Blackbody radiation is a fascinating concept that shines some light (hehe) on quantum phenomena and stars. Planck’s research into the ultraviolet catastrophe at the beginning of the 20th century jumpstarted quantum physics.
An ideal blackbody is an object that absorbs all radiation incident on it, and then emits 100% of it. This means that the rate at which it absorbs energy is the same as the rate of energy radiation. What makes blackbodies particularly interesting to study is that the spectrum of the emitted light is dependent only on its temperature.
The Ultraviolet Catastrophe
The way to model the energy of blackbody radiation was using the Rayleigh-Jeans law.
I won’t go too much into the details of its derivation, but the main thing we should take away is that it assumes that the energy of light E(f) is independent of the frequency and that it can take on all values. This law matched experimental predictions for large wavelengths, however, it failed miserably at shorter wavelengths like ultraviolet. The values for energy diverged to infinity!
Many physicists tried to solve the ultraviolet catastrophe, but it was Max Planck who finally cracked the mystery. His brilliant insight was that photons were only able to take on discrete (quantized) energy values. The equation for E(f) was hence
The entire equation looks very similar, however, this time it doesn’t diverge. The u(f, T) notation means that this is a function with two variables as its inputs. This is why the temperatures are always labeled or color-coded next to the curves.
The graph we looked at in the beginning is of course the intensity graph, which can be derived from Planck’s law. It is a lot of variables to look at, but the beauty in it all is that thinking about discrete energy levels completely changed physics.
Planck called this a mathematical trick at first since he was unable to explain this quantized nature of light. In 1905, Einstein showed that light also behaves like a particle with quantized energy in his Nobel prize-winning photoelectric effect experiment.
Stefan-Boltzmann law
Blackbody radiation is a very important concept in astrophysics. It can help us understand HR diagrams, habitable zones for exoplanets and the life cycles of stars. Let’s look at the basic equations for stellar radiation and power.
For practical applications, we can assume that object act as ideal blackbodies. This approximation works surprisingly well. Now, when we integrate the spectrum given by Planck’s law, we arrive at a very important result in astrophysics: the Stefan-Boltzmann law.
What this means is that the intensity of radiation is proportional to the fourth power of temperature. This is a key result for HR diagrams and figuring out information about distant stars. I’ll cover HR diagrams in the next post!
References
Blackbody radiation. Brilliant Math & Science Wiki. (n.d.). https://brilliant.org/wiki/blackbody-radiation/.
Blackbody radiation: Cosmos. Blackbody Radiation | COSMOS. (n.d.). https://astronomy.swin.edu.au/cosmos/b/blackbody+radiation#:~:text=As%20the%20temperature%20of%20the%20blackbody%20increases%2C%20the%20peak%20wavelength,(Stefan%E2%80%93Boltzmann%20Law).
The Editors of Encyclopaedia Britannica. (2019). Stefan-Boltzmann law. Encyclopædia Britannica. https://www.britannica.com/science/Stefan-Boltzmann-law.
The Editors of Encyclopaedia Britannica. (2020). Blackbody. Encyclopædia Britannica. https://www.britannica.com/science/blackbody.