FANDOM


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} $