A spacecraft is transferred from a low circular parking orbit of radius r1 to a higher circular operational orbit of radius r2 using a single-impulse maneuver that places it on an elliptical transfer orbit. The impulse is applied at point A (radius r1), which becomes the periapsis of the transfer orbit. Point B (radius r2) is the apoapsis.
Assume r2>r1, the planet’s mass is M, and the spacecraft’s mass is m.
(a) Rank, from greatest to least, the spacecraft’s kinetic energy at the following three points and justify each step of your ranking:
• immediately before the impulse at point A (circular orbit)
• immediately after the impulse at point A (elliptical transfer orbit)
• at point B on the transfer orbit
(b) For the same three moments listed in part (a), rank the (negative) gravitational potential energies U from greatest magnitude to least magnitude. Provide reasoning that explicitly cites how U depends on radius.
(c) At point A, the impulse changes the spacecraft’s total mechanical energy from Einitial to Etransfer. Explain whether Etransfer−Einitial is positive, negative, or zero, and discuss how your answer is consistent with the rankings established in parts (a) and (b).