 Stephans Boltzmans Law

If Stephan-Boltzman’s Law gives us the total amount of radiation throughout the spectrum, and our camera only works in the LWIR region how can it display the correct temperature?

This is an excellent question.

Firstly, yes Stephan-Boltzmans Law gives us the total radiation throughout the spectrum. And yes, our camera is calibrated using Stephan-Boltzman’s Law. And yes, our cameras have a limited spectral response.

Stephan-Boltzmans Law can be divided into two parts;

Stephan-Boltzmans Law for blackbodies

Stephan-Boltzmans Law for greybodies

Stephan-Boltzmans Law for blackbodies.

This states that the radiation emitted by a blackbody is proportional to the temperature to the fourth power. It only works with an absolute temperature scale like kelvin. It gives us this formula; Where

σ=(the Stephan-Boltzman constant) and T= temperature in kelvin

With this formula we can calculate how much radiation is emitted by a blackbody at a given temperature.

Stephan-Boltzmans Law for greybodies

This states that the radiation emitted by a blackbody is proportional to the emissivity times temperature to the fourth power. It only works with an absolute temperature scale like kelvin. It gives us this formula; Where

σ=(the Stephan-Boltzman constant), ε= emissivity and T= temperature in kelvin

With this formula we can calculate how much radiation is emitted by a greybody at a given temperature.

Next we must look at the difference between a blackbody and a greybody.

A blackbody, is theoretical object that absorbs all radiation incident on it. A blackbody will have an emissivity of 1, and it will be constant throughout the spectrum, in other words it emissivity will always be 1, everywhere on the spectrum.

A greybody, is another ideal definition, it is an object having a constant emissivity less than 1. In other words its emissivity will always be less than 1, but the value will be the same throughout the spectrum.

Most objects are neither blackbodies, nor greybodies, they are realbodies. A realbody is an object that has an emissivity that will vary with temperature angle and wavelength. in other words, the emissivity is not constant, it will be different in different parts of the spectrum, it will vary with viewing angle and temperature.

So why is Stephan-Boltzmans law used for calibration?

The camera is calibrated to blackbodies, and the amount of energy received by the detector is never the same as the amount of energy being emitted by the blackbody, it is always less. The offset between the emitted energy and the energy sensed by the detector is programmed into the camera. This is done by pointing the camera at a blackbody, and telling it that this much radiation is equal to this temperature etc. The camera then uses Stephan-Boltzmans Law for greybodies to calculate the temperature, and it needs a grey emissivity value to do this.

What about emissivity measurements?

Well, the camera assumes that we are looking at a greybody, and gives greybody equivalent values. So when we measure the emissivity of a material, the value we get is a greybody equivalent emissivity value (an average value within the spectral range of our camera). The actual emissivity might vary within the limited spectral range of our camera. Also as the object heats up the Planck curve will shift and the average value might change, thus a change in the measured emissivity.