A convenient way to solve the problem: let the temperature T_c and emissivity \epsilon_c of the cold side of the satellite be parameters; then the heat and entropy lost at T_c can be expressed in terms of these parameters. For Carnot’ cycle, the total entropy is conserved, hence the entropy and heat obtained at the hot side can be expressed if we introduce additional parameters, the temperature T_h and emissivity \epsilon_h of the hot side. There is one more condition, though: the heat balance on the hot side due to heat absorption and emission. What is left to do is solving and optimizing for the maximal usable power.
Please submit the solution to this problem via e-mail to firstname.lastname@example.org. The next hint for the Problem 5 will be published at 13:00 GMT, 17th April 2022, together with the next updates of the intermediate results.