Consider a slight improvement on the two-layer toy model of Earth’s atmosphere shown in the figure in Example 34.2. Assume that there are two atmospheric layers, each transparent to incoming solar radiation with incoming solar radiation intensity I = 〈I_in〉 ≅ 0.84I_⊙/4. Assume that all outgoing infrared radiation (from the surface and from each atmospheric layer) consists of two equal parts. 

One part is in a frequency range that is completely absorbed by each atmospheric layer, and the second part is in a frequency range that passes through all atmospheric layers. Thus, for example,1/2 of the IR radiation emitted by the surface is absorbed completely by the lower atmosphere layer(layer 2) and the other 1/2 passes through both atmospheric layers and escapes to space. Set up the three linear equations for the temperatures T_s, T₁, T₂ of the surface and the two atmospheric layers. What is the temperature at the surface? To simplify the algebra you may want to first solve for the values τ_i = σT_i⁴, for i = s, 1, 2.