1) The Magalons have lost the war with Zargon, and are headed back to their home planet, which has a mass of $M = 8.20 \times 10^{33}$ kg, and a radius of $R = 5,500$ km. Because the gravitational field of Magalon is so large, the mother ship has to undergo a four-step process to get into a low enough orbit for the Magalons to use small shuttle craft to get from the mother ship down to the surface of Magalon. The mother ship has a mass of $m = 29,500$ kg. In the first step of the process, it initially enters a circular orbit around Magalon of radius, $r_1 = 33,000$ km. In the second step of the process, it drops into a lower circular orbit of radius, $r_2 = 31,000$ km, by transitioning into an elliptic orbit, and then transitioning into the lower circular orbit, as shown in the figure. The rocket engines used to make the transition have a thrust of, $F_{th} = 1.08 \times 10^{14}$ N.
Figure 1 - To get back to their home planet, the Magalons enter a large radius orbit of $r_1 = 33,000$ km, and then use an elliptic transfer orbit to drop into a circular orbit of radius, $r_2 = 31,000$ km. (Not to scale.)
a) For how long should the mother ship fire its rockets at Point A, shown in Fig.-1, to leave its original circular orbit and enter the elliptical transition orbit?
b) For how long should the mother ship fire its rockets at Point B, to leave the elliptical transition orbit and enter the smaller, second circular orbit?
c) In the third step of the process, the mother ship drops from the second orbit with radius, $r_2 = 31,000$ km, into a much smaller circular orbit with radius, $r_3 = 13,000$ km, and in the fourth step of the process, it drops from the orbit with radius, $r_3 = 13,000$ km, into an even smaller circular orbit with radius, $r_4 = 6,000$ km, by again entering an elliptical transition orbit. Now how long should it fire its rockets at each of the transition points similar to Point A and Point B? (Two-part answer.)
d) What did you neglect that makes your answer to Part c inaccurate, other than what's discussed in the note below?
e) Considering your answer to Part c, in what ways did the Magalons learn how to exceed the laws of physics as we know them?
Note: In your answers to Parts a, b, and c, neglect the distance the mother ship travels during the time the rockets are firing.
