# Part II: What Ångström didn’t know

When the product of the absorption factor times the amount of CO_{2} encountered equals one, then the amount of light is reduced by a factor of 1/e, i.e. 1/2.71282… . For this, or larger, amounts of CO_{2},the atmosphere is *optically thick *at the corresponding wavelength. If you double the amount of CO_{2}, you reduce the proportion of surviving light by an additional factor of 1/e, reducing the proportion surviving to about a tenth; if you instead halve the amount of CO_{2}, the proportion surviving is the reciprocal of the square root of e , or about 60% , and the atmosphere is *optically thin*. Precisely where we draw the line between "thick" and "thin" is somewhat arbitrary, given that the absorption shades smoothly from small values to large values as the product of absorption factor with amount of CO_{2} increases.

The units of absorption factor depend on the units we use to measure the amount of CO_{2} in the column of the atmosphere encountered by the beam of light. Let’s measure our units relative to the amount of CO_{2} in an atmospheric column of base one square meter, present when the concentration of CO_{2} is 300 parts per million (about the pre-industrial value). In such units, an atmosphere with the present amount of CO_{2} is optically thick where the absorption coefficient is one or greater, and optically thin where the absorption coefficient is less than one. If we double the amount of CO_{2} in the atmosphere, then the absorption coefficient only needs to be 1/2 or greater in order to make the atmosphere optically thick.

The absorption factor, so defined, is given in the following figure, based on the thousands of measurements in the HITRAN spectroscopic archive. The "fuzz" on this graph is because the absorption actually takes the form of thousands of closely spaced partially overlapping spikes. If one were to zoom in on a very small portion of the wavelength axis, one would see the fuzz resolve into discrete spikes, like the pickets on a fence. At the coarse resolution of the graph, one only sees a dark band marking out the maximum and minimum values swept out by the spike. These absorption results were computed for typical laboratory conditions, at sea level pressure and a temperature of 20 Celsius. At lower pressures, the peaks of the spikes get higher and the valleys between them get deeper, leading to a broader "fuzzy band" on absorption curves like that shown below.

We see that for the pre-industrial CO_{2} concentration, it is only the wavelength range between about 13.5 and 17 microns (millionths of a meter) that can be considered to be saturated. Within this range, it is indeed true that adding more CO_{2} would not significantly increase the amount of absorption. All the red M&M’s are already eaten. But waiting in the wings, outside this wavelength region, there’s more goodies to be had. In fact, noting that the graph is on a logarithmic axis, the atmosphere *still* wouldn’t be saturated even if we increased the CO_{2} to *ten thousand* times the present level. What happens to the absorption if we quadruple the amount of CO_{2}? That story is told in the next graph:

The horizontal blue lines give the threshold CO_{2} needed to make the atmosphere optically thick at 1x the preindustrial CO_{2} level and 4x that level. Quadrupling the CO_{2} makes the portions of the spectrum in the yellow bands optically thick, essentially adding new absorption there and reducing the transmission of infrared through the layer. One can relate this increase in the width of the optically thick region to the "thinning and cooling" argument determining infrared loss to space as follows. Roughly speaking, in the part of the spectrum where the atmosphere is optically thick, the radiation to space occurs at the temperature of the high, cold parts of the atmosphere. That’s practically zero compared to the radiation flux at temperatures comparable to the surface temperature; in the part of the spectrum which is optically thin, the planet radiates at near the surface temperature. Increasing CO_{2} then increases the width of the spectral region where the atmosphere is optically thick, which replaces more of the high-intensity surface radiation with low-intensity upper-atmosphere radiation, and thus reduces the rate of radiation loss to space.

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