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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?

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The term origin is misleading. In your title you use the correct term focus. Every ellipse has two foci. The term origin is likely to be confused with the geometrical centre of the ellipse.

You are misinterpreting your textbook. It probably says that the Sun resides at one focus of the ellipse, not at the middle (= centre). But this is a simplification anyway. Both the Earth and the Sun orbit their common centre of mass. The significance of this focus is that it is where the centre of mass is located. The second focus is empty and has no significance for the gravitational orbit.
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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.

In the Sun-Earth system, the centre of mass is inside the Sun which is very much more massive than the Earth. So the orbit of the Sun about the centre of mass is usually ignored. In the Earth-Moon system the centre of mass is just below the surface of the Earth.

See animations in wikipedia article Gravitational Two Body Problem.

See also :
Why is the center-of-mass of 2 bodies at the focus of their elliptical orbits?
Significance of the second focus in elliptical orbits

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  • $\begingroup$ So I guess my question is, how does the math show that the focus $\textit{is}$ the position of the center of mass? $\endgroup$ – Ptheguy Mar 24 '17 at 2:00
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    $\begingroup$ To answer that, you need to go through the math, which is probably given in your textbook. The usual procedure is to introduce new co-ordinates which are the distances from the centre of mass. In the new co-ordinates the orbits are ellipses with the centre of mass at one focus. $\endgroup$ – sammy gerbil Mar 24 '17 at 2:14

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