Is there any simple experiment done on Earth proving that the Earth is rotating around the Sun? Something in the same spirit of what Foucault did proving that the Earth is rotating around itself.
P.S. I know that actually the system rotates around its center of mass. My point is to prove, by an experiment on Earth that we are rotating around some point.
P.S.2 - Done on Earth. One cannot watch the skies!
The most straightforward observation to show that the Earth moves is stellar parallax. If you take photographs of a groups of stars over a period of six months (half an orbit), some of the stars will seem to shift in position compared to the others. These stars are much closer to Earth and so seem to move more. This is similar to how, when you are riding in a car, the road under you seems to move much faster than far away hills.
The picture below shows an example of parallax with the Big Dipper providing a background of far away stars. The red circle indicates the position of a start much closer to Earth. The two smaller boxes show the view from Earth, with the bigger dot indicating the apparent position of the nearby star.
There is! It corresponds just to the tides due to the Sun.
Let us suppose that the Earth is not rotating around the Sun, that is, we are not in free fall towards the Sun. In this case the liquid from oceans would accumulate nearest the Sun. The effect would be only one daily tide.
On the other hand when we fall towards the Sun there are accumulation of liquid nearest and farther from the Sun causing two daily tides.
An easy way to see this is to remember that closer point to the Sun is pulled more (by gravity). The side closer the Sun is attracted more than the center of the Earth, which is attracted more than the farther side. Another way to see the effect is to compute the effective gravity generate by the real gravity plus a small correction due to non inertial effects.
This proves that we are falling towards the Sun and since we don't reach it, the only possibility is that we are in circular motion. Notice I am talking about tides due the Sun which correspond to a tiny effect compared to tides due to the Moon.
Another interesting point that someone could ask is whether that does contradict General Relativity or not. According to GR it would be impossible distinguish between a gravitational fields and accelerate frames by any local experiment. A body in free fall should not experience gravity. The thing is that the above mentioned experiment is non local. It compares results at a Earth's diameter distant from each other.
P.S.2 - Done on Earth. One cannot watch the skies!
If you insist on this, it cannot be done.
Your condition is equivalent to us living on a permanently clouded world. In that case, we would experience alternating light and dark periods and would have no way of hypothesizing what was happening. Understanding the universe would have to wait for the invention of aircraft that could pierce the cloud layer.
P.S. by watch, I assume you mean at any frequency.
However... If you relax this condition and allow naked-eye observations with no measuring equipment (so quite a bit less than the 16th century), then it's quite easy: In mid-winter, at midnight, you look at the North Pole region of the sky (Big Dipper, Polaris, etc.) and make a sketch (or just commit it to memory). Then, six months later, you look again (mid-summer, midnight, somewhere South of the Artic Circle). Compare the stars with what you saw in winter. Everything is half-a-turn round the sky.
Go into a dark room, paint some fake stars on the wall and ceiling, light a candle in the middle of the room and play around with an apple or orange or other spherical fruit. You should soon get it.
