## Effective radiating temperature of planets

The Sun radiates essentially as a black body, with an effective surface temperature Ts = 5,750 K and radius Rs = 700,000 km. Applying Stefan-Boltzmann's law we find that the total power radiated by the Sun in all directions is

where a is the Stefan-Boltzmann constant. The flux of sunlight (W m~2) arriving at a planet at distance D from the Sun is simply this power divided by the surface area of a sphere of radius equal to the distance to the Sun: that is,

At the Earth's distance from the Sun the solar constant Fs is equal to 1.37 kWm-2. The total amount of sunlight absorbed by the planet is then equal to

where RP is the planetary radius; kR2 is the projected disk area; and AB is the Bond albedo, which is the fraction of sunlight reflected by the planet in all directions. Assuming no other sources of heat, in order for the temperature of the planet to be in equilibrium, this absorbed power must be balanced by the thermal radiation emitted to space. Since the observable temperatures of the giant planets are so much smaller than the Sun's, the Planck functions of incident solar and emitted thermal radiation show negligible overlap and thus the two fluxes may be considered separately. The total power emitted by the planet (of effective radiating temperature TE) to space is equal to its total surface area multiplied by the infrared flux Fir = aTe; that is,

Equating the total absorbed power to the emitted power we find ct(4kR2)TE = (1 - AB)kR2Fs (3-32)

and rearranging for TE, the effective radiating temperature, we obtain

4a or by substituting Fs from Equation (3.29)

For the Earth TE is estimated to be approximately 255 K. The mean surface temperature on the Earth is fortunately significantly higher than this due to the greenhouse effect provided by the IR absorptivity of carbon dioxide and water vapor.

The albedo used here for effective radiating temperature calculations is the Bond albedo (or planetary bolometric albedo) and is defined as the fraction of incident solar radiation scattered in all directions. When measuring the reflectivity of planets, in particular the giant planets, all we see is the radiation reflected in the direction back to Earth. If the Earth, Sun, and planet are all in the same line, the albedo measured is called the geometric albedo. Should the surface of the planet reflect light equally well in all directions (and thus be a perfect Lambert reflector) then the Bond and geometric albedos are in fact identical. Otherwise, they differ depending upon the scattering properties of the planet's surface or cloud layers.