Why is the Sun Yellow?

This classic childhood question actually has the same answer as, "Why the sky is blue?'', which is actually not the same answer as many teachers give when teaching introductory Astronomy courses, including myself.  This was pointed out in an article, written by Jonathan M. Marr and Francis P. Wilkin at Union College, which appeared on arXiv today.

The usual explanation for why the Sun is yellow is given by the use of Wien's Law which describes where the peak of a blackbody spectrum is given the temperature of the body.  Using Wien's Law and the surface temperature of the Sun, 5800 K, we end up with a wavelength of 500 nm, which we perceive as green.  However, the Sun isn't green by yellow.  Thus the explanation goes that the sky preferentially scatters blue light, via Rayleigh scattering, which shifts the color towards yellow.  Plus the way the human eye perceives the light and the shape of the blackbody curve naturally favor a yellow color rather than green.

This explanation turns out to be partially correct.  While Wien's Law does give the peak emission for a blackbody curve, it only does so for the intensity, B_{\lambda}, and only when done in wavelength.  If one does the emission in frequency, which is inversely related to wavelength one ends up with the blackbody peak being at 880 nm, which is red.  One can't have two peaks so what is going on here?

This ends up being a pretty subtle, but power piece of radiative transfer (I apologize in advance this gets a little technical).  It turns out what matters is the units one looks at.  If one looks at the proper units for the emission, the peaks line up.  The proper units to look at are not intensity but what is termed the spectral energy distribution (SED).  Whereas intensity, B_{\nu} in this case, is in ergs/sec/cm^2/str/Hz, the SED is measured in \nu B_{\nu} which is ergs/sec/cm^2/str.  With the per frequency dependence gone, the peak now lines up properly with the peak gotten from doing the analogous thing with \lambda B_{\lambda}.  This is because the frequency and wavelength are inversely related.  This shifts the respective intensities in different ways.  The wavelength version shifts the intensity down to smaller wavelength, higher frequency.  The frequency version does the opposite by shifting the frequency down and increasing the wavelength.  The SED formulation removes this dependency, reconciling the two.

Using the SED version of the Planck function, another name for a blackbody distribution, the peak is now in between the two at around 633 nm, which is red.  The Sun isn't red though so what's the deal?  It turns out that astronomers have been abusing Wien's Law.  The Planck function isn't sharply peaked around its peak.  Rather it slowly falls off in the immediate region surrounding the peak wavelength.  Over the visual range, which is actually quite small in terms of wavelength space at 390-750 nm, the SED of light between the red and blue end only varies by about 1% which is less than the eye can discern.  Thus the eye would see the unmodified light of the Sun as white, and not any color.

Then why does the Sun look yellow?  Well the atmosphere preferentially scatters blue light via Rayleigh scattering as noted before.  This gives the sky its blue color.  All the blue light that comes from the sky has to be removed from the light coming directly coming from the Sun.  As a result this leaves only the red and green light, which our eyes perceive as yellow.

Thus from now on whenever I, or any other astronomer for that matter, are asked why the Sun is yellow or why the sky is blue, we can give the same answer from now on.