A spherical satellite in near-earth orbit is exposed to solar irradiation of \(1368 \mathrm{~W} / \mathrm{m}^{2}\). To maintain a desired operating temperature, the thermal control engineer intends to use a checker pattern for which a fraction \(F\) of the satellite surface is coated with an evaporated aluminum film ( \(\left.\varepsilon=0.03, \alpha_{S}=0.09\right)\), and the fraction \((1-F)\) is coated with a white, zinc-oxide paint \(\left(\varepsilon=0.85, \alpha_{S}=0.22\right)\). Assume the satellite is isothermal and has no internal power dissipation. Determine the fraction \(F\) of the checker pattern required to maintain the satellite at \(300 \mathrm{~K}\).