The focus’s offset relative to size
Ellipses of the same size can have different focal separations and shapes. Dividing c by a makes eccentricity a ratio rather than a kilometre distance.
Coincident foci produce a circle with e=0; noncircular ellipses have 0<e<1. Uniformly enlarging an ellipse leaves the ratio unchanged, so a larger orbit need not be more eccentric.
Connecting shape with near and far distances
For an ideal ellipse, periapsis distance is a(1−e) and apoapsis distance a(1+e). At fixed a, increasing e to 0.5 gives distances of 0.5a and 1.5a.
Speed also varies: equal areas in equal times imply faster motion near periapsis and slower motion near apoapsis. Eccentricity affects this geometry without itself being an instantaneous speed or period.
Closed and open conics
In ideal two-body conic classification, e=1 is parabolic and e>1 hyperbolic. These open paths do not inherit an ellipse’s apoapsis and repeating orbital period.
Real objects experience perturbations or thrust, and reported elements refer to a model and epoch. A hyperbolic solution describes that modeled trajectory rather than ruling out changes during later encounters.