You are going to build a very simple energy balance model for the earth. The model is based on the fact that, on average, the incoming energy (solar) is equal to the outgoing energy (from the earth). The variables are:
S --- the solar constant.
A --- planetary albedo of the earth.
T --- temperature of the earth's surface (Kelvin).
You will assume that the earth has no atmosphere. Thus, the greenhouse effect, the transfer of energy by conduction and convection, and the transfer of latent heat need not be included.
1. Develop your energy balance model. Your model will be an equation which expresses the energy balance using the above variables S, A, and T. You will use the Stefan-Boltzmann Law for the outgoing energy. The Stefan-Boltzmann constant σ is 5.67 x 10-8 W m-2 K-4.
2. Solve the equation algebraically for T. As a check that you've set up your "simple" model and solved the equation correctly, you may treat the earth as a blackbody. That is, it receives all radiation impinging on it. Set albedo to zero and calculate a temperature. You should get 278.6 K (5.4 C).
3. Using the mean planetary albedo, calculate earth's surface temperature. You may report the temperature in units of Kelvin, celsius, or fahrenheit. If solved correctly you should see in the equation that the average incoming solar radiation at the top of the atmosphere (units of W/m2) is the solar constant scaled by a factor of 4. State why the temperature appears inconsistent with reality.
NOTE: For this assignment, the variables, equations, and all algebra steps must be shown, handwritten on paper that you'll submit. Answers to the following questions may be handwritten or done in a word processor and printed for submission.
4. Using this equation for T:
a) If albedo increases, what happens to T?
b) If incoming solar radiation increases, what happens to T?
c) If the snow and ice on the earth decreases, resulting in an albedo of .17, what is the temperature?
5. To calculate the temperature of Mars (still assuming no atmosphere), what input values would you need to change?