Greenhouse Effect

Let us consider a rocky planet with an atmosphere. The scale height of the atmosphere is assumed to be much smaller than the radius of the planet. The planet is irradiated by its host star. The photons from the star penetrate the atmosphere and hit the ground, which re - emits the same energy in a different wavelength, for which the atmosphere is opaque.

Let us first consider what happens when the optical depth of the atmosphere it $\tau = 1$. We model this atmosphere as a single thin, perfectly absorbing layer. The flux emitted out into space by this layer has to be equal to the incoming flux from the star $F$. This layer must also be emitting the same flux downward to the ground. To balance the radiation from the star and from the upper layer, the ground has to emit $2 F$. If the flux scales with the temperature as $F \propto T^4$, then the temperature on the ground is greater by $\sqrt[4]{2}$ than the temperature at the upper layer.

When there are $n$ such shells, then the temperature on the ground is $n^{1/4}$ larger then the temperature at the uppermost layer.

It is also possible to model this problem in a continuous way. From the condition that the net flux emitted from each layer has to be equal to the flux absorbed, we get

$\frac{d^2 F}{d \tau^2} = 0$

Integrating yields

$F \propto T^4 \propto \tau \Rightarrow T\propto \tau^{1/4}$