## Sunday, March 10, 2013

### How Long to Freeze in Space

A common Sci-Fi trope is a ship losing power and the crew slowly freezing to death.  I decided to do some calculations to see how long this would take to happen.  This turns out to be pretty hard to actually do, mainly due to the large number of unknowns.

In the vacuum of space, only one of the three types of heat transfer applies: radiation.  The formula for heat output of a body is:
P = {dE}/{dt} = epsilon sigma A (T^4 - T_a^4)
Where: P is power in watts, E is energy in Joules, t is time in seconds, A is surface area in square meters, and T is temperature of the object, T_a is ambient temperature in Kelvin.  sigma is a constant which is 5.67 times 10^-8 {"w"}/{"m"^2 "K"}.  epsilon is the emissivity of the object, which ranges from 0-1.

The energy contained by an object is given by:
E = mCT
Where E is energy, m is mass, C is specific heat, and T is temperature.  If you're going to do any calculations be careful with specific heat, as it is usually given in J/(g K), not J/(kg K).

Luckily, it's pretty easy to estimate just with a spreadsheet.  We can put each degree from say 70F to 32F and use the first formula to find the heat power output (the change between each degree is less than 1%).  That output is just in joules per second.  We can find the total heat energy of an object at a given temperature in joules.  Then you can find how long it will take to drop one degree.

However, I wanted to to solve it algebraically and get a general formula.  I played around with making a differential equation of these for a while.  Eventually I came up with a solution that ignores the ambient temperature (which at 3K in space is pretty ignorable).

Take the derivative with respect to time of the heat energy:
{dE}/{dt} = mC {dT}/{dt}

Now note that we already have a formula for {dE}/{dt} (and ignore the ambient temperature):
epsilon sigma A T^4 = mC {dT}/{dt}

Now we have one of those separable differential equations all the kids are always talking about:
int {mC}/{epsilon sigma A} {dT}/{T^4} = int dt

Evaluating the definite integral gives the formula:
t(T_f) = {- m C}/{3 cdot A sigma epsilon}(1/T_0^3 - 1/T_f^3)
Where: T_0 is initial temperature, and T_f is final temperature, both in Kelvin.

Four of the above variables need to be estimated.  To give some idea how accurate the estimates are, all the variables are liner.  This means if you double one it'll double the time (or half it).  Most of the variables are hard to estimate to within even an order of magnitude.  The formula also mainly serves as a worst case estimate.

Being near a star would add enough heat that the problem could be over heating (indeed it is for the ISS).  As noted on that page, our current spacecraft have mylar insulating layers which reduce the effective emissivity to 0.03.  This increases the cooling time by a factor of 33 over what one would assume with a simple dull paint.

The formula also assumes the object is of uniform material and temperature.  In reality the specific heat of materials varies quite a bit.  Most metals are < 0.5, compared to water at 4.2.  It's probably fair to assume most the thermal mass comes from the metal in the ship, so I've used 0.5 in my estimates.

The formula also ignores the fact that as the outside of the ship cools heat must be transfered from the interior (via conduction and convection) before it can radiate into space.  This slows the process down more, by an amount that I don't even want to estimate.  If the ship is designed to passively hold heat (which seems like it might be a good idea), it could easily be several orders of magnitude.

Mass and surface area of a fictional ship are pretty hard to estimate, to say the least.  I got my estimates from the infamous Star Trek vs Star Wars site.  If you think the estimates are wrong, feel free to head over there and start an argument about it.

Finally, it's also worth noting here these all assume there is no internal power generated.  The power output of the runabout is about 150 kW.  Anything generating power inside would subtract from that rate and increase the time to freeze.  A human body only gives off about 100 watts each, and lights would probably be around 1 kW max.  So, it would seem that in a situation where a small ship loses all power, freezing to death would be a serious concern.  Of course humans can survive temperatures well below freezing.  It would seem like a good idea to equip small ships with survival suits with an internal power source.

The time it takes to go from 70F to 0F varies from hours to months.  The general trend is larger the object the longer it takes.  A human body would only take a few hours.  A small ship (runabout, max crew: 15) could be anywhere from 12 hours to a couple weeks.  A large ship (galaxy class, crew: about 1200) would take months to years.

I made a table with some estimates, keep in mind these are probably underestimates.  I used 0F as the final temp because I think it's reasonable for a clothed human to survive down to that.

 Name Length (m) Crew Surface Area ("m"^2) Metric Tons Time 70F to 0F Death Star II 160,000 2,500,000 80,400,000,000 1,500,000,000,000,000 1199 years Super Star Destroyer 17,600 300,000 201,911,000 8,322,000,000 2.65 years Borg Cube 3,036 130,000 55,325,235 60,000,000,000 70 years Enterprise-D 642 1,200 524,742 10,200,000 1.25 years Enterprise 310 430 145,901 1,000,000 161 days Runabout 23 8 2,082 1,000 11.3 days Type 6 Shuttlecraft 6 2 92 50 12.8 days TIE Fighter 6 1 190 6 17.8 hours Naked Human 2 1 1.80 0.090 0.96 hours (98.6F to 84F)