(a) Use Eq. (3.17) and simple geometry to derive Eq. (19.5) for the temperature Tp of a planet at a distance D from the Sun.
(b) Imagine the greenhouse gases in Earth's atmosphere to be a single layer that is completely transparent to the visible wavelengths of light received from the Sun, but completely opaque to the infrared radiation emitted by the surface of Earth. Assume that the top and bottom surface areas of the layer are each equal to the surface area of the planet and that the temperature at the top of this atmospheric layer, T⊕, is just the blackbody temperature found in Example 19.3.1. Show that the blackbody radiation emitted by this atmospheric layer results in a warming of Earth's surface to a temperature of Tsurf=21/4T⊕. Compare this result with Earth's average surface temperature of 15∘C=59∘F.