Consider an opaque, gray surface whose directional absorptivity is 0.8 for \(0 \leq \theta \leq 30^{\circ}\) and 0.1 for \(\theta>30^{\circ}\). The surface is horizontal and exposed to solar irradiation comprised of direct and diffuse components.

(a) What is the surface absorptivity to direct solar radiation that is incident at an angle of \(45^{\circ}\) from the normal? What is the absorptivity to diffuse irradiation?

(b) Neglecting convection heat transfer between the surface and the surrounding air, what would be the equilibrium temperature of the surface if the direct and diffuse components of the irradiation were 500 and \(200 \mathrm{~W} / \mathrm{m}^{2}\), respectively? The back side of the surface is insulated.