Blackbody Radiation
In the 19th Psalm, King David wrote that "The Heavens declare the glory of God; the skies proclaim the work of His hands. Day after day they pour forth speech; night after night they reveal knowledge." (bold added)
How much "knowledge" can really come from a series of points of light in the sky? The answer is A LOT! We will discuss some of the knowledge constantly streaming down from the heavens in this chapter.
A blackbody is an ideal absorber of light. Being a perfect absorber also makes it a perfect emitter of light. One complication is that objects don't generally emit the same wavelengths of light that they absorb. Think of a black t-shirt in sunlight. At least in the visible spectrum the shirt absorbs very well. That's why it's black. It doesn't emit any of the light that strikes it. Contrast that with a red t-shirt. The red shirt absorbs all colors coming from the sun but only emits red light.
For the black shirt in sunlight, it gets hotter and hotter and will start radiating more and more light back to the environment on account of getting hotter. The light coming off will, however, be mostly infra-rad (IR) and not visible light. When the rate that light energy gets absorbed by the t-shirt (power) matches the rate of light energy being radiated out, the shirt reaches thermal equilibrium (if we assume for now that there is very little heat transfer due to other mechanisms) and stops getting hotter.
Since this discussion will relate to stars, realize that in the vacuum of space, radiation is the whole story of heat transfer. Even though space is not a perfect vacuum, for these purposes we can assume it's quite good.
A black t-shirt does a good job of absorbing many wavelengths, but a blackbody is assumed to be 100% effective at absorbing all wavelengths. An even better black body than a black t-shirt is something like a cardboard box with a little hole cut in it. If the hole is small, looking into the hole, you see black even if light is entering. In this sense, it is the hole, and not the box that is the ideal blackbody. This hole is much like the pupil of our eyes. Your pupil is a hole to your inner eye, and while the surface of the inner eye is red, the pupil appears black because visible light enters but very little manages to escape. In the rare instance of "red-eye" in a photo, there is enough light going in from a camera flash that you see the inner eye for its true color.
The idea of light entering a small hole into a larger volume is that light enters the hole, and then bounces around the inside of the box (or eye) and little by little gets absorbed before having a chance to ever exit the small hole. If we don't use a cardboard box or a human eye, but maybe a clay cavity, then we can see what happens as the clay gets hotter without worrying about starting a fire.
Regardless of temperature, what we find is that there is broad-spectrum light being emitted from the hole that depends only on the temperature of the clay and not on the wavelengths of light that originally entered the hole. The shape of the curve can be seen below.
Characteristics of Blackbody Radiation
While the theory that leads to the shape of the curve is beyond the scope of our course, we will still be a little quantitative. You will notice if you play with the slider, that with higher temperatures, the intensity peaks at lower wavelengths, which have higher frequencies. The relationship is simple, and is called Wien's displacement law. You can think about the peak moving, or being displaced as a function of temperature. The peak wavelength given off by a blackbody is given by:
where, , and temperature T is in kelvin. From this simple relationship, we can determine the temperature of everything from the rubber on a race car's tire to the surface of the sun - never needing contact as with a standard thermometer.
Another characteristic of this curve is that the area under it represents the total energy given off by an object per unit time per unit surface area. This is shown in the interactive graphic in watts per square meter. While you can see from playing with the slider that the value rises with temperature, what probably isn't obvious is that it rises in proportion to the 4th power of temperature in kelvin. This is referred to as the Stefan-Boltzmann Law, and written in terms of the power given off by an object is: In this equation, is the emissivity of the object, and characterizes how good a black body the object really is, where is an ideal black body. The surface area of the object is A. Lastly, , which is known as the Stefan-Boltzmann constant, which is derived from other constants of nature.
The nature of the power equation being proportional to temperature to the fourth power means that the power radiated by an object - if we double its temperature - is times higher.
EXAMPLE: The peak wavelength given off by our sun is almost exactly 500nm. What is the surface temperature of the sun?
SOLUTION: Using Wien's displacement law, needs to be algebraically rearranged to give Plugging in numbers gives