Chapter 13.2, Problem 13.116P

A spacecraft of mass mdescribes a circular orbit of radius r₁ around the earth, (a) Show that the additional energy ΔE that must be imparted to the spacecraft to transfer it to a circular orbit of larger radius $r_2$ is $\Delta E=\frac{GMm(r_2-r_1)}{2r_1{r}_2}$ where Mis 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 ΔE_A and ΔE_Bwhich must be imparted at Aand Bare_: respectively, proportional to r₂and r₁ :
   $\Delta E_A=\frac{r_2}{r_1+r_2}\Delta E$

   $\Delta E_B=\frac{r_1}{r_1+r_2}\Delta E$