Why do the 2 planets in a two-planet system orbit around the center of mass?

Question

When reading the Feynman Lectures (chapter 7, volume 1), Feynman was explaining the formation of tides on the Earth and said the following:

The moon does not just go around the earth, the earth and the moon both go around a central position, each falling toward this common position

I got confused by that statement. Does that mean that the 1st Kepler's Law actually means that a planet orbits around the center of mass and that this center of mass is the one located at one of the focus of the elliptical orbit? Or is the focus at the center of the massive body?

Moreover, if the Earth is orbiting around the sun, does that mean that it would be doing kind of small orbits in the large orbit? What I mean is that, if it is also orbiting around that center of mass from the Earth-Moon system, then its orbit around the sun would not be a perfectly elliptical path, it would be like going overall in an elliptical path but doing small circles through that path, right?

I would appreciate if someone can clear my doubts.

Answer 1

“Using Newtonian gravity only (i.e. ignoring any relativistic corrections) it can be shown that two bodies will orbit about their common centre of mass, and their orbits will be ellipses with the centre of mass at one focus.

Three or more bodies will also orbit about their common centre of mass, but in the general case their orbits are no longer ellipses and may be chaotic - determining the orbits for three or more bodies is known as the three body problem.

However, if one body is much more massive than the others (as is the case with the Sun and the planets) then to a first approximation that body can be taken to be stationary and the other bodies will orbit in ellipses with the most massive body at a focus. Kepler’s laws apply to this first approximation situation.

Over long periods of time the gravitational interactions between the less massive bodies (which are ignored in the first order approximation) must be taken into account and the orbits may become chaotic. It is not known for certain that the orbits of the planets in the solar system will still be stable hundreds of millions of years in the future - see this Wikipedia article.

The mutual orbits of the Earth and the Moon about the Sun are discussed and illustrated at the end of this Wikipedia article.