### What is the shape of the Sun's orbit around the Earth taking into account elliptical orbits?

• Keshav Srinivasan

6 years ago

Consider the non-inertial frame that is at rest with respect to the Earth's revolution around the Sun. (Ignore the Earth's rotation around its own axis.) My question is, what is the shape of the Sun's orbit around the Earth in this reference frame?

Now if we assume the Earth moves in a circular orbit around the Sun, then the Sun's orbit around the Earth will also be circular. (Just use geometric the definition of a circle - the set of all points a fixed distance from a given point.). Specifically it's the great circle formed by the intersection of the ecliptic plane with the celestial sphere.

But the Earth does not move in a circular orbit - Kepler's first law states that the Earth moves in an ellipse around the Sun, with the Sun at one of the foci of the ellipse. So if we consider the Earth's elliptical orbit around the Sun, what is the shape of the Sun's orbit around the Earth. I doubt it's an ellipse, so would be it a more complicated-looking curve.

Note that I'm not interested in gravitational influences from the moon and other planets - this is pretty much a purely mathematical question: if we assume the Earth moves in an ellipse, what would be the shape produced? It's also an ellipse. It's simple geometry. @DavidHammen Could you spell out the logic? Because it's not obvious to me. For an object orbiting around some center (technically the barycenter rather than the center of the Sun), the standard type of transformation into a rotating reference frame in Newtonian physics would give you a coordinate system where both the object *and* the center were at rest, not one where the center is orbiting around the object...to get the latter I think you'd have to use two of these types of transformations in a row (equivalent to a single, different form of coordinate transformation) around different axes.

• 6 years ago

The Earth moves in an elliptic orbit around the sun (or around the barycenter). If, in helocentric coordinates the Earth is at position (x,y), then in Geocentric coordinates the position of the sun is in position (-x,-y)

So the locus of the Sun in Geocentric coordinates exactly matches the locus of the Earth in Heliocentric. The path of the sun in geocentric coordinates is an Ellipse. Thanks, I missed the fact that there's an isomorphism between the set of points (x,y) and the set of points (-x,-y).