On average, the total power of the thermal radiation emitted by Earth is equal to the power absorbed from sunlight. As CO levels rise, the emitted radiation can maintain this equilibrium only if the temperature increases, so that the total emitted radiation is unchanged (averaged over enough time, in the order of few years so that diurnal and annual periods are averaged upon).
According to Stefan–Boltzmann law, the total emitted power by Earth per unit area is:
where &sigma; is Stefan–Boltzmann constant and &epsilon; is the emissivity in the relevant wavelengths. T is some average temperature representing the effective radiation temperature.
CO content changes the effective T, but instead one may treat T to be a typical ground or lower-atmosphere temperature (same as T or close to it) and consider CO content as changing the emissivity &epsilon;. We thus re-interpret &epsilon; in the above equation as an effective emissivity that includes the CO effect;, and take T=T. A change in CO content thus causes a change d&epsilon; in this effective emissivity, so that  is the radiative forcing, divided by the total energy flux radiated by Earth.
The relative change in the total radiated energy flux due to changes in emissivity and temperature is:
Thus, if the total emitted power is to remain unchanged, a radiative forcing relative to the total energy flux radiated by Earth, causes a 1/4-fold relative change in temperature.
Thus: