The average must lie on the straight line that goes through the centers of the Earth and Sun (one way to see that velocity behaves nicely wrt symmetry about this axis is using Kepler's equal-areas law). However, the other symmetry of the ellipse as a static object isn't respected by the dynamics (the Earth moves faster when it's nearer to the Sun) so we can't use a second symmetry argument to nail down the exact location of the average. I'm hoping that Veit (or others) will see how to proceed from here. Jim Propp On Thursday, July 14, 2016, James Propp <jamespropp@gmail.com> wrote:
Ignoring the effects of bodies other than the Earth and the Sun, and treating the Earth as a point mass, what is the average position of the Earth over the course of a year as it travels an elliptical orbit with the Sun at one focus?
If you ask me "Are you interested in the average position of Earth relative to the the Sun, or relative to the position of the center of mass of the two-body system?", my answer is "I'm interested in both".
I'm hoping that there's an embarrassingly easy way to see what the answer is using principles from physics that I must've been taught but haven't fully absorbed.
I'm also interested in the limiting case where the ratio of the two masses in a two-body system goes to 0. Even if the answer to my original questions is "It's messy in either coordinate system", this limiting case (in which the two original questions coincide) might behave nicely.
Jim Propp