It always bugged me that the orbits of the planets about the sun were elliptical, but when discussing the two-body problem, the orbits were circular about the center of mass. My physics teachers sloughed off my question, but it stuck in my memory as a minor mystery. Now that I’m retired I decided to look into it. The relationship is simple now that I have swept away all the confusing side investigations.
Center of Mass
In physics the circular motion about the common center of mass is called barycentric motion. I’ll focus on the earth’s movement about the sun and I’ll start with the barycentric motion.
Next convert to algebraic notation and then shift attention to observation positions, Earth to the Sun’s center.
This last equation is the formula of an ellipse with a focus at (f,0), the center of the sun. It arises because astronomical measurements are more naturally made with the center of the sun than the center of mass of the system.
Note: I had to use graphic images to show the equations, this broke the hypotenuse link in the image. The hypotenuse distance formula between two points is nicely developed at the MathWarehouse site.