An alternative derivation of the total orbital angular momentum can be obtained by applying the conservation laws of angular momentum and energy.
(a) From conservation of angular momentum, show that the ratio of orbital speeds at perihelion and aphelion is given by
vavp=1−e1+e.(b) By equating the orbital mechanical energies at perihelion and aphelion, derive Eqs. (2.33) and (2.34) for the perihelion and aphelion speeds, respectively.
(c) Obtain Eq. (2.30) directly from the expression for vp (or va).