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:
(x/a)**2 +(y/b)**2 = 1 ,"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.
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 from a extreme-mass-disparity 2-body system since the other less massive bodies just cause gravitational perturbations.
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