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**How the Greenhouse Gas Effect Works**

We all know by now that the greenhouse gas effect depends upon absorption by greenhouse gas molecules of infrared energy radiated from the Earth’s surface, and which is then re-radiated by these molecules in all directions. Effectively, half the energy is emitted to outer space, and half is emitted to the Earth’s surface where it is absorbed again. Together with the additional energy continually being received from the Sun, this warms the surface more than would be the case without the GHG, and so causes increased radiation from the surface.

Consider an area of 1 square metre of the surface, and let us take a value of 235 Watts for the power incident upon this area. Assume an emissivity of 1.0 for the atmosphere. Follow the process through second by second as in the table below.

Sun’s radiation to____Surface radiation____Back radiation____Radiation to

Earth’s surface(W)___to atmosphere(W)___to surface(W)_____space(W)

235________________235_______________117.5___________117.5

235________________352.5______________176.25__________176.25

235________________411.25_____________205.625_________205.625

235________________440.625____________220.3125________220.3125

235________________455.3125___________227.65625_______227.65625

235________________462.65625__________231.328125______231.328125

235________________466.328125_________233.1640625_____233.1640625

235________________468.1640625________234.0820313_____234.0820313

235________________469.0820313________234.5410156_____234.5410156

235________________469.5410156________234.7705078_____234.7705078

235________________469.7705078________234.8852539_____234.8852539

235________________469.8852539________234.942627______234.942627

235________________469.942627_________234.9713135_____234.9713135

235________________469.9713135________234.9856567_____234.9856567

235________________469.9856567________234.9928284_____234.9928284

235________________469.9928284________234.9964142_____234.9964142

235________________469.9964142________234.9982071_____234.9982071

235________________469.9982071________234.9991035_____234.9991035

235________________469.9991035________234.9995518_____234.9995518

235________________469.9995518________234.9997759_____234.9997759

We see that the radiation to space converges towards 235 Watts, which in fact is the limit set by the power received from the Sun since, in equilibrium, the power emitted must equal the power received.

Also, the power radiated to the atmosphere converges towards 470 Watts, which is twice the power emitted from the atmosphere to space. Now, the power radiated from a body is proportional to the fourth power of the absolute temperature, by the Stefan-Boltzmann Law.

Let Ts be the absolute temperature of the Earth’s surface.

Let Ta be the absolute temperature of the top of atmosphere.

Therefore, we have

( Ts )^4 = 2 × ( Ta )^4

and Ts = (2)^0.25 × ( Ta)

Finally,

**Ts = 1.189 × Ta**

That is, the surface temperature of the Earth is 18.9% greater than the temperature of the GHG emitting to space. This will be true provided that there is at least sufficient greenhouse gas present to be able to cope with the amount of energy being received from the Sun.

Note that this analysis can be applied to other planets which have an atmosphere containing at least sufficient greenhouse gases, like Venus. It also means that there cannot be a runaway greenhouse gas effect, because the ToA temperature is governed by the emission which must be equal to the energy received from the Sun, and the surface temperature is limited to twice the ToA temperature. This reasoning, of course, assumes that no energy is entering the atmosphere from other sources, such as geothermal energy, and it assumes an emissivity of 1.0 for the atmosphere.

**Aubrey E Banner**

Sale, Cheshire, UK