We show how the CO2 contribution to the Earth’s greenhouse effect can be estimated from relatively simple physical considerations and readily available spectroscopic data. In particular, we present a calculation of the “climate sensitivity” (that is, the increase in temperature caused by a doubling of the concentration of CO2) in the absence of feedbacks. Our treatment highlights the important role played by the frequency dependence of the CO2 absorption spectrum. For pedagogical purposes, we provide two simple models to visualize different ways in which the atmosphere might return infrared radiation back to the Earth. The more physically realistic model, based on the Schwarzschild radiative transfer equations, uses as input an approximate form of the atmosphere’s temperature profile, and thus includes implicitly the effect of heat transfer mechanisms other than radiation.

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We note that there are also a number of technical discussions available in online forums, at varying levels of rigor and complexity. We have found the “Science of doom” blog (<http://scienceofdoom.com/>) especially useful for its thoroughness and pedagogic bent.
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See also, for instance, Sec. 1.3.2 of Ref. 5.
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In climate science, the term “feedback” is a bit technical and designates any climate variable that can affect the “Earth’s energy budget” [the difference between the right- and left-hand sides of Eq. (2)], when it changes in response to a change in the Earth’s surface temperature. For our purposes, we define the “no feedback” case as a situation in which we allow T to change in response to a change in xCO2, but keep x' (the fraction of radiation blocked by other greenhouse gases) and α (the Earth’s albedo) constant. (See also Sec. 8.4 of Ref. 5 for a precise definition of climate feedback parameters and some example calculations.)
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A slightly more sophisticated model for the atmospheric density, which included the effect of the temperature lapse rate, was used in the numerical calculations in Ref. 6.
22.
We are still considering only a one-dimensional model, so “upwards” here means “precisely vertically.” This, and other limitations of our model, will be revisited at the beginning of the next section.
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The product N(ν)ξ(z) is related to the “optical depth” χν(z) at wavenumber ν, measured downward from the top of the atmosphere (in Andrews 5, Chap. 3), by N(ν)ξ(z)=χν(0)-χν(z). It is similarly related to Pierrehumbert’s τν, which increases with altitude (Ref. 3, Sec. 4.2) as τν(p(z),p(0)).
25.
In fact, Eq. (18) is equivalent to the first Eq. (4.9) of Ref. 3, if the dependence of N(ν) on height is neglected there. The reader is referred to Chap. 4 of Ref. 3 for a much more in-depth treatment of the radiative transfer equations, with many examples.
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We note that the climate literature, by convention, actually considers the radiative forcing at the tropopause, rather than at z=, so in fact Iν(+)|ξ=ξ1=B(ν,T(0))e-N(ν)ξ1+N(ν)0ξ1e-N(ν)(ξ1-ξ')B(ν,T(ξ'))dξ' would be a better quantity to compare with standard climate models. Interestingly, although this obviously has the same large and small N(ν) limits as Eq. (20), we have not been able to find a simple approximation to it that works as well as Eq. (22) does for Eq. (20).
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