A spacecraft of mass m describes a circular orbit of

Chapter 13, Problem 13.116

(choose chapter or problem)

A spacecraft of mass m describes a circular orbit of radius \(r_{1}\) around the earth. (a) Show that the additional energy \(\Delta E\) which must be imparted to the spacecraft to transfer it to a circular orbit of larger radius \(r_{2}\) is

\(\Delta E=\frac{G M m\left(r_{2}-r_{1}\right)}{2 r_{1} r_{2}}\)

where M is the mass of the earth. (b) Further show that if the transfer from one circular orbit to the other is executed by placing the spacecraft on a transitional semielliptic path AB, the amounts of energy \(\Delta E_{A}\) and \(\Delta E_{B}\) which must be imparted at A and B are, respectively, proportional to \(r_{2}\) and \(r_{1}\):

\(\Delta E_{A}=\frac{r_{2}}{r_{1}+r_{2}} \Delta E \ \ \quad \Delta E_{B}=\frac{r_{1}}{r_{1}+r_{2}} \Delta E\)