How to measure the effects of the orbit around the Sun? It is well know that we can measure the spinning of the Earth with a Foucault pendulum.
But, is there a similar experiment for the orbit of the Earth around the Sun? I would like to know if we could prove that the Earth orbits the Sun without astronomical observation.
 A: You can only know that a Foucalt pendulum demonstrates the rotation of the earth by way of astronomical observations - that is, it is by observing the motion of the stars that you can tell how long the day is. 
That being so, you can determine that the motion of a Foucalt pendulum corresponds to the position of a distant star, that is, it corresponds to a sidereal day. Since this is 4 minutes shorter than a solar day, the orbital period of the earth can be determined.
It's true that this does not exactly meet your stated requirements, but without some astronomical observations your first statement is untrue. So, fair is fair.
A: You could argue that the cycle of the seasons can be used as an argument that the Earth and Sun rotate around each other. This is caused by the tilt of Earth's axis of rotation, relative to its orbital plane. Because relative to the direction between the Earth and the Sun this tilt rotates around in one year. However you could argue that this is caused by some axial precession (which the Earth does, but very slowly). However this can be ruled out since the orbital plane of the Moon remains fixed to Earths tilt, unless you want to argue that this orbital plane experiences precession as well. This method to rule out very strong axial precession does involves some astronomical observation, namely our Moon. But this can be done without any telescopes.
A: There are two big problems with this question:


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*Some kind of astronomical observation is needed.
Imagine intelligent aliens living on a habitable exoplanet that has a very thick and very opaque atmosphere and that has a slightly higher value of g than our 9.80665 m/s2. The thick, opaque atmosphere means they cannot see the stars or even their sun. They don't even have day and night. The higher value of g coupled with the thick atmosphere means they can't build rockets to escape their prison. Explaining the tides, the seasons, and the behavior of a Foucault pendulum on that exoplanet is going to be a very difficult task.

*You can't "prove" that the Earth orbits the Sun.
Ultimately, you can't disprove those pesky Aristotelian geocentricists who insist that the Earth is the center of the universe and that it's the universe that is rotating rather than the Earth. After all, all frames of reference are equally valid. Some frames are just a bit tougher to work with than are others.


I'll address the second problem by noting that except for some rather amusing crackpot websites, we've gotten past those pesky Aristotelian geocentricists a long time ago. Just because their point of view is valid doesn't mean it's "right". It's okay to use a geocentric point of view to explain the weather. (Challenge: Try explaining a hurricane from the perspective of an inertial frame. It's not easy.) It's not okay to use a geocentric point of view to explain the solar system, the galaxy, or the universe.
I'll address the first problem by interpreting "no astronomical observations" as meaning "not observing the stars or planets." I'm still allowed to observe the Sun and Moon. I don't see how the question is answerable otherwise. One simple approach is to observe the times at which the Sun rises and sets. This varies over the course of a year, and pretty much repeats year after year. There's still a problem here: What if the Sun is of negligible mass and is powered by something much stronger than fusion? Maybe those pesky geocentricists are right, after all! Maybe the Sun does orbit the Earth! We need to see a force (and we can't use parallax).
My solution: Use a gravimeter (better: a superconducting gravimeter), a Foucault pendulum (better: a precision rate integrating gyroscope), and a good clock. The clock will give us an independent measure of time. The Foucault pendulum will give us a measure of the Earth's rotation rate with respect to a non-rotating frame (the sidereal day).
The gravimeters will show a number of fluctuations due to direct and indirect tidal gravity effects. Some of those effects will be attributable to the Moon, others to the Sun. (This is why I need to know about the Moon.) After we get rid of those lunar effects, we'll see:


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*Semidiurnal and diurnal changes in sensed gravitational acceleration, with periods of 12 and 24 hours. Compare with the Foucault pendulum, which will have a period of  has a period that is consistently 3.93 minutes shy of 24 hours per our good clocks. This gives a good clue that the Earth is orbiting the Sun (or maybe vice versa).

*Those semidiurnal and diurnal changes in gravitational acceleration will vary in in intensity over of the course of a year. We're measuring tidal gravity, which varies with the inverse of the cube of the distance to the Sun. This makes this measurement rather sensitive to the slight eccentricity of the Earth's orbit.

*The period of these semidiurnal and diurnal changes in gravitational acceleration will not be exactly 12 and 24 hours. That too will vary over the course of a year. The phase difference changes by over 30 minutes over the course of a year, and the day-to-day changes will be greatest mid-December. We've just measured the equation of time!


So now we have a measurable force, and from that we can compute the gravitational force. It's rather large, more than large enough to be able to conclusively say that it's the Earth that's orbiting the Sun rather than vice verse.
