentric view inexact as well: the static point in his model is the center of mass of the whole Solar System, not the center of the Sun. And now we know even that is false as the Solar System moves relative to the center of the Milky Way, which itself moves relative to the local group. More than that, we know that there is nothing special with regards to the physical laws of gravitation about the center of the Sun or the center of the Earth: the physical laws are the same in both points, and we know how to describe the system from both referentials (although one description is simpler than the other, they are two views on the same reality). That is not to say that choosing the geocentric referential is not useful, interesting or efficient to solve a whole class of problems, but if your goal is to understand the movement of planets and stars, it’s not the most efficient and has actually been in the way of figuring it out for many centuries.
The law of universal gravitation enables us to make calculations on the evolution of a system with any number of masses and to go beyond the approximation provided by Kepler's laws. It's been so efficient that with just that tool and the observation data of planet trajectories, which had some anomalous divergences from what Newton's theory predicted, Le Verrier was able to predict the existence and position of a new planet, Neptune, before anyone had seen it. And sure enough, when astronomers pointed their telescopes in the direction given by Le Verrier, there it was.
So to summarize what happened here, scientists were seeing a deviation from the theory, and that could mean three things.
First, that the observations were wrong. That can happen, and the normal procedure is to repeat the observations by as many independent observers as possible. The observations were right.
Second, that the theory was wrong. We'll see something like that happened a little later.
Third, that the anomaly resulted from the perturbations within the theory of an as yet unseen object.
Le Verrier tried the third hypothesis and deduced from it the existence of Neptune. All that had to be done was to check against observation, and the planet was discovered (and Newton's theory confirmed once more).
Encouraged by this success, Le Verrier tried to apply the same method to another anomaly: the orbit of Mercury. He also predicted the existence of a planet between Mercury and the Sun that could account for the perturbations, but this time no such planet was observed. This time, we were in the second case, and it wasn't until Einstein came up with General Relativity that it was confirmed and that Newton's theory was replaced.
This is a great illustration of how science works: observation is the ultimate judge of the validity of a theory. Not common sense, not faith, not hearsay, not authority, not personal preferences. Reality doesn't care what we believe.
So to get back to the original question, no, I do not believe that the Earth revolves around the center of the Sun. I know that the law of universal gravitation -or General Relativity if you need the extra precision- is the most efficient way of explaining the movements that we observe in the Solar System. I know it because I can and have verified it and because innumerable people have done so as well, independently and consistently. I have actually verified myself quite a lot more than that about gravitation when I was a PhD student in a General Relativity lab.
You can also verify it yourself with little effort: grab one of the available open source astronomy programs or solar system simulators out there and look at their algorithms. Then take one of the predictions it’s making about the position of a planet in the sky for tonight and check it for yourself.
This brings me to an important point. While observing the skies is within the reach of almost anyone (I'm saying "almost" because I live in Seattle), verifying all scientific results yourself is impractical when not plain impossible. So how do we know something without having to rely on some form of authority? That is of course a very relevant question that all scientists must have asked themselves at a point of their career (hopefully at the very beginning). The answer to that is that the principle that a scientific assertion can be verified independently is absolutely essential. You cannot qualify a result as scientific unless it is verifiable and falsifiable. And it’s not going to be trusted until it has actually been independently verified. Yes, I’m saying that you have to evaluate and verify scientist’s claims. You absolutely have to. That’s the point. There’s no science if nobody does.
Of course, that doesn’t exclude the possibility of error and fraud, but it provides a set of tools that while imperfect is very efficient and powerful. The scientific community is so large that all results, especially the spectacular and important ones are going to receive lots of scrutiny. They’re going to be challenged in every way people can think of as the reward to overthrowing a previous result grows with the exposure that result has received. That is why Einstein’s glory is greater today than Newton’s.
Don’t get me wrong, there are cases where scientists benefit professionally from defending the status quo, but those can’t stop good new science from appearing and from eventually prevailing.
This is why I trust scientific results more than our primate’s intuition that would have me believe that the Earth doesn’t move, that there’s a point in playing the lottery or that an electron can’t be at two places simultaneously.