In every derivation of Kepler's Laws that I have seen, we assume that the sun is stationary. However, in other places I have read that celestial bodies move about their barycentre (center of mass).
So are planets actually moving in elliptical orbits around the Sun or do they move in circular orbits around their center of mass?
Newtonian Mechanics – Do Planets Move in Elliptical Orbits Around the Sun or Circular Orbits Around Their Center of Mass?
inertial-framesnewtonian-gravitynewtonian-mechanicsorbital-motionreference frames
Best Answer
In an ideal two body system (say a sun and a planet), both bodies would move around their barycenter. An ideal periodic orbit would be an ellipse or a circle.
EDIT : See comment by @user11153 regarding the barycenter of the solar system and related links.
In a more complex system like our solar system, to a good approximation the planets can be modeled by a two body system (i.e. the Sun being so massive it is the dominant effect) and for many practical purposes the motion of the Sun around the barycenter is not significant, as the barycenter is actually inside the Sun.
More precise calculations the motion of a planet requires allowing for the gravitational perturbation of other planets as well as allowing for the center of mass and relativistic effects. The net effect is that no planets actually orbit in ideal elliptical orbits.
I have the impression from this question that you think the elliptical orbits are a result of using the barycenter as a center of motion and that otherwise a circle would be the orbit's shape.
This is not the case. The general shape for an orbit in an ideal two body system with a Newtonian gravitational force is an ellipse. A circle is a special case of an ellipse.