Problem 6. In space, thermal equilibrium is achievedwhen incoming radiation (e.g. from the Sun) is balanced againstoutgoing radiation (e.g. from the surface of Earth). Theequilibrium achieved is a dynamic one, because there is still a netflow of heat in and out of the system, and the Sun and Earth neverreach the same temperature (thankfully) because much of theradiation leaves the system.
(a) The Sun provides a heat to the surface of Earth withan intensity (power per unit area) of about 1000 W/m2 . Compute thetotal power received. (Hint: The correct area to use is thecross-sectional area of the Earth, because that is the size of the‘shadow’ of solar radiation that is absorbed.) (In reality, about1400 W/m2 reaches Earth and about 30% is reflected.)
(b) Suppose the Earth is a perfect black-body absorberand emitter of radiation, and has a uniform surface temperature.(This is not a great assumption.) Find the equilibrium temperatureT of the surface in Kelvin and in Celsius, where Earth radiatesexactly as much power as it receives from the Sun. Is it anywhereclose to Earth’s average surface temperature?
(c) In fact, Earth’s atmosphere is not transparent tothe outgoing radiation, which makes the emissivity of Earthimperfect. The result is a delicate balance that preserves alife-friendly temperature. What emissivity e is required to achievethe current 15◦C average surface temperature? What emissivity ewould cause the temperature to rise by 2◦C? This is a vastlyoversimplified model of Earth’s climate. More accurate modelsinclude multiple coupled layers with independent temperatures andemissivities; these models can fairly accurately predict thesurface temperature as a function of greenhouse gas emissions(which determine the emissivity of the atmosphere). The net effectof adding carbon dioxide to the atmosphere is to reduce the amountof infrared emission at a given temperature, lowering e and raisingthe temperature.
