Figure: Ellipses, eccentricity, and orbits

Caption: Two diagrams illustrating how the elongation of an ellipse and the location of its 2 focuses depend on eccentricity e. The second diagram is just an elaboration of the first one.

Features:

1. An ellipse is a plane curve and, if you neglect scale, a geometric shape.

2. The standard ellipse formula in Cartesian coordinates is
   ( x a ) 2 + ( y b ) 2 = 1

   where "a" is the semi-major axes, "b" is the semi-minor axis, and a ≥ b without loss of generality since one can just flip the names if a < b.

   If a = b, then the ellipse specializes to the circle with radius a = b.

3. The two focuses of an ellipse have special geometric significance. Most obviously, a triangle with vertices at the focuses and on the ellipse curve has the two sides touching the ellipse curve having summed length = 2a (see Wikipedia: Ellipse: Definition of an ellipse as locus of points).

4. In Newtonian physics, a bound 2-body system interacting through a inverse-square law force (e.g., a gravitationally bound 2-body system) has the 2 bodies orbiting their mutual center of mass in ellipses with the center of mass being at a focus of each elliptical orbit. The other focuses being just empty points in space.

   For a gravitationally bound 2-body system as the mass of body_1 (i.e., M_1) goes to infinity relative to the mass of body_2 (i.e., M_2), body_1 goes to being centered on the center of mass. Thus, body_2 effectively orbits body_1 in an elliptical orbit when M_1 >> M_2.

   The extreme mass disparity situation is pretty common: e.g., (1) a star is usually much more massive than any of its planets, (2) a planet is often much more massive than any of its moons.

   When the extreme mass disparity situation holds between a massive body (usually called the primary) and a group of less massive bodies, then usually to 1st order approximation the massive body and each less massive body form a extreme-mass-disparity 2-body system since the other less massive bodies just cause gravitational perturbations.