Understanding the Greenhouse Effect – part 1

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This is the first post in a series to explain how greenhouse gases work and what evidence we have that it is indeed the greenhouse gases that cause the observed warming of the earth.

This first post will deal with the physics and the natural greenhouse effect.

Black Body Radiation

A black body is a theoretical body that absorbs all electromagnetic radiation that falls onto it. A black body emits radiation with a spectrum that can be calculated with a equation, and it depends only on its effective temperature. The amount of energy emitted per surface area also depends only on the temperature. The hotter it is, the more energy it emits at shorter wavelengths. In reality, no perfect black body exists, but for most purposes it is close enough. The spectrum of the sun is close to that of a black body, and emits most of its energy at visible wavelengths.

A black body that absorbs electromagnetic radiation also gets a temperature that depends on the amount absorbed, and can be calculated according to the Stefan-Boltzmann law.

The surface of the earth can also be seen as a black body, although it reflects a large part of the incoming radiation. The figure below shows the spectrum of the sun over frequency in red, and the theoretical black body radiation of the earth. Because it is a plot over frequency, shorter wavelengths are to the right.

We now have the information we need to calculate a rough estimate of the temperature of the earth:

The amount of energy the earth gets from the sun is 1367 W/m2. The cross section of the earth is its radius squared times pi, but the surface of earth is 4 times that. So on average, each square metre gets a quarter of the energy from the sun, 341.75 W/m2. The albedo is the fraction that is reflected, and the value for the earth is about 0.3. The reflected amount cannot heat the earth, so we just take the remaining part into account. All this together gives us a value of 254.8 Kelvin, which is -18.3 C (or -1 Fahrenheit) – clearly, something is wrong here. The average temperature of the earth is more like15 degrees Celsius.

So where is the error? To account for the observed difference, we would need an additional 150 W/m2.

Transparency of the Atmosphere

The clue lies in the different wavelength regimes where the radiation from the sun comes to the earth, and that of the earth leaving from the ground. As explained above, the wavelength depends on the temperature. The radiation from the sun is mostly in the visible domain, for which the atmosphere is nearly transparent (safe clouds). But the radiation going up from the earth, being much cooler than the sun, is in the infrared and for that the atmosphere is not completely transparent. But to maintain a balance (as much energy as is coming in has to get out for an equilibrium), the temperature rises. Or in other words, the atmosphere, being warmed by the absorption of the infrared radiation, radiates back to the earth. And this is the missing radiation to get a temperature of 15 degrees.

Not all components of the atmosphere absorb infrared radiation. The atmosphere consists of 78% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, and trace amounts of other gases plus around 1% water vapor. Nitrogen and oxygen are symmetric molecules, and do not absorb in the infrared. Argon is a monatomic gas, and does not absorb either. On the other hand, CO2 and H20 can bend and swing in modes which do absorb in certain wavelengths in the infrared.

The graphic above shows the radiation from the atmosphere going down to the surface in the infrared. The graphic has been calculated from a model by David Archer, using this nice web interface. Most of the radiation comes from water vapor and carbon dioxide.

In reality, the radiation budget of the earth is a lot more complicated than described. The figure above shows the different flows of radiation between the sun, the atmosphere and the surface. The image links to an interesting paper examining the radiation budget.

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