Space is very cold-on average, about 3 K. So, how come our astronauts do not freeze to death?

On Earth, heat transport is mostly done by conduction of heat through a substance, and convection. Both of these transport mechanisms however depend on the presence of mass to do their heat transport, something in which space is sorely lacking-typical densities are perhaps a few dozen atoms per cubic meter. Compare this to 1025 per cubic meter for air, and you quickly see that even though space is on average 3 K, you still won't have a lot of cooling from this. To get a nice demonstration of this phenomenon, try the difference between licking 0-degree air, and a 0-degree metal bar*.

No, astronauts lose their heat mainly through blackbody radiation. It so happens that this process can be simply described by the Stefan-Boltzmann law:

Φ=(1-ε) σ T4

Here, Φ is the radiative flux, ε is the albedo, or whiteness, and σ is the Stefan-Boltzmann constant, numerical value 5.67x10-8 J K-4 m-2 m-1. By integrating the flux over the area of his space suit we can now compute the amount of heat he loses by radiation.

Now, we'll use some reasonable numbers to get a rough idea how much heat our astronaut loses. His space suit has an area of about 2 square meters. It's white, so the albedo is high, say 0.9. The temperature of our astronaut is 310 K, or body temperature. This gives a power output of 100 W, which is close to the power your average human puts out.

The T4 has another neat implication: if the power varies by a factor of 2, the temperature will only vary by a factor of 1.19. So, you can change the power by quite a bit, while only varying the temperature by a modest amount. Say our astronaut gets very active, and his power becomes 50 percent higher. The temperature will eventually become 343 K, or 70 C. However, with the isolating properties and heat capacity of the spacesuit he's likely to be fine.

There is however one large assumption made in this whole story: that space doesn't radiate back. This is all fine if you consider only the background radiation of 3 K, which will contribute a measly 0.9 microwatt in our example. However, if the astronaut gets into the sun, without protection, it becomes a different story. I won't do the maths, but suffice it so say the sun heats the earth up to an average temperature close the body temperature of our aforementioned astronaut, so this amount of heat is definitely not negligible. I imagine it will get sweaty in there.

As a final remark, you might have wondered why the power dissipation of suit and astronaut match so closely. I mean, why would the Stefan-Boltzmann constant have a value that allows a human to survive in space? Before you start blaming your favorite deity, let me point out that there is one parameter that we have control about: the albedo. By setting it high, we put the dissipation at a comfortable value. If we were to paint the suit black, the albedo would be higher, so there would be more blackbody radiation. This would mean it would become quite chilly for our astronaut. In short, spacesuits are white because space is cold. As a bonus, the white color reflects the sunlight, so the astronaut is less likely to get baked.

Sources:

• http://scienceworld.wolfram.com/physics/Stefan-BoltzmannLaw.html
• http://scienceworld.wolfram.com/physics/Stefan-BoltzmannConstant.html
• m_turner made a good point about the sun being reflected of the spacesuit, and about me mixing up Nernst coeffient and albedo.

*Disclaimer: This will likely cost you your tongue. So, actually, you'd better not do this and just believe me.