= Infrared emissivity inferred from a one-layer greenhouse model
{title2=$\epsilon_a=2(1-\epsilon_{\rm eff})$}
A surface black in the thermal infrared beneath a shortwave-transparent, isothermal grey layer has atmospheric balance $\epsilon_a\sigma T_s^4=2\epsilon_a\sigma T_a^4$. For nonzero layer emissivity, $T_a^4=T_s^4/2$. Direct surface transmission plus upward layer emission then gives $F_{\rm out}=(1-\epsilon_a/2)\sigma T_s^4$, proving the title. This model only permits $1/2\le\epsilon_{\rm eff}\le1$ for $0\le\epsilon_a\le1$. A real vertical opacity/temperature profile is not uniquely determined by this one fitted parameter.
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