Consider the diagram we presented in Extreme Greenhouse, Part Three. The diagram shows the atmosphere divided into three layers. The Upper and Lower Gas are transparent to long-wave radiation. The Filter Gas is opaque to long-wave radiation.
Let us suppose that all the gas we portray in our diagram is air with an impurity that gives it an absorption length of 3 km for long-wave radiation when at standard temperature and pressure (STP). As we will see in later posts, this choice of 3 km will be useful when we consider CO2 in the atmosphere. Let us give our Lower Gas a thickness of 3 km. Two-thirds (1−e−1) of long-wave radiation will be absorbed in the Lower Gas. If we made the Lower Gas 30 km thick, it would be black to long-wave radiation, but we have assumed it is transparent. On the other hand, if we made the Lower Gas only 300 m thick, the lower layers of the Filter Gas would be transparent to long-wave radiation, and we have assumed them to be black. Thus our choice of 3 km for the Lower Gas is our way of approximating the 3-km absorption length of our gas.
Now consider the Upper Gas. As we present in Atmospheric Pressure, the Upper Gas is not at standard temperature and pressure. Indeed, the Upper Gas continues out to infinity as the atmosphere becomes thinner and thinner. Nevertheless, we can assign the Upper Gas a thickness in terms of the mass of air it contains. A 3-km high column of air 1 m2 in cross-section, and at STP, weighs 3000 kg and absorbs two-thirds of long-wave radiation. To the first approximation, this same mass of air will absorb two-thirds of long-wave radiation regardless of its pressure and temperature. So let us say that the Upper Gas contains 3000 kg of air per square meter. As we showed in our previous post, 3,000 kg of air above every square meter means a 30 kN weight for every square meter, so the pressure at the base of the Upper Gas is 30 kPa. The Upper Gas allows one-third of long-wave radiation from the Filter Gas to pass out into space.
The Upper Gas continues to infinity, but its base is at an altitude where the pressure is 30 Pa. According to our previous calculations, this altitude will be roughly 12 km. Given that our Lower Gas extends to 3 km and our Upper Gas begins at 12 km, this leaves our Filter Gas with a thickness of 9 km.
Now suppose we double the concentration of the long-wave absorbing impurity in our gas. The absorption length of the gas will halve. The Lower Gas will be 1.5 km thick. The Upper Gas will begin at 15 Pa pressure, or 19 km altitude. Our Filter Gas will now be 17.5 km thick. Decreasing the absorption length of our gas causes the Filter Layer to grow. As we determined previously, a thicker Filter Layer means a warmer Body.
From now on, we will refer to the boundary between the Upper Gas and the Filter Gas as the tropopause of our atmosphere. At this altitude, our atmosphere stops getting colder as we ascend. The Earth's atmosphere has a tropopause as well, and we will see in subsequent posts that it is the altitude of the tropopause that dictates the strength of the greenhouse effect.