Did we ever actually see the earth revolving around the sun? Is the geocentric model completely disproved?
Did we ever actually see the Earth revolving around the Sun?
Also, is the geocentric model completely disproved or was it sidelined because the heliocentric model made things easier to understand?
(BTW I know Earth revolves around the Sun and am just asking out of curiosity.)
What you're asking, basically, is whether there are any proofs for the heliocentric model of the Solar System.
A literal naked-eye observation of the Earth revolving around the Sun would be rather difficult, since human beings have never gone to another planet yet, and have only been to the Moon briefly, decades ago.
Here are several proofs; some of them are historically relevant also.
Kepler's laws of planetary motion
This became one of the earliest proofs, as soon as Newton figured out the law of universal gravitation, and the "fluxions" (what we would call today differential equations). When you assume a heliocentric model, and the inverse square law for gravity, then Kepler's laws in a heliocentric model come out of the equations naturally, as soon as you do the math.
This is like saying: "if it's heliocentric, and knowing that the law of gravity is correct, then Kepler's laws should be such-and-such". And then: "oh, but the theoretical calculations for Kepler's laws match the actual observations with great precision. Therefore, our hypothesis (heliocentric, inverse square law) must be correct."
It was the earliest strong indication that the heliocentric model is in natural accord with the basic laws of science, whereas the geocentric view was becoming more and more contrived as evidence accumulated.
Tycho Brahe in late 1500s provided the enormous mass of observations of planetary motion. Johannes Kepler in early 1600s used Brahe's observations to come up with his laws empyrically (and also arguing for the heliocentric model). Isaac Newton in late 1600s said "yes, Kepler is right, because of mathematics and the law of gravity, and here's the proof from calculus".
An early argument against heliocentrism was that, if the Earth was really revolving around the Sun, then very distant objects, such as the stars, would appear to be ever-so-slightly bobbing back and forth around their average positions. Since that's not the case, it was argued, therefore the Earth must be fixed.
You can see this argument in historical archives, proposed by theologians in the late 1600s, in favor of the geocentric model, and against the growing consensus then of "natural philosophers" (what we would call today scientists) that the heliocentric model was correct.
In reality, parallax does exist, it's just very small. It was measured experimentally in the 1800s, and was then quickly used to determine, for the first time, the distance to the nearest stars.
Aberration of starlight
The direction where we see a distant star also changes when Earth's speed vector changes during its revolution around the Sun. This is different from parallax; it's more akin to the way raindrops on the side windows of a car leave diagonal traces when the car starts moving (even though the raindrops fall vertically).
It's essentially a relativistic phenomenon (when applied to light), but it can be partially explained in a classic framework. It was actually observed before parallax in the late 1600s (the heyday of Newton), but it went unexplained until early 1700s.
Orbital mechanics of interplanetary probes
Landing a probe on Mars or Venus would simply not work if you assumed a geocentric model. A geocentric description of the Solar System might remain valid in a purely kinematic perspective (just the geometry of motion) as long as you remain on Earth. But the illusion breaks down quickly as soon as you start to consider dynamics (see Kepler's laws), and/or when you try to actually leave Earth (space probes).
Let me reinforce this point, since several answers and comments got it wrong: the geocentric and the heliocentric models are not completely interchangeable, or a matter of relativity. You could build an "explanatory" geocentric model, and it would be "correct", purely in a kinematic fashion (the geometry of motion), and only as seen from Earth. But the model breaks down as soon as you consider dynamics (forces and masses); it would also reveal itself as incorrect even from a kinematic perspective as soon as you leave Earth.
This is not just an artifice to simplify calculations. The dynamic calculations are wrong in a geocentric model.
In order to compute the very high precision trajectory of the space vehicle carrying the Curiosity rover and successfully place it on Mars, you must operate from a heliocentric perspective. The dynamics are all wrong otherwise. You would not miss the target only a little, in a geocentric approach, you would miss it by a lot - the vehicle would not even go in the general direction of Mars.
When observed in a telescope, Venus has phases like the Moon, and also grows and then shrinks in size, synchronized with its phases (it's large as a thin crescent, it's small when it's gibbous). In a geocentric model, the size changes could be explained by an elliptic orbit of Venus around Earth, but the phase changes synchronized with that are harder to explain. Both phenomena become trivial to explain in a heliocentric model.
It should be noted that this does not necessarily prove the heliocentric model, just the fact that Venus is orbiting the Sun, not the Earth. So it's an argument against pure (or strict) geocentrism.
Jupiter, when observed in a telescope, clearly has its own satellites. This was an early blow against a strict geocentric model, which assumed that everything must orbit the Earth. It opened the door to the idea that orbits could be centered on other celestial bodies too, and to the idea that things in orbit around larger objects could have their own smaller satellites (and therefore the Earth could orbit the Sun without losing the Moon).
The list could continue (and the full list is very long) but these arguments should suffice. You don't necessarily have to see something with your own eyes in order to know with certainty that it's there. In the case of Earth's revolution around the Sun, it was simply a matter of an overwhelming amount of evidence piling up in favor of it.
Geocentrism simply doesn't make any sense whatsoever in modern science and space exploration.
Can someone provide an example of a dynamics calculation that goes wrong in a geocentric model?
Well, epi-cycles were introduced to correct for the retrograde motions of the plants in a geocentric model; they never really worked super well, and even if one could make them work mathematically, they represent really complicating and unphysical orbits... Not the best argument, but I thought it should be included here.
Well, at a simple level, in a geocentric model you're assuming that the frame of the earth is an inertial frame, so using classical dynamics F = ma. But in that frame F *doesn't* equal ma, there are centrifugal and coriolis forces. The heliocentric model describes that frame as non-inertial, accounting for the extra terms in the dynamics. The "it's all just choice of frame of reference" claim has to face the issue that the only frame of reference in which those terms disappear from your dynamics equations is that of the Sun. So what does that mean, physically? :-)
Mind you, general relativity means you'll still miss Mercury using heliocentric Newtonian dynamics. And maybe other planets, I'm not sure. But that's a much smaller error than treating the earth as a Newtonian inertial frame, and therefore failing to go even in the general direction of the planet you want.