# Boltzmann Breakfast

A few days ago Phil Plait at Bad Astronomy wrote a beautifully clear post about why there are no green stars. If I can summarize, it's because anything that's been heated up enough to emit green light is also hot enough to emit red, yellow, and some blue as well. The combination appears to us as a brilliant white. The equation that tells you just how much light is being dumped out by an object at a given temperature is called Planck's law, after the legendary Max Planck. The radiance per frequency emitted by a blackbody (most hot things you meet in daily life are blackbodies to a good approximation) is

That will tell you proportionally how much energy is being dumped into each slice of the spectrum. Phil's post has some graphs for different values of the temperature T, and not only does the emitted spectrum shift more toward the blue as temperature increases, but the size of the graph increases as well. In other words, not only does more energy proportionally get allocated toward higher frequency light, the total amount of power being dumped into space in the form of light increases quickly as the temperature goes up.

Very quickly, it turns out. To find the total power radiated we'll have to integrate the above equation over the entire frequency spectrum. Let's not bother to be rigorous, we'll just do some voodoo and see if we can get an estimate. We see that the power is proportional to ν3. Integrating over ν pretty much just adds another factor of ν as far as the units are concerned. It's a start, but we'd like to have a rough idea of how the "typical" ν varies with temperature. See the scaling factor involving 1 over the exponential expression in the denominator? It's just that - a dimensionless scaling factor - and so if we wanted to we could rename the thing in the exponent as x, solve that for T, and write the whole equation in terms of T and x. Try it out yourself! You'll eventually find that instead of ν4, the whole equation could just as well be written in terms of T4. And that's just the Stefan-Boltzmann law.

What does that mean? Make an object twice as hot, and it radiates sixteen times as much light. So when you turn on your toaster oven and the heating elements are a blazing orange, they initially cool down very quickly when the timer pops up. They're radiating away an enormous amount of power because of their high temperature. But they cool, and correspondingly a reduction of 2 in the temperature means it's now cooling (at least via radiation) 16 times slower. This is why the heating elements will continue to glow a dull red for a longer time, and then the no-longer-glowing elements will still remain hot for several minutes.

Physics. Even toast can't escape it!

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Nor are there any "gold stars" as in the Green Lantern comic books.

At one point, Larry Niven recommended that I be the Science Advisor to that comic book line, then took the position himself.

Still later, I assisted a Writers Guild of America screenwriter on a Green Lantern feature screenplay, and my son (then about 15) suggested that a solid gold asteroid vaporized in the chromosphere of a star, and that accounts for the comic book premise.

When I was Adjunct Professor of Astronomy at Cypress College, I made my students work long and hard on star colors -- and the lovely application of "color curves" to dating globular clusters.

Nice blog; keep it up!

Speaking of Phil Plait, did you hear that he's president of the JREF now?

By Max Fagin (not verified) on 04 Aug 2008 #permalink

Green star? No problem! Boltzmann assumes a continuum blackbody source. Use an atomic line source. A (second generation) star whose visible surface is rich with thallium strongly emits at 535 nm. Thallium is volatile so atomic thallium emission could overpower the contingent thermal emission background.

Introducing enough thallium, Z = 81, into a star's atmosphere is left as an exercise for the alert reader.

Nice post. I think it would be nice to add that the color of the light, as we perceive it has to do mostly with the way our eyes work and only under extreme circumstances has to do anything with the real color of the light. If you have a place totally lit by incandescent light bulbs, the light is mostly yellow, but you do not feel it more than very slightly tinted. Yet, if you take a picture with an old style, non digital film camera, everything comes out an ugly yellow.

I really would not mind seeing a physicist telling us the real color of the sun, measured through spectroscopy, not by looking up and saying "yes, it is yellow".

From what you say, it should be bluish white, since that is the color of the photosphere, due to its 5,780 degrees Kelvin temperature.

By Andres Villarreal (not verified) on 04 Aug 2008 #permalink

A.V., the sun is in fact white. See the link at the top. I've heard that the intensly white cumulus clouds are the color of the sun, they have recombined the yellowish sun with the atmospheric blue that has been scattered out of the sunlight. White couds are used to calibrate sun observing instruments, or so I've heard. rb