Color temperature is the color of light that would be emitted by an object (technically a black body, which is an ideal and diffuse emitter. Most solids are extremely close to a black body. Feel free to ask for more details) at a given temperature.
For example, if you get an LED light bulb that has a color temperature of 2700K, it’s giving you the color that an incandescent light bulb would give off at 2700 kelvin.
Brightness temperature is not measuring an actual temperature, but instead takes the intensity of radiation from a source and provides the temperature of a black body that would emit that radiation.
For example, if you use an infrared thermometer, it measures the emissivity of a surface, and returns the brightness temperature, which, as most surfaces are very close to black bodies, is reasonably accurate to the actual temperature.
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