Attached is a Sage worksheet on the simplest global energy balance model. An energy balance model looks at energy in and energy out:

change in temperature = energy in - energy out

Pretty straightforward, eh? Let be the incoming solar radiation (insolation) and be albedo (percentage of sunlight the earth reflects) -- our "energy in" will be . Then we can use Boltzmann's black-body radiation to get "energy out" -- it's , where is our global average surface temperature in kelvins and is Boltzmann's constant. So we get the equation

It is not too hard to use calculus to find a linearization near the equilibrium point of this equation and then do some analysis.

The attached worksheet can be loaded into sagenb.org if you want to work through it without installing Sage:

MAA-NCS Climate Modeling -- Boltzmann

I'm trying to figure out how to put up a full interactive worksheet; it's not so hard to put up cells, but I'm not sure about a whole Sage file... let me know if you know!