Fermi Paradox: Understanding the Greenhouse Effect Part 1 is attempting to explain the greenhouse effect to a lay person. I chimed in at Atmoz: Understanding the Greenhouse Effect. However, it got me thinking about all the people (who should know better) that say the greenhouse effect is due to CO2 reflecting solar radiation back towards the earth. I want to scream.

Let’s assume a plane-parallel atmosphere where there are only three processes taking place: reflection due to CO2, absorption into the surface, and reflection from the surface as illustrated in the following cartoon. Each of the numbers will be explained below. While I’m here, let’s define some constants and variables.

R^* = reflectivity of CO2
R = reflectivity of the ground
S= solar constant
A = total absorbtivity
R_T = total reflectivity

The yellow ball is the sun. The blue is the atmosphere. The brown is the earth. The black dots are the CO2. The yellow lines are the sun’s rays. Everything is drawn to scale. CO2 molecules are huge, and reside in a layer near the stratopause.

1 - the solar radiation from the sun, the solar constant.
2 - the amount of the incoming solar radiation that is reflected by the CO2.
3 - the amount of the solar radiation that is transmitted through the CO2.
4 - the amount of transmitted radiation absorbed into the surface.
5 - the amount of transmitted radiation reflected from the surface.
6 through 10 should be obvious what they are based upon the previous explainations.

The values of the 12 numbers in the cartoon are:
1 = S
2 = R^*S
3 = (1-R^*)S
4 = (1-R^*)S(1-R)
5 = (1-R^*)SR
6 = (1-R^*)SR(1-R^*)
7 = (1-R^*)SRR^*
8 = (1-R^*)SRR^*(1-R)
9 = (1-R^*)SRR^*R
10 = (1-R^*)SRR^*R(1-R^*)
11 = (1-R^*)SRR^*RR^*
12 = (1-R^*)SRR^*RR^*(1-R)

The amount of total reflected radiation is equal to the sum of 2, 6, and 10 (plus all the other smaller terms).
(Note: the left-hand side of the following three equations should be R_T, for total reflectivity, not R.
Simplifying… and recognizing the geometric series, since 0 < RR^* < 1…

Let’s do the same with the absorbed radiation. The amount of total absorbed radiation is equal to the sum of 4, 8, and 12 (plus all the other smaller terms). Simplifying… and recognizing the geometric series, since 0 < RR^* < 1…

Now let’s plot the absorptivity as a function of the reflectivity of CO2.

And now the corresponding total reflectivity as a function of the reflectivity of CO2.

I set the reflectivity of the ground, R, equal to 0.3. If there were no reflection due to CO2 the total reflectivity would be 0.3, and the absorptivity would be 0.7, exactly as the graphs show when R^* equals 0. When R^* > 0 the absorptivity falls monotomically to 0. Thus, any reflection in the visible due to CO2 will cool the surface.

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