By Joshua N. Winn, Princeton University
The molecules in the air around us are moving at speeds of hundreds of meters per second. Of course, they don’t get very far before knocking into another molecule, but still, that’s an impressive speed. What happens when we consider photons? What would happen if we trap some photons from the Sun, in a reflecting box, and measure their wavelength?
Planck Spectrum
Let’s think of a chart on which we choose the case of nitrogen molecules at 300 Kelvin; basically, air at room temperature. It rises from zero to a peak at around 400 meters per second, then it falls off again. If we turn up the temperature, the peak spreads out and moves to higher velocities, and if we turn it down, the peak moves to the left, to lower velocities. In general, the most probable speed is the square root of 2kT over m.
Now, let’s switch to photons. Let’s imagine trapping some photons from the Sun, in a reflecting box, and measuring the wavelength of each and every one. We’ll draw a wavelength scale and divide it up into bins, like ticks on a ruler, and then keep a tally of how many photons have a wavelength within each bin. Once we collect enough photons, we see there’s a peak at around 0.6 microns. That is the most popular wavelength to have in sunlight.
The shape of the function looks like the Maxwell-Boltzmann distribution, but it’s different in detail because photons are not our everyday particles. This is called a Planck spectrum. This curve is for 5800° Kelvin, approximately the temperature of the Sun’s outer layers. The bright star Vega is hotter than the Sun—it’s closer to 9500 Kelvin—so its spectrum is shifted toward higher energies, which means, shorter wavelengths. And the faint, nearby star Proxima Centauri is only about 3000 Kelvin, so its photons generally have lower energies, and longer wavelengths.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Flux Density
The Planck spectrum is usually expressed as the flux per unit wavelength, the so-called flux density. Flux density is power per unit area per unit wavelength.
When we measure the flux density of the Sun, as a function of wavelength, we find it’s a pretty good fit to a theoretical Planck spectrum. It peaks at around 2 kilowatts per square meter per micron, at a wavelength of half a micron.
Why doesn’t it fit exactly? The Planck spectrum describes the radiation we get from particles that have been knocking around long enough to reach a constant temperature: they’re in thermodynamic equilibrium. It’s often called a ‘blackbody’ spectrum, because technically, the derivation relies on the material being a perfect absorber of photons, and therefore ‘black’.
The Sun, or any other real object, does not meet those criteria exactly. The Sun is not all at one temperature; it gets hotter as we go deeper. And the Sun’s material is not perfectly absorbing. But the spectrum of the Sun and other stars are nevertheless reasonably well described by the Planck function.
Planck Spectrum on Logarithmic Axes
Let’s look at the Planck spectrum on logarithmic axes. That way, we can let the wavelength scale range over a factor of a 1000, from ultraviolet to infrared, and we can let the flux density scale over a factor of a trillion.
As we increase the temperature, the curve lifts up vertically. Hotter sources produce more radiation at all wavelengths. The area under each curve—the integral of flux density over wavelength—is the total flux, which is equal to sigma times T to the 4th. That’s the Stefan-Boltzmann law. We double the temperature, and the flux rises a factor of 16.
Also, as we increase the temperature, the peak of the spectrum shifts to shorter wavelengths. At room temperature, 300 Kelvin, almost all the energy comes out in the infrared; it peaks at around 10 microns. As we dial up the heat, the peak shifts to shorter wavelengths. That makes sense because we expect the typical photon energy, hc/ over lambda, to be on the order of kT; that implies lambda should be of order hc over kT: it should be inversely proportional to temperature.
Wein’s Law
When we do the math exactly, we find that the peak of the spectrum occurs when lambda is about one-fifth of hc over kT. That’s called Wien’s law. We can also write it as a scaling relation: lambda peak equals 10 microns times T divided by 300 Kelvin to the minus one power.
We are constantly bathed by photons whose spectrum follows the Planck function with an accuracy better than one part in 10,000, and a temperature of 2.7° Kelvin. According to Wien’s law, that corresponds to a wavelength of 1 millimeter, in the microwave band of the spectrum.
Why is the universe permeated with this microwave blackbody radiation? It’s a clue that at some point in the past, the universe was itself was a ‘gas’ of particles at a single temperature in thermodynamic equilibrium, long before it became the place we know today, with tiny pockets of extreme heat and vast expanses of frigid cold. This so-called cosmic microwave background radiation is some of the best evidence we have for the Big Bang.
Common Questions about the Planck Spectrum
The bright star Vega is hotter than the Sun—it’s closer to 9500 Kelvin—so its spectrum is shifted toward higher energies, which means, shorter wavelengths. And the faint, nearby star Proxima Centauri is only about 3000 Kelvin, so its photons generally have lower energies, and longer wavelengths.
The Planck spectrum is usually expressed as the flux per unit wavelength, the so-called flux density. Flux density is power per unit area per unit wavelength.
The Planck spectrum describes the radiation we get from particles that have been knocking around long enough to reach a constant temperature: they’re in thermodynamic equilibrium. It’s often called a ‘blackbody’ spectrum, because technically, the derivation relies on the material being a perfect absorber of photons, and therefore ‘black’.
