Radiative forcing by carbon dioxide depends on the difference between surface and stratospheric temperature scaled by the logarithm of its concentration (Wilson and Gea-Banacloche 2012; Jeevanjee et al. 2021). This relationship arises due to the cooling-to-space theory or the \ (=1\) law, where all emission of infrared radiation originates from the atmospheric pressure level where the gas reaches sufficient optical thickness (in the case of CO2, in the stratosphere). Here we develop theoretical understanding of forcing by other well-mixed greenhouse gases including methane (CH4), nitrous oxide (N2O), and chlorofluorocarbons (CFCs). Radiative forcing by an optically thin absorber (e. g. , CFC-12) is governed by emission throughout the troposphere and scaled by the total change in gas concentration, such that a linear increase in gas abundance yields a linear increase in forcing. We examine the factors that control the magnitude of radiative forcing, finding that CFC-12 is a stronger per-molecule absorber than CO2 due to its larger average cross-section. Application in idealized atmospheres with simplified lapse rates illustrates how radiative forcing by optically thin gases depends almost linearly on lapse rate. Finally, gases that are both optically thin and optically thick across their absorption spectrum, such as N2O and CH4, can be understood as a combination of the two regimes, yielding a super-logarithmic relationship to concentration. Our theory is in excellent agreement with full-physics line-by-line calculations in atmospheres with and without spectral overlap by water vapor.
Czarnecki et al. (2025) studied this question.