Blackbody Radiation

The first question is, what exactly is a blackbody? Briefly, it’s a hypothetical object that absorbs all the radiation directed at it, without reflecting any (so it would appear black in the visible range, as well as every other part of the electromagnetic spectrum). A blackbody also radiates energy “perfectly”, which is to say that it emits the maximum amount of radiation possible for its temperature, and that it radiates energy equally in all directions — or as physics would say, it radiates isotropically.

Now that that’s cleared up, we move onto the second question: why do we even care about blackbodies in astronomy? The answer is that many cosmic objects can be modeled as blackbodies because they are close enough to being perfect blackbodies that we can approximate, even though perfect blackbodies are thought to only exist in the realm of theoretical physics. Planets, stars, the universe itself…

Wait, the universe?

Indeed.

In 1989, NASA launched the Cosmic Background Explorer satellite to, well, explore cosmic background radiation (they’re very creative with naming things, it seems). The Far Infrared Absolute Spectrophotometer was designed to measure the Cosmic Microwave Background, and it found that the CMB matches a blackbody curve for T=2.725 K almost perfectly! This might not seem like much of an accomplishment, but the FIRAS data is very strong evidence supporting the “hot Big Bang” theory — which basically states that the universe used to be really hot, until it expanded and cooled off.

COBE FIRAS data for CMB

Credit: NASA JPL at Caltech

Radiation curves show how much energy is given off by a blackbody at varying wavelengths. In the graphic below, you can see that the curves climb sharply, peak, and then decay away towards zero as the wavelength increases (however, they will never actually reach zero, since blackbodies radiate energy at all wavelengths).

So what can these radiation curves tell us?

Firstly, we can see that the peaks are at different wavelengths for blackbodies of different temperatures — the peak for a higher-temperature blackbody has a shorter wavelength than the peak for a cooler blackbody. This explains why we see cooler stars as reddish in color and hotter stars as bluish, but more on that later in a separate post. We can also see that as the temperature increases, so does the total energy emitted (the area under the curve), so a star at 6000 K will emit more energy than one at 4000 K.

Radiation Curves

Credit: University of Michigan

At this stage, you may be thinking, it’s all well and good that radiation curves can tell us things, but is there a way to be quantitative about how much the peak wavelength shifts, or how much energy is emitted? For the answers, we must once again venture into the dark depths of mathematics, which is unfortunately unavoidable in astrophysics. Remember that units must always match and that significant figures count.

You may remember from one of our history posts that Rayleigh and Jeans tried to describe the shape of a radiation curve, but they ran into the “ultraviolet catastrophe” when they got to short wavelengths. Wien tried to do the same thing, but his equation ran into problems with long wavelengths. Then Planck did some thinking and combined the Rayleigh-Jeans Law and Wien’s Approximation into Planck’s Radiation Law, which works at all wavelengths. This equation also allows us to calculate the amount of energy given off at a certain wavelength by a blackbody at a certain temperature.

Neither of us have ever needed Planck’s Radiation Law for a test (and one of us has a testwriter who would be the type to try and include it), so we won’t give you a practice problem for it, but the equation is included above just in case.

Wien’s Law, also known as the Wien Displacement Law, allows us to calculate the peak wavelength from the temperature or vice versa. From this equation, it’s easy to see that higher temperatures result in shorter peak wavelengths, and lower temperatures result in longer peak wavelengths.

Ex. Assume the temperature of the Sun is 5800 K. What is the Sun’s peak wavelength?

Answer: 5.0 * 10^-7 m

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The Stefan-Boltzmann Law, derived independently by Josef Stefan and Ludwig Boltzmann, allows us to calculate the area under the curve described by Planck’s Radiation Law. This is the flux of the blackbody — the power per area. You can see that a small change in temperature will cause a large change in flux, since temperature is raised to the fourth power.

The Stefan-Boltzmann Law can also be found in an alternate form that calculates the TOTAL energy emitted by the star or other blackbody.

(The geometrically astute will realize that 4πr^2 is the formula for surface area of a sphere, and that we are simply multiplying “power per area” by area to get total power.)

Ex. Assume the temperature of the Sun is 5800 K. What is the Sun’s flux?

Answer: 6.4 * 10^7 W/m^2

Ex. What is the luminosity of the Sun? Its radius is 6.9599 * 10^8 m.

Answer: 3.9 * 10^26 W

———-

TL;DR — Blackbodies are objects that absorb and radiate energy “perfectly”. Many astronomical objects (most importantly stars) can be modeled as blackbodies. Radiation curves can tell us a lot about the objects they come from — Planck’s Radiation Law lets us plot the shape of the radiation curve, Wien’s Law lets us calculate the peak wavelength, and the Stefan-Boltzmann Law allows us to calculate the flux and/or the luminosity.

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Sources and links for further reading:

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