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Effective temperature is a term used in physics and astronomy to quantify the temperature of an object by comparing its luminosity to that of a black body.

A black body radiator will give off a certain amount of energy from its surface per unit area per unit time (called the flux). The Stefan-Boltzmann Law states that the flux emitted by a black body is dependent only on the temperature, according to

Ftotal = σ × (Teffective)4

If we know the size of an object we can compute the luminosity, which is just the flux times the object's surface area. For a sphere, the luminosity would be

L = 4 π R2 × F = 4 π R2 × σ T4

So if we want to measure the temperature of a black body, we can solve this relation for the effective temperature if we know the luminosity and the size of the object.

The effective temperature is very important when solving radiative transfer problems, particularly in stellar and planetary atmospheres. The problem is that there are no perfect black bodies in nature, though some objects come close. Any hot, dense object (like the filament in a light bulb or an electric stove) is approximately a black body. Stars are almost black bodies, though not quite. The Sun is nearly a black body, and has an effective temperature of 5774 kelvins.

The concept of effective temperature doesn't make much physical sense for nonthermal radiation sources. For those, we could measure the brightness temperature, or else ignore the concept of temperature altogether. For example, a synchrotron radiation source may emit relativistic electrons, and we would probably talk about the electrons' kinetic energy rather than their temperature.

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brightness temperature -- color temperature -- effective temperature
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Sources:
Radiative Processes In Astrophysics, G. Rybicki and A. Lightman, Wiley Interscience
Class lecture notes

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