Let us consider a particle of mass  m orbiting a point mass  M at some eccentric orbit. Suppose that said particle can lose energy, but cannot rid itself of its angular momentum. Under this restriction there is a minimal energy that a particle can attain. The angular momentum is

 L = m v r

In the terminal circular orbit, the relation between the radius and the angular velocity is

 v = \sqrt{\frac{G M}{r}}

Solving for  v and  r yields

 v = \frac{G M m}{L}

 r = \frac{L^2}{G M m^2}

and the terminal energy is

 E = -\frac{1}{2} m v^2 = - \frac{1}{2} \frac{G M^2 m^3}{L^2}