Chapter 1, Problem 129CP

Let's stop ignoring the greenhouse effect and incorporate it into the previous problem in a very rough way. Assume the atmosphere is a single layer, a spherical shell around Earth, with an emissivity e 0.77 (chosen simply to give the light answer) at infrared wavelengths emitted by Earth and by the atmosphere. However, the atmosphere is transparent to the Sun's radiation (that is, assume the radiation is at visible wavelengths with no infrared), so the Sun's radiation leaches the surface. The greenhouse effect comes from the difference between the atmosphere transmission of visible light and its rather strong absorption of infrared. Note that the atmosphere's radius is not significantly different from Earth's, but since the atmosphere is a layer above Earth, it emits radiation both upward and downward, so it has twice Earth's area. There are three radiative energy transfers in this problem: solar radiation absorbed by Earth's surface; infrared radiation from the surface, which is absorbed by the atmosphere according to its emissivity; and infrared radiation from the atmosphere, half of which is absorbed by Earth and half of which goes out into space. Apply the method of the previous problem to get an equation for Earth 's surface and one for the atmosphere, and solve them for the two unknown temperatures, surface and atmosphere.

a. In terms of Earth's radius, the constant $\sigma$ , and the unknown temperature T_s of the surface, what is the power of the infrared radiation from the surface?

b. What is the power of Earth 's radiation absorbed by the atmosphere?

c. In terms of the unknown temperature T_e of the atmosphere, what is the power radiated from the atmosphere?

d. Write an equation that says the power of the radiation the atmosphere absorbs from Earth equals the of the radiation it emits.

e. Half of the power radiated by the atmosphere hits Earth. Write an equation that says that the power Earth absorbs from the atmosphere and the Sun equals the power that it emits.

f. Solve your two equations for the unknown temperature of Earth.

For steps that make this model less crude, see for example (https://openstaxcollege.org/l/21paulgormlec) by Paul O'Gorrnan.