Caption: Two diagrams illustrating how the elongation of an ellipse and the location of its two 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  , 

   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 (AKA their barycenter) 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 barycenter. 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.

5. For absolute value of the x distance of the focuses from the origin, we have the formula

        c = sqrt(a**2-b**2) 

   (see Wikipedia: Ellipse: Ellipse in Cartesian coordinates). Note c = 0 for circles.

6. The ellipse eccentricity e is a parameter that is an alternative to c and that is a simple measure of the deviation of an ellipse from a circle. The eccentricity formula:

        e = c/a =  sqrt(a**2-b**2)/a = sqrt[1-(b/a)**2]  , 

   and so e = 0 implies c = 0 (i.e., a circle) and e = 1 implies c = a (i.e., a straight line).

7. For orbits, the closest/farthest approach to the barycenter is the periapsis/apoapsis. The periapsis distance and the apoapsis distance are, respectively,

        r_per = a-c = a(1-e) and  r_apo = a+c = a(1+e) 

   Now the standard definition of the mean orbital radius is

        r_mean = r_per + r_apo = a , 

   i.e., r_mean is just the semi-major axis.

   So we see that orbital eccentricity is the maximum relative deviation of the radius of an elliptical orbit from the standard mean orbital radius.

   Note that (100*e) % is the maximum relative deviation as a percentage.

Images:

1. Image link: Itself.
   
2. Image link: Itself.