You’re the only star in heaven You’re the only star that shines You’re the only star in heaven Now that only star is mine — Frankie Goes to Hollywood

Snuggler’s lament

These days of northern winter seem endless. It’s been brutally cold here in Boulder, causing much snuggling at night, both between humans and with Canises Major and Minor. Snuggling is fun, of course, but also useful: body heat shared is body heat doubled.

After a while it can get too hot, and even Mrs. BA with her ice cold feet will move away to cool off a bit. When that happens, of course, my mind turns to matters scientific. Our bodies generate a lot of heat. And with the Sun making only a desultory appearance every day, I was thinking recently about the energy generated by the Sun, versus that emitted by humans. I remember reading once that if you compare the heat coming a single square centimeter from the Sun to the same area on a human being, you’d find we actually put out more energy! As a skeptic I’m used to analyzing such claims; as a scientist I have the mad math skillz to work out the numbers; and as a communicator, I have the soapbox upon which I can talk about the whole thing.

So let’s get to it. Are humans more energetic than the Sun?

energetic

Yar, thar be math below, and plenty of it. Be ye fairly warned, says I.

Glowing places

As it happens, the math isn’t that hard. Objects that are warm (really, anything warmer than absolute zero) have a characteristic way they emit energy, called black body radiation. Both humans and the Sun are pretty close to being such radiators, and it’s not too bad to just assume they’re blackbodies. There’s a simple equation to calculate the amount of energy emitted per second, called the luminosity:

Luminosity = area of object x σ x temperature4

where σ, the Stefan-Boltzmann constant, is just a number (if you wanna check my math, it’s 5.67 x 10-5 erg cm-2 s-1 K-4, and K is Kelvins, the unit of temperature). That equation makes sense: at a given temperature, a bigger object will emit more energy. And if two objects are the same size, the hotter one will give off more energy (in other words, and to be a bit more vernacular, it’ll be brighter).

The Sun is big, so even if it were colder than a human it would win that fight! So we want to compare apples to apples here, taking one square centimeter of each and seeing who wins. That means we want the luminosity divided by the area, giving us the energy emitted per square centimeter in one second. Rearranging, we get

Luminosity / area = σ x temperature4

Hey, wait a sec! This makes something clear right away: if you want to compare the energy emitted per square centimeter from any two objects, all that matters is their temperatures. The hotter one wins.

That means the story I read — that humans emit more per square cm than the Sun — is wrong. The Sun is a lot hotter than a human, so it emits vastly more energy than a person does! In fact, it’s the ratio of the temperatures raised to the 4th power. The Sun’s temperature is 5780 Kelvins, and a human is 310 Kelvins. Plugging and chugging shows that the Sun gives off a whopping 121,000 times as much energy per square centimeter as a person does!

Yegads. And the Sun is a whole lot bigger. If you’re not careful, you may get the impression the Sun gives off quite a bit of energy.

Pump up the volume

cubic sun

A cubic Sun.

Anyway, far be it from me to simply say the story is wrong and drop it. There’s more science here! Instead of using the area, what about the volume? In other words, assume both a human and the Sun have the same temperature throughout (I mean, every chunk of a human is 310K, and the Sun is 5780 K). Would a cubic centimeter of the Sun outshine a cubic centimeter of human?

This is a little bit tougher to calculate. We need the total luminosity of the Sun and its volume, and the same for a human. For the Sun that bit’s actually easy, since we just use that luminosity equation above (knowing the Sun’s area is 6.1 x 1022 square centimeters, which I leave up to you to calculate if you want). The Sun’s energy emitted per second is then about 4 x 1033 ergs/second, where an erg is just a unit of energy astronomers like to use. It’s a small unit, but 4,000,000,000,000,000,000,000,000,000,000,000 of them is still a lot.

What about a human? Well, we need the area of a human to plug into the equation to get the luminosity, and we’ll have to estimate that. Let’s use me as an example, and assume I’m a big rectangular solid, like a shoebox (or a monolith). I’m 177 cm tall, about 50 cm across, and about 15 cm deep. That gives me an area of very roughly 25,000 square centimeters. That’s an estimate, but good enough — it won’t matter if I’m off by a factor of two either way.

Plugging away, I get that my luminosity is then 1.3 x 1010 ergs/sec. That’s a lot smaller than the Sun. But then, I’m not all that hot*.

I am your density

OK, almost there! All we need to do now is divide those numbers by the respective volumes and compare them. The Sun’s volume is 1.4 x 1033 cubic centimeters. That means that each cubic centimeter gives off 4 x 1033ergs/second / 1.4 x 1033 cc = 2.8 ergs/second/cc. So every second, each cc of the Sun emits 2.8 ergs. OK then. What about me?

water density

My volume is easy to estimate: I know humans are the same density as water, which is 1 gram/cc. I also know my mass is about 75,000 grams, so my volume must be 75,000 cc! Easy peasy.

Finally, dividing my luminosity by my volume yields 170,000 ergs/sec/cc.

Hey, wait a sec! That means not only am I brighter than the Sun, I’m a lot brighter! About 60,000 times brighter!

Make a gas of U and ME

So in that sense, the legend is right. If you want to think of it this way, a cubic centimeter of human gives off a lot more energy than the same volume of the Sun does!

But hold on there. Is this really a fair statement?

Well, not really. First, there are a whole lot more cubic centimeters in the Sun (about 1028 times as many, or ten billion billion billion times as many), so when you divide by such a big number the energy per cc for the Sun drops drastically. So even if we say, sure, humans are more luminous per cubic centimeter, it’s best not to get cocky. The Sun can still vaporize us with lots of cubic centimeters left to spare.

Second, remember the assumption I made, that the Sun has the same temperature everywhere? That’s not even close to being true. In fact, it’s whoppingly untrue. The core of the Sun is 15 million Kelvins hot, so each cc there is blasting out vast amounts of energy: about 5 quadrillion times what a cc of human flesh does. But outside of that region the Sun is much cooler, and each cc doesn’t contribute nearly as much. Over the entire Sun, that dilutes the amount of energy per cc quite a bit. Averaging over the volume of the entire Sun is not a great way to think about it, and makes comparisons difficult, if not really meaningless.

Like this exercise has any profound, deeper truths to it in the first place. Actually, it’s just an excuse to have some fun and do some mental gymnastics. And, in the end, it really comes down to this: the Sun is bright, and we are not.

But, you knew that. Of course, some humans are hotter than others. But I’m not sure I can do the math for that.

Tip o’ the pound of flesh to BABloggee Brad Stacey.