Infrared emissivity inferred from a one-layer greenhouse model (${ϵ}_a=2(1-{ϵ}_{\mathrm{eff}})$)

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