The amount of energy you receive from the earth is the projected area times the solar constant times 1-albedo, or pi r^2 * K * (1-albedo), the amount you emit is going to be proportional to this boltzman equation times the surface area of the earth, or 4 pi r^2.  The earth in other words receives the projected area in solar radiation, which makes sense and radiates it out from the entire earth a sphere.   If you set the E(emitted) = E(absorbed) notice that the pi r^2 factors cancel out and you are just left with the factor of four.  The boltzman equation just means that the temperature is proportional to the temperature of the object to the 4th power.  Scientists knew about this back in the 1800’s, and in fact did this same calculation I’m doing now.  so let’s set up the equation, albedo is .31, solar constant at the earth’s radius is 1361 watts per meter squared and the stefan-boltzman constant is 5.67e-8 watts/m^2K^4

K  * (1-albedo) * pi r^2 = boltzman constant * T^4 * 4 pi r^2

1361 * .69 = 4 * 5.67e-8 T^4

T = 254K or -3F

Without greenhouse gases the earth would be -3F.  This is a very simple calculation and something we have known for hundreds of years.

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