Science Clock Series: Part X

It's been a while since my last post, and I figured you all deserve an explanation. I was in Nebraska all last week attending my grandfather's funeral, immediately after serving as a groomsman for a wedding in Kona. It's been a hectic fortnight. Between helping out around the family farm and catching up with relatives I haven't seen in years, this blog got put on the back burner. I'm now back in Hawaii, however, and getting back into the swing of things.

Anyway, today's number comes from both physics and biology, and is given by:

\[\lambda\ \text{of human body heat (in \(\mu\)m)}\] The Greek letter lambda \((\lambda)\) in this equation stands for “wavelength,” specifically the wavelength of the electromagnetic radiation given off by an object at average human body temperature. The symbol \(\mu\) is the Greek letter mu, and is an SI prefix denoting micro-, or one-millionth of something. So a micrometer is a millionth of a meter.

To understand this subject fully requires a short history lesson, and an understanding of the term “black body.” A black body is an idealized object that absorbs all electromagnetic energy that hits it, no matter what angle it hits at or how powerful it is. Being an ideal object, true black bodies don't exist in nature, but many things come pretty close. It's important to realize that a black body may not (and in reality will not) be actually black. A black body at any temperature above absolute zero (that is, all of them) will emit electromagnetic radiation in a characteristic energy distribution called “black-body radiation” that depends only on the temperature of the object: not what it's made of, not its shape, just its temperature. If the temperature is hot enough, a black body will emit light in the visible portion of the spectrum and visibly glow.

One example of a class of objects that approximate black bodies is stars. This may seem strange to you, because stars are anything but black – but it goes back to what I said earlier. When you think about it, stars really don't reflect any of the electromagnetic radiation that hits them. They absorb it, then re-emit it in a characteristic black-body spectrum. (It's not perfect because the presence of various elements in stars modifies it somewhat, but it's close enough to be able to measure the temperature of stars by measuring the amount of light they give off over all parts of the spectrum.)

Now, although there are no true black bodies in the universe, most things actually come pretty close. Some brilliant physicists in the late 19th and early 20th centuries formulated a couple of equations that describe the amount of electromagnetic radiation given off by a black body, per either unit wavelength or frequency. Once you know that, you can figure out where exactly on the spectrum an object is radiating most strongly at. A German physicist by the name of Wilhelm Wien deduced what is now known as Wien's Displacement Law in 1893. In symbols, it looks like: \[\lambda_{\text{max}}T=b\]In words, it says that as a black body (which is a good approximation for most objects remember) gets hotter, the wavelength at which it radiates most strongly becomes shorter and shorter; or alternatively, it produces more energetic light.  The Greek letter \(\lambda\) (lambda) stands for wavelength, the T stands for the absolute temperature in kelvins, and b is a quantity known as Wien's Displacement Constant, approximately equal to \(2.8977685\times10^{-3}\,\text{m}\cdot\text{K}\).

Now that we have an equation to use, it's a simple matter of algebra and plugging in numbers. Rearranging it to solve for the wavelength we get \(\lambda_{\text{max}}=b/T\). Despite what we all learned growing up, there isn't actually such a thing as a “normal” human body temperature; it varies by individual, time of day, state of the body itself, and many other factors. However, a value of 98.6 ºFarenheit works well enough as a good “average” temperature, so let's go ahead and use that. Converting to the Kelvin scale, we have:
\begin{align} [\text{K}]&=([^\circ\text{F}]+459.67)\times\frac{5}{9}\\
&=(98.6+459.67)\times\frac{5}{9}\\
&=310\ \text{kelvins}\end{align}Plugging that number in for the temperature in Wien's Displacement Law gets us:
\begin{align}
\lambda_{\text{max}}&=\frac{b}{T}\\
&=\frac{ 2.8977685\times10^{-3}\,\text{m}\cdot\text{K}}{310\,\text{K}}\\
&=9.34\times10^{-6}\,\text{meters}
\end{align}This is great and all, but our answer's in meters, not micrometers (\(\mu\)m) as it needs to be. One micrometer is \(1\times10^{-6}\) meters, however, so we need merely convert to get \(9.34\,\mu\)m. Which is fairly close to ten (within 10%), but actually closer still to nine. Oh, well. It was approximate.

This 9.34 \(\mu\)m, by the way, is a wavelength that corresponds to the far infrared. We humans constantly radiate at a range of wavelengths – near infrared on the shorter side, for example, sub-millimeter on the longer – but that wavelength (or near to is) tends to be where we radiate most strongly. And there's really not much we can do about it, since as endothermic creatures our bodies try to keep themselves in a tight specific temperature range. Though I can't think of any reason you'd want to change your peak radiating wavelength. We can't see in infrared, so it doesn't really matter. (If you're curious, it requires being a few hundred degrees hotter before the peak wavelength shifts into the visible portion of the spectrum and things start visibly glowing. If you have an electric stove, you can see this effect in action as it heats up: the point where it starts to glow is when it becomes hot enough to start radiating significantly in the visible region.)

Anyway, that was quite the lengthy post. Tune in next time for a much shorter one where we look at a number from astronomy again! Click here to jump directly to the next part.