Question

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/m^2 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 area of Earth?

a) Calculate the total intercepted radiation and absorption rate.
b) Determine the Earth's temperature and its dependence on area.
c) The greenhouse effect is essential for the solution.
d) The problem is unsolvable without knowing the Sun's temperature.

Answer 1

To estimate the temperature of the Earth, we need to consider factors such as the area intercepted by the Sun's radiation, the absorption rate of solar energy, and the temperature at which Earth radiates energy. The answer does not depend on the area of Earth.

Step-by-step explanation:

To estimate the temperature of the Earth, we need to consider several factors. (a) Assuming the Sun's rays are parallel, we need to determine the area that must be multiplied by the solar constant, S, to get the total radiation intercepted by Earth. This area is given by A = πR^2, where R is the radius of the Earth. (b) Assuming that Earth reflects about 30% of the solar energy it intercepts (albedo A = 0.3), the rate at which Earth absorbs energy from the Sun can be calculated using Absorption rate = (1 - A)S. (c) To find the temperature at which Earth radiates energy at the same rate, we can use the Stefan-Boltzmann law, which states that the power radiated by a blackbody is proportional to the temperature to the fourth power. Equating the energy absorbed from the Sun to the energy radiated, we can solve for the temperature, T. (d) The answer does not depend on the area of Earth, as it cancels out in the equations.