= Solution
Let absorbed solar flux per unit surface area be $F_\odot=\sigma T_0^4$. The <single-layer greenhouse model> emits infrared upward and downward. Top-of-atmosphere balance gives $\sigma T_a^4=F_\odot$, hence $T_a=T_0$. Surface balance is $\sigma T_s^4=F_\odot+\sigma T_a^4$, so
$$
\boxed{T_s=2^{1/4}T_0\simeq333\,\mathrm K.}
$$
The layer simultaneously satisfies $\sigma T_s^4=2\sigma T_a^4$. This ideal radiative model excludes convection, spectral windows and atmospheric <temperature> structure; it is not a quantitative prediction for the real atmosphere.
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