Tutorial on the Greenhouse Effect

This page presents a simple physical model of the greenhouse effect that demonstrates how the blanketing effect of greenhouse gases in the atmosphere can elevate the surface temperature of a planet. The model is an instructional "toy model," meaning it strips the process down to its essential elements so that the basic ideas are easy to convey. Models that are used to make predictions by climate experts are substantially more sophisticated, but at root the physics are similar to what is described below.

The first key idea is that hot objects lose heat faster than cold objects. This is obvious from everyday experience (you can feel the heat coming from a fire). Detailed observations show that the rate of heat loss is very sensitive to temperature -- specifically, if the temperature is doubled (on an absolute scale), the rate of heat loss is not twice as high -- it is sixteen times as high.

The second key idea is that planets are near an equilibrium where heat lost to space almost exactly equals sunlight gained. Because hot objects lose heat rapidly, they tend to cool off if they have no energy source to maintain their temperature. On the other hand, because cold objects only lose heat slowly, they tend to warm up in the presence of energy sources. In both cases, the objects converge toward a condition where they lose heat at exactly the same rate that it is supplied by energy sources. In the case of planets, the energy source is sunlight.

Let's see how this works for a planet with no atmosphere. At the position of Earth, the absorbed sunlight is 240 Watts/meter2. In equilibrium, this means that the planet would lose heat to space -- as infrared radiation -- also at a rate 240 Watts/meter2. How can we calculate the temperature from this? Detailed measurements show that, mathematically, the relationship between heat loss and temperature can be described by the equation F = &sigma T4, where F is the rate of heat loss (the "heat flux") and &sigma is a fundamental physical constant (called the Stephan-Boltzmann constant) with a value of 5.67 x 10-8 Watts/meter2 Kelvin4. We can rearrange this equation to state that, for a planet with no atmosphere,

T = (F/&sigma)1/4.

Plugging in F=240 Watts/meter2 and &sigma=5.67 x 10-8 Watts/meter2 Kelvin4, we find that T=255 K, which corresponds to a temperature of -18oC or 0oF.

Thus, if Earth had no greenhouse effect, the average surface temperature would be 0oF -- far below the freezing temperature! The oceans would be totally frozen and life would not exist on Earth.

How does having an atmosphere with greenhouse gases affect this situation? The greenhouse effect only works if the atmosphere is transparent to sunlight but opaque to infrared (heat) wavelengths. Many gases -- CO2, water vapor, methane -- behave just this way. These are the greenhouse gases.

In this case, the Earth still gains 240 Watts/meter2 from the sun. It still loses 240 Watts/meter2 to space. However, because the atmosphere is opaque to infrared light, the surface cannot radiate directly to space as it can on a planet without greenhouse gases. Instead, this radiation to space comes from the atmosphere.

However, atmospheres radiate both up and down (just like a fire radiates heat in all directions). So although the atmosphere radiates 240 Watts/meter2 to space, it also radiates 240 Watts/meter2 toward the ground! Therefore, the surface receives more energy than it would without an atmosphere: it gets 240 Watts/meter2 from sunlight and it gets another 240 Watts/meter2 from the atmosphere -- for a total of 480 Watts/meter2 in this simple model.

Now like the atmosphere, the Earth's surface is near an equilibrium where it gains and loses energy at almost the same rate. Because the surface gains 480 Watts/meter2 (half from sunlight and half from the atmosphere), it also must radiate 480 Watts/meter2. Unlike the atmosphere, however, the ground can only radiate in one direction -- upward. Thus, the surface radiates 480 Watts/meter2 upward, and because the atmosphere is opaque to this infrared light, it is absorbed by the atmosphere rather than escaping to space. Notice that the atmosphere, the surface, and the planet as a whole each gain energy at exactly the same rate it is lost.

We can again use the simple expression T = (F/&sigma)1/4 to calculate the temperature of the surface. Using F = 480 Watts/meter2 and &sigma=5.67 x 10-8 Watts/meter2 Kelvin4, we find that T=303 K, which corresponds to 30oC or 86oF.

Key points:

&diams Without greenhouse gases, we calculated that the surface temperature would be 255 K (0oF), whereas with greenhouse gases we calculated the surface temperature would be 303 K (86oF). Therefore, the blanketing effect of atmospheric greenhouse gases has caused an elevation of the surface temperature. This is the greenhouse effect!

&diams The greenhouse effect is NOT a situation where "heat is trapped and can't escape." The above calculation makes clear that the opposite is true: the greenhouse effect is how the atmosphere adjusts so that it CAN lose heat when greenhouse gases are present in the atmosphere. About the same amount of heat escapes to space regardless of whether a greenhouse effect exists.

&diams In our simple model, we predicted an elevation in surface temperature of 48oC (86oF). This is an overestimate. On the real Earth, the current average surface temperature is 288 K (59oF), not 303 K, so the actual greenhouse effect causes a warming of only 33oC (59oF) relative to an atmosphere without a greenhouse effect. Thus, the crude model presented here overestimates the strength of the greenhouse effect by 50%. This discrepancy is caused by several factors that we neglected. For example, some sunlight is absorbed in the atmosphere rather than at the surface, and some infrared radiation from Earth's surface can escape to space rather than being absorbed in the atmosphere. These effects are all included in real climate models. Properly taking these effects into account would lead to a predicted temperature much closer to the actual temperature.

&diams An increase in the abundance of CO2, water vapor, methane, and other greenhouse gases causes a decrease in the fraction of infrared radiation from the surface that can escape to space. This forces the surface temperature to increase as the Earth strives to reach the new equilibrium. More greenhouse gases mean a stronger greenhouse effect and a hotter planet.

&diams When the greenhouse gas abundance is increased, it takes time for the system to warm to the new equilibrium temperature. During these times, the Earth absorbs slightly more sunlight than it loses heat, which is what allows the warming. Thus, during these times, the Earth is slightly out of equilibrium. What this means is that even if the abundances of greenhouse gases became constant right now, the Earth would continue to warm by another 0.5-1oC (1-2oF) over the next 50-100 years as it reached the new equilibrium temperature. This delayed warming has already been caused and is unavoidable. Of course, additional warming will occur if greenhouse gas abundances continue to increase.