As the very first rudiment of climatology, estimate the temperature of Earth. Assume it is a...

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As the very first rudiment of climatology, estimate the temperature of Earth. Assume it is a perfect sphere and its temperature is uniform. Ignore the greenhouse effect. Thermal radiation from the Sun has an intensity (the "solar constant" S) of about 1370 W/m2 at the radius of Earth's orbit. (a) Assuming the Sun's rays are parallel, what area must S be multiplied by to get the total radiation intercepted by Earth? It will be easiest to answer in terms of Earth's radius, R. (b) Assume that Earth reflects about 30% of the solar energy it intercepts. In other words, Earth has an albedo with a value of A = 0.3. In terms of S, A, and R, what is the rate at which Earth absorbs energy from the Sun? (c) Find the temperature at which Earth radiates energy at the same rate. Assume that at the infrared wavelengths where it radiates, the emissivity e is 1. Does your result show that the greenhouse effect is important? (d) How does your answer depend on the the area of Earth? a. In terms of Earth's radius, the constant o, and the unknown temperature Ts 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 Te 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 power 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 the lectures (https://openstaxcollege.org// 21paulgormlec) by Paul O'Gorman. As the very first rudiment of climatology, estimate the temperature of Earth. Assume it is a perfect sphere and its temperature is uniform. Ignore the greenhouse effect. Thermal radiation from the Sun has an intensity (the "solar constant" S) of about 1370 W/m2 at the radius of Earth's orbit. (a) Assuming the Sun's rays are parallel, what area must S be multiplied by to get the total radiation intercepted by Earth? It will be easiest to answer in terms of Earth's radius, R. (b) Assume that Earth reflects about 30% of the solar energy it intercepts. In other words, Earth has an albedo with a value of A = 0.3. In terms of S, A, and R, what is the rate at which Earth absorbs energy from the Sun? (c) Find the temperature at which Earth radiates energy at the same rate. Assume that at the infrared wavelengths where it radiates, the emissivity e is 1. Does your result show that the greenhouse effect is important? (d) How does your answer depend on the the area of Earth? a. In terms of Earth's radius, the constant o, and the unknown temperature Ts 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 Te 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 power 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 the lectures (https://openstaxcollege.org// 21paulgormlec) by Paul O'Gorman.