Does the orbits of stars in a binary system have a common focus, which is the center of mass of the stars?

Question

I know that when a planet orbits a star (and assume the star is stationary because its mass is >> the mass of the planet), the orbit is an ellipse, and the star sits at one of the foci of the ellipse. What I am wondering is if instead of a planet surrounding a star but a binary system where the two stars's mass are of the same order of magnitude, are their orbits ellipses and share a common focus at the center of mass?

Intuitively I think it is true because this is a result generalized from the planet surrounding the star case but strictly speaking I have a hard time to mathematically prove it.

Answer 1

Elliptic orbits with the focus at the centre of mass (the barycentre) are the solution to the "two body problem".

The wikipedia page on the central force problem shows how the two body problem can be reduced to solving two central force problems. Therefore the elliptical solution to the central force problem also solves the two body problem, and in moreover Kepler's laws apply to such bodies.