Editing Effective temperature (section)

===Surface temperature of a planet===

The surface temperature of a planet can be estimated by modifying the effective-temperature calculation to account for emissivity and temperature variation.

The area of the planet that absorbs the power from the star is {{math|''A''<sub>abs</sub>}} which is some fraction of the total surface area {{math|''A''<sub>total</sub> {{=}} 4π''r''<sup>2</sup>}}, where {{mvar|r}} is the radius of the planet. This area intercepts some of the power which is spread over the surface of a sphere of radius {{mvar|D}}. We also allow the planet to reflect some of the incoming radiation by incorporating a parameter {{mvar|a}} called the [[albedo]]. An albedo of 1 means that all the radiation is reflected, an albedo of 0 means all of it is absorbed. The expression for absorbed power is then:

:<math>P_{\rm abs} = \frac {L A_{\rm abs} (1-a)}{4 \pi D^2}</math>

The next assumption we can make is that although the entire planet is not at the same temperature, it will radiate as if it had a temperature {{mvar|T}} over an area {{math|''A''<sub>rad</sub>}} which is again some fraction of the total area of the planet. There is also a factor {{mvar|ε}}, which is the [[emissivity]] and represents atmospheric effects. {{mvar|ε}} ranges from 1 to 0 with 1 meaning the planet is a perfect blackbody and emits all the incident power. The [[Stefan–Boltzmann law]] gives an expression for the power radiated by the planet:

:<math>P_{\rm rad} = A_{\rm rad} \varepsilon \sigma T^4</math>

Equating these two expressions and rearranging gives an expression for the surface temperature:

:<math>T = \sqrt[4]{\frac{A_{\rm abs}}{A_{\rm rad}} \frac{L (1-a)}{4 \pi \sigma \varepsilon D^2} }</math>

Note the ratio of the two areas. Common assumptions for this ratio are {{sfrac|[[Area of a disk|1]]|[[Sphere#Surface area|4]]}} for a rapidly rotating body and {{sfrac|1|2}} for a slowly rotating body, or a tidally locked body on the sunlit side. This ratio would be 1 for the [[subsolar point]], the point on the planet directly below the sun and gives the maximum temperature of the planet — a factor of {{sqrt|2}} (1.414) greater than the effective temperature of a rapidly rotating planet.<ref>Swihart, Thomas. "Quantitative Astronomy". Prentice Hall, 1992, Chapter 5, Section 1.</ref>

Also note here that this equation does not take into account any effects from internal heating of the planet, which can arise directly from sources such as [[radioactive decay]] and also be produced from frictions resulting from [[tidal forces]].