Effective Temperature - Planet

Planet

The effective temperature of a planet can be calculated by equating the power received by the planet with the power emitted by a blackbody of temperature T.

Take the case of a planet at a distance D from the star, of luminosity L.

Assuming the star radiates isotropically and that the planet is a long way from the star, the power absorbed by the planet is given by treating the planet as a disc of radius r, which intercepts some of the power which is spread over the surface of a sphere of radius D. We also allow the planet to reflect some of the incoming radiation by incorporating a parameter 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:

The next assumption we can make is that the entire planet is at the same temperature T, and that the planet radiates as a blackbody. The Stefan–Boltzmann law gives an expression for the power radiated by the planet:

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

Note that the planet's radius has cancelled out of the final expression.

The effective temperature for Jupiter from this calculation is 112 K and 51 Pegasi b (Bellerophon) is 1258 K. A better estimate of effective temperature for some planets, such as Jupiter, would need to include the internal heating as a power input. The actual temperature depends on albedo and atmosphere effects. The actual temperature from spectroscopic analysis for HD 209458 b (Osiris) is 1130 K, but the effective temperature is 1359 K. The internal heating within Jupiter raises the effective temperature to about 152 K.