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Since at equilibrium, A î= A 4, we have i 2 S Î = *T4 = —-- (1 - ap) So = ---aT4

Therefore,

So in the limit e ^ 0 (transparent atmosphere), Ts = Te, and for e ^ 1 (opaque atmosphere), Ts = 21/4Te, as found in Section 2.3.1. In general, when 0 < e < 1, Te < Ts < 21/4Te. Thus partial transparency of the atmosphere to IR radiation—a "leaky" greenhouse—reduces the warming effect we found in Eq. 2-9.

To find the atmospheric temperature, we need to invoke Kirchhoff s law,3 such that the emissivity of the atmosphere is equal to its absorptivity. Thus

We can now use Eqs. 2-14, 2-10, 2-11, and 2-12 to find

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