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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.

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  • $\begingroup$ Also in the case that you do not consider m1 >> m2 both bodies orbit in ellipses around their common centre of mass. If m1 >> m2 the ellipse of m1 is quite small and in star/planet situation the centre of mass lies well within the star. $\endgroup$ – Hartmut Braun Jan 12 at 17:19
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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.

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