Based on new radiative transfer numerical evaluations, we reconsider an argument presented by Schack in 1972 that says that saturation of the absorption of infrared radiation by carbon dioxide in the atmosphere sets in as soon as the relative concentration of carbon dioxide exceeds a lower limit of approximately 300 ppm. We provide a concise brief and explicit representation of the greenhouse effect of the earth's atmosphere. We find an equilibrium climate sensitivity (temperature increase $\Delta T$ due to doubling of atmospheric $CO_2$ concentration) of $\Delta T \simeq 0.5 ^0C$. We elaborate on the consistency of these results on $\Delta T$ with results observationally obtained by satellite-based measurements of short-time radiation-flux versus surface-temperature changes.