The Human Eye, Optimized For Sunlight. Maybe. [Built on Facts]

The human eye is sensitive to a portion of the electromagnetic spectrum that we call visible light, which extends from around 400 to 700 nanometer wavelength, peaking in the general vicinity of greenish light at 560 nanometers:

Here’s the intensity (formally: power per area per unit solid angle per unit wavelength – whew!) of the radiation emitted by an object with the temperature of the sun, plotted as a function of wavelength in nanometers according to Planck’s law:

Spectral radiance (W/(sr m^3)) vs. wavelength (nm)

You’ll notice it also peaks around the same place as the spectral response of the human eye. Optimization!

Or is it? That previous equation was how much light the sun dumps out per nanometer of bandwidth at a given wavelength. But nothing stops us from plotting Planck’s law in terms of the frequency of the light:

Spectral radiance (W/(sr m^2 Hz)) vs. frequency (Hz)

In this case what’s on the y axis is power per area per unit solid angle per frequency. Ok, great. But notice it’s not just the previous graph with f given by c/λ. It’s a different graph, with different units. To see the difference, let’s see this radiance per frequency graph with the x-axis labeled in terms of wavelength:

Spectral radiance (W/(sr m^2 Hz)) vs. wavelength (nm)

Well. This is manifestly not the same graph as the radiance per nanometer. Its peak is lower, in the near infrared and outside the sensitivity curve of the human eye. This makes some sense – there’s not much frequency difference between light with wavelength of 1 kilometer and light with wavelength of 1 kilometer + 1 nanometer. But light of 100 nanometer wavelength has a frequency about 3 x 10¹³ Hz more than light with wavelength 101 nanometers.

So what gives? Is the eye most sensitive where the sun emits the most light or not? The simple fact of the matter is there’s no such thing as an equation that just gives “how much light the sun puts out at a given wavelength”. That’s simply not a well-defined quantity. What is well defined is how much light the sun puts out per nanometer or per hertz. In this sense our eye isn’t optimized so that its response peak matches the sun’s emission peak, because “the sun’s peak” isn’t really a coherent concept. The sensitivity of our eyes is probably more strongly determined by the available chemistry – long-wavelength infrared light doesn’t have the energy to excite most molecular energy levels, and short-wavelength ultraviolet light is energetic enough to risk destroying the photosensitive molecules completely.

This wavelength/frequency distribution function issue isn’t just a trivial point – it’s one of those things that actually gets physicists in trouble when they forget that one isn’t the same thing as the other. For a detailed discussion, I can’t think of a better one than this AJP article by Soffer and Lynch. Enjoy, and be careful out there with your units!