I have recently been challenged to come up with an experiment that demonstrated the heleocentricity of the Earth-Sun system (that the Earth goes around the Sun) like experiments such as Foucault's Pendulum demonstrated the rotation of the Earth.
Demonstrating that you are accelerating is not as easy as Newton supposed. For instance, if you are sitting in a room and noticed that you were being pushed back in your chair, your cup slides towards you across the table and the lamp hangs from the ceiling at an angle, you could surmise that you are accelerating. OR that the room is tilting. OR that another mass has moved into position (somehow) to change the total gravitational field. OR that an invisible spirit is playing a joke on you. See what I mean?
There is nothing special, in physics, about heleocentricism - it is just another frame of reference. Its not even inertial since the Sun, with the entire solar system, accelerates around the local center of mass of the Galaxy and that about the local galactic cluster.
We only teach it in class because students seldom have the math to handle the arbitrariness of reference frames and because we favor, for philosophical reasons, those ideas that have the least amount of extraneous assumptions and fiddly bits.
So what am I to do?
Fortunately the phrasing of the challenge gives me a way forward ... you see, all the same arguments are valid for a spinning/not-spinning Earth. Classically we figure the Earth is spinning in the sense that we can show how coriolis and centrifgal effects vary over the surface. The spinning Earth is the least fiddly model that accurately accounts for what we can measure.
One of the consequences that leads to Foucault's pendulum also means it is easier to launch a satellite into a particular orbit if we have to account for the angular momentum of the Earth as it spins.
We have spacecraft that have visited the inner planets, and observation platforms about the Sun. In order to get them there, we have had to account for the orbital angular momentum of the Earth as it turns about the Sun. Here, then, is the experiment equivalent to the likes of Foucault's pendulum, that demonstrates heleocentricity in much the same way that other experiments demonstrate a spinning Earth.