PX282 - J7b - greenhouse effect

the atmosphere is transparent to the heating from the sun in optical light
but it is more opaque to outgoing IR radiation
this is due to molecular absorption (especially from water and carbon dioxide)

image: E. M. Smith

$L_{i n}$ must equal total $L_{o u t} :$

T_{s}^{4} = 2 T_{p}^{4}

with $N$ optically thick layers:

T_{s}^{4} = (1 + N) T_{p}^{4}

considering continuous absorption:

T_{s}^{4} = (1 + \frac{3}{4} τ) T_{p}^{4}

where, $τ$ is the optical depth

comparing the two equations, $τ ≃ 1.33$ is equivalent to one optically thick layer
for earth, $τ_{I R} \sim 0.8$ , $T_{p} = 255$ K $⟹ T_{s} ≃ 287$ K $≃ 14 °$ C
for venus, $T_{p} = 230$ K, $T_{s} = 735$ K $⟹ τ_{I R} ≃ 100$
in practice, the wavelength dependence of absorption needs to be accounted for