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## Problem

Consider the following information.

Solar constant:

Albedo of the Earth:

Emissivity of the Earth: (Blackbody)

Stefan-Boltzmann constant:

Part 1: Calculate the surface temperature of the Earth with the absence of the atmosphere. Note that the surface temperature is labelled in the above diagram.

Part 2: Calculate the surface temperature of the Earth with the atmosphere present.

## Solution

Part 1

Without an atmosphere the power of solar radiation absorbed by the surface of the Earth is equal to the power radiated by the Earth.

This temperature is , which is an ice world!

Part 2

Add an atmosphere. The power of solar radiation absorbed by the atmosphere of the Earth is equal to the power radiated by the atmosphere. Thus,

.

This is the same calculation as in part 1, so .

Now, the power of the solar radiation still reaches the surface of the Earth alongside and the power radiated by the atmosphere. Therefore, we tweak the equation from part 1 by adding an atmospheric radiation term.

Since ,

.

This temperature is around , which is too hot! Sure, some places on Earth can be hotter than this temperature, but the average surface temperature is only around . One possible source of error in our calculation in part 2 is the amount of solar radiation that reaches the surface of the Earth, but there could be a plethora of other causes such as other thermal transfer effects.