I am learning about orbits and getting very confused as to what is exactly true and what is an approximation.
Namely the following points:
- It is often said that the planet has an elliptical orbit with the sun at one focus. I think this is an approximation in assuming that the mass of the sun is so much greater than that of the planet that its motion is negligible compared with the planet, but in fact both have an elliptical orbit about their common center of mass (as in this SE post).
But the derivation we used in our classes for the elliptical motion of a planet is about the sun, and not for both bodies about a common center of mass. The other derivations I have seen also show this orbit about the sun. The derivations all initially assume the sun as a center, and use the gravitational potential around the sun. I am guessing that the approximation comes in because, if you assume the sun at one focus, then the kinetic energy is not simply $0.5mv^2$ because the sun is actually a non inertial reference frame. Is this correct?
Finally, is angular momentum really conserved if we place the sun at a new focus? Why/why not? Linear momentum is clearly not conserved. It would be if we considered the whole system (the planet and the sun). So is linear momentum conserved?