Atmosphere Contents

4. Energy Budgets

Model of average Earth surface temperature

Using the information that the solar constant is 1365 Wm2 and the Earth can be treated as a black body, we can work out an approximation for the Earth's average surface temperature from the Stefan-Boltzmann Law. This states that the energy emitted by a body (E) equals the temperature (T) of the body to the power of 4 times by the Stefan-Boltzmann constant, s = 5.6704 x10-8 Js-1m-2K-4.

                   E = sT4      (1)                                                                      

Now since we are treating the Earth as a black body (perfect emitter), the radiation absorbed and emitted by the Earth are equal.

                                          Ein = Eout               (2)                                

Ein is the solar constant, acting on 1 x 1 m2 areas across half the globe (the side of the Earth facing the sun -- essentially a circular disc). The approximate area of the disc the solar radiation can hit is pr2. But some of this is reflected away by the Earth, where the average Earth albedo is 0.3 (30%).

The Earth is also emitting radiation across its entire surface area, which is 4pr2.

This leads to the relationships,
                                        Ein = So(1-a)pr2       (3)
                                                               Eout = 4pr2sT4               (4)
and therefore,                                           So(1-a)=4sT4                  (5)           (the pr2 cancel on both sides)

where a and So are the Earth surface albedo and the solar constant.

If we rearrange for T and sub in the values we get:

T=(So(1-a)/4s)(1/4) = (1365*(1-0.3)/ 4*5.6704 x10-8)(1/4) = 254.8 K = -18.34 °C.

For the average Earth surface temperature this seems a bit low. The reason being that this simple model does not have an atmosphere and the greenhouse effect.

Why is this answer an underestimation of the Earth's real temperature?