Conservation of Energy with an Ellipse

[SprucerMoose]

Gday,

This may be a trivial questiom, but I just cannot figure it out.

Why is mechanical energy conserved with an elliptical orbit? I understand that the mechanical energy of the system, i.e. both bodies, is conserved, but how is this isolated to the satellite's mechanical energy being conserved?

 

[SprucerMoose]

Hang on, I think I figured it out.

Is it simply because gravity is a conservative force, thus any change in K.E. = change in Ug, therefore mech energy conserved?

Does this get more complicated if we consider the little motion of the larger planet, or does this simple relationship still hold?