Freezing in Space I - Blackest Night

In the last post I made, I discussed the fact that humans radiate energy.  In that post I calculated that we actually radiate quite a lot of power.  This immediately raises a few questions, the most obvious one being: How long would it take you to freeze in space?  This question is multifaceted, and I'm going to split it between two parts.  This first part, 'Blackest Night' is how quickly we'd freeze if we were completely lost in space, nothing anywhere near.  The second part, 'Turn On The Sun!' will address what would happen in near earth orbit.



We need to clarify what we mean when we say freezing in space.  The fatal temperature change for a human is (according to the all knowing internet), roughly a drop of 7C.  If you remember my previous post, we calculated that we would radiate about 860 W.  Now, we have to ask how much energy it takes to change our temperature by 7 C.  Well, as I've discussed a few times, the energy it takes to change the temperature is given by
\[Q=mc\Delta T\]
The mass of a human is ~75 kg.  We're mostly water, so let's just assume that we are all water.  This gives us a specific heat of 4.2 kJ/kg*K.  The energy needed for our 7 C temperature change is then
\[Q=(75kg)(4.2kJ/kg*K)(7K)=2.2 MJ\]

By the time we have dropped to 30 C, we are only radiating a power of
\[P=eA\sigma T^4\=(.97)(1.7m^2)(5.67\cdot10^{-8}W/m^2K^4)(303K)^4=790W\]
Let us assume that the average power radiated is the geometric average of these two powers,
\[P_{avg}=\sqrt{(860W)(790W)}\approx825W\]
This gives us a time to freeze of
\[t=\frac{2.2MJ}{825W}\approx2700s\]
This is ~45 minutes.  So you've got 45 minutes until a deadly freeze in deep space.  Seems a rather long time, does it not?  I'm fairly certain that you'd freeze much faster in antarctica than deep space.  Why?  Because in Antarctica you have more cooling mechanisms that just radiation, you have conduction in the air around you and convection of that warmer air away from your skin.  If you're adrift in space, for all that it is rather cold, the good news is that you'll asphyxiate before you freeze!

All of this was done assuming that there's no energy gain from anywhere.  That is, that we're stuck somewhere in the deepest space, the blackest night.  Tomorrow we'll consider what would happen if you were in near earth orbit, with all of these lovely energy sources around, particularly the sun.