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 balance models, which we will discuss shortly, might refine this assumption somewhat. Our modified model should decrease the value of $$S$$ (the amount of energy absorbed by the Earth) by a factor that is proportional to the albedo: as the albedo of the planet increases it absorbs less energy, and as the albedo decreases it absorbs more. Let's try this, then:

In the special case where the Earth's albedo $$\alpha$$ is 0, (4d) reduces to (4c), since $$1-\alpha$$ is just 1. OK, so once again let's fill in our observed values and see what happens. We'll approximate $$\alpha$$ as being equal to .3, so now we have:

Which gives us a result of:

This is far more accurate, and the remaining difference is well within the margin of error for our observed values.

So now we're getting somewhere. We have a simple model which, given a set of observed values, manages to spit out a valid equality. However, as we noted above, the purpose of a model is to help us make predictions about the system the model represents, so we shouldn't be satisfied just to plug in observed values: we want our model to tell us what would happen if the values were different than they in fact are. In this case, we're likely to be particularly interested in $$\mbox{T}_\mbox{p}$$: we want to know how the temperature would change as a result of changes in albedo, 116