# Meaning of the focus of an elliptical orbit

The following assumes we are dealing with a two-body system where no external forces exist and the two bodies interact via a conservative, central force that is proportional to the inverse square of the separation between them.

Under these assumption, we can show that the bounded orbits are ellipses. However, we arrive at the conclusion that the origin of these ellipses is not their center, but a position $\epsilon a$ away. ($\epsilon$ is the eccentricity of the ellipse, and $a$ is the factor in the denominator of the $x$ term when one writes the equation of the ellipse.)

What is so special about this origin? My classical mechanics textbook says that if we are considering a planet orbiting about the sun (assuming infinite mass of the sun), that the sun resides in the middle. Why is this the case? What would happen if the masses of the two bodies are comparable and one cannot be approximated as infinite compared to the other?

The lighter body does not orbit about the heavier body. Regardless of their relative masses, both bodies orbit about their centre of mass (the barycentre). They move along different ellipses with the same eccentricity, always on opposite sides of the centre of mass, which is at one focus $F$ of each ellipse.
• So I guess my question is, how does the math show that the focus $\textit{is}$ the position of the center of mass? – Ptheguy Mar 24 '17 at 2:00