Human Radiation

Things are still busy here at the Virutosi.  Hopefully in a week or so we'll be back to normal, and much more active than we've been recently  Anyways, today I'd like to consider human radiation.  It is well known that any object will radiate energy based on its temperature.  Even more interesting, we radiate at all wavelengths, though at the human body temperature our radiation is sharply peaked in the infrared.  Even so, we still put out some x-ray radiation.  As a professor of mine once said, consider that next time you sleep with someone!  Given all this, the question on my mind today is:  how does the energy we radiate daily compare to the energy we consume?  That is, why don't I lose weight sitting here typing on the computer?

We physicists call perfect radiators black bodies (something that radiates perfectly also must absorb perfectly).  For perfect radiators, the power radiated is given by
\[P=\sigma A T^4\]
where sigma is the stephan-boltzman constant, A is the surface area of the radiator, and T is the temperature of the radiator.  For objects that are not perfect emitters or perfect absorbers, we through in a fudge factor, e the emissivity, which is between 0 and 1.  This makes the power emitted
\[P=e \sigma A T^4\]

To figure out the power radiated by a human, we need to know three things.  The first is the emissivity of human skin.  It turns out this is .97.  The second is the temperature of a human body, ~37 C.  The third is the surface area of a human.  This requires a little estimation.  I'm about 180 cm tall, and I wear 35" waist pants, so my radius is ~14 cm.  Modeling myself as a cylinder, I have a surface area of ~1.7 m^2.  Now we can estimate my power output:
\[P=.97*5.67\cdot 10^{-8}W/m^2K^4 * 1.7 m^2 * (310K)^4\]
\[P \approx 860W\]
I'm powerful!  That's about 14 (60W) lightbulbs!

We'd like to compare that to our daily energy intake, so we need to turn this power into an energy.  Well, there are 86400 s in a day.  So we radiate 74*10^6 J per day.  If you read my beer diet post, you'll know the conversion between J and Calories (note the capitol C).  If not, suffice it to say that 1 Calorie = 4.2*10^3 J.  If I consume about 2000 Calories a day (typical, right?), then I take in about 8.4*10^6 J per day.  So, dear reader, why haven't I lost weight while typing this post?

There are a few answers.  I'm wearing insulating layers, clothing which keeps in some of my radiated heat.  Also, our skin temperature is lower than our internal temperature of 37 C.  But more than that, I'm absorbing energy from the surroundings.  The earth is a fairly good blackbody radiator, with an average temperature of ~20 C.  This means that my net power loss, with no clothing, would be about
\[\Delta P \approx (5.67\cdot 10^{-8} W/m^2K^4)(1.7m^2)((310K)^4-(293K)^4)\]
\[\Delta P = 180W\]
This is ~15*10^6 J per day.  While still more than my intake, this is much closer, and you can imagine that the rest of the difference can be made up by our lower skin temperature, and clothing and such instruments of men.  Of course, these numbers are rough, so I don't recommend the 'naked diet', where you try to lose weight by walking around naked.  Or if you do try it, don't say I told you to when you're taken in for indecent exposure!