### Radiative Equilibrium

We have already mentioned the idea of radiative equilibrium, where the incoming energy and the outgoing energy are in balance, resulting in a steady temperature, but now we are in a position to combine a few other ideas to express this notion in a simple equation that is at the heart of all climate models. Before we begin, we introduce the solar constant which is the amount of incoming solar electromagnetic radiation per unit area. Just for your information, this amount is measured on a plane perpendicular to the Sun's rays and at the mean distance from the Sun to the Earth.

We begin with the energy (in units of W/m^{2}):

E in =S( 1−a )

Here, S is the solar constant — 1370 W/m^{2}, and a is the albedo, which is about .31 based on satellite measurements. Then we deal with the energy out, using the Stefan-Boltzmann law:

E out =σ T^{4}

Combining energy in (E_{in}) and energy out (E_{out}), we get:

S( 1−a )=σ T^{4}

Now, we can solve this to find what the equilibrium temperature of our planet is:

T^{4} =(S( 1−a ) σ ), so T= (S( 1−a ) σ )^{1/4} adding numbers, T = (343( 1−.31 ) 5.67e−8 )^{1/4} = 254° K=− 19° C=− 2.2° F

Yikes! This is too cold — we know the mean temperature of the Earth is more like 15°C (288°K or 59°F). What have we left out? The simple answer is the emissivity, which makes sense since we know the Earth is not an ideal blackbody. (Remember that emissivity is a measure of how good an object is at emitting (giving off) energy via electromagnetic radiation; in the above, we have effectively assumed an emissivity of 1, which is for a perfect black body material). Using the equation above, let’s see what that emissivity number should be:

S(1−a )=εσ T^{4}

ε= S(1−a ) σ T^{4} = 343(1−.31 ) ( 5.67e−8 )( 288 4 ) =0.606

So, then, even if all of these equations have you seeing stars, what does this basically mean? There is something about the Earth that prevents it from emitting as much energy as it should. What is this something? It is the *greenhouse effect *— the key that makes our planet a nice place to live.