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emitted energy, or received energy. Fortunately, it’s only a trivial matter of algebraic manipulation to rearrange our last equation to solve for ${\mbox{T}}_{\mbox{p}}$:

${\sqrt[{4}]{\tfrac {(S_{o}(1-\alpha )}{4\sigma }}}={\mbox{T}}_{\mbox{p}}$

(4g)

We’re now free to plug in different values for incoming solar radiation and planetary albedo to see how the absolute temperature of the planet changes (try it!). But wait: something is still amiss here. By expressing the model this way, we’ve revealed another flaw in what we have so far: there’s no way to vary the amount of energy the planet emits. Recall that we originally expressed F—the total energy radiated by Earth as a blackbody—in terms of the Stefan-Boltzmann law. That is, the way we have things set up right now, the radiated energy only depends on the Stefan-Boltzmann constant σ (which, predictably, is constant) and the absolute temperature of the planet T_p. When we set things up as we did just now, it becomes apparent that (since the Stefan-Boltzmann constant doesn’t vary), the amount of energy that the planet radiates depends directly (and only) on the temperature. Why is this a problem? Well, we might want to see how the temperature varies as a result of changes in how much energy the planet radiates^[1]. That is, we might want to figure out how the temperature would change if we were to add an atmosphere to our planet—an atmosphere which can hold in some heat and alter the radiation profile of the planet. In order to see how this would work, we need to understand how atmospheres affect the radiation balance of planets: we need to introduce the greenhouse effect and add a parameter to our model that takes it into account.

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1. In fact, there’s another clue that something’s not right here. Solving the equation using the values we’ve got so far gives us a temperature of 255K, which is significantly below the freezing point of water (it’s around 0 degrees F, or -18 degrees C). As you can easily verify, this is not the temperature of the planet’s surface, at least most of the time. Something is wrong here. Hang in there: we’ll see the explanation for this anomaly soon, in Section 4.1.3.

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