Now I remember reading somwhere that for spaceship, its not keeping warm in the cold of space that's a problem. Because the lack of atmosphere, means they can't lose heat through convection/conduction. Just radiation. So once you got some heat it was hard to get rid of it with out intensionally using radiators.

But someone recently pointed out to me that during the Apollo 13's accident on the mission, one of the the dangers they faced was the cool temperature brought on by not having power.


So was I just totally wrong? If a theoretical starship does lose all power, how quickly would it freeze to death? IS it just a size thing?

I'm just guessing here...

No power = no heat to dissipate?

Spaceships only generate heat when they're working, yeah?

Yeah but you should have the heat you generated before hand?

This is a Stefan–Boltzmann law question. You solve it with,

P = A*s*e*T^4

Where A is the area in square meters of the object dissipating heat, T is the temperature you want to keep the inside at, P is the amount of power being dissipated, and s is the stephan-boltzman constant, (roughly) 5.67×10^-8 W*m^-2*K^-4, and e is the thermal emissivity of the 'skin' of the object radiating the stuff.

In this case, we want to keep the inside of our tin can at 300 kelvin or so, which gives us T. The three men on the Apollo 13 mission were likely producing about 500W of power between the three of them. Estimating roughly from a photograph gives me about 30 square meters of surface area for the command module. We can infer by the observation that the command module was cooling down: 500W < P. Therefore, the emissivity of the Apollo 13 command module must be greater than, roughly, e>0.04.

Well yeah, but it's only going to get colder once the power's out.

Apollo was designed to radiate heat very efficiently because is it always easier to make something warmer than making something colder. They used heaters normally to keep the craft at a comfortable temperature, but under the power restricted conditions of Apollo 13 the heaters were turned off, hence the flying refridgerator.

Thanks for the detailed response, Feil. That's great. Sort of proves how very rusty my physics is, since I used Stefan-Boltzmann stuff in relation to stars in my degree (only just scraped it though)

So I take those terms are only equal when there is no temperature change. And its basically an surface area vs amount of energy being radiated? A ship with a lot of higher volume compared to its surface area is going to cool slower than something with a higher surface area/volume ratio?

ghetto edit to add: Thanks to you others too.

originally this was in response claiming that life support on a star trek ship was important to stop them freezing to death. I think it'd take quite a while for a ship that size to cool down/ and of couse if other system were still active it wouldn't at all.

The terms are equal all the time - it's what changes force what other changes that tell us interesting things about what's going on. If the temperature drops, and area is held constant, then we know the power being radiated will drop, too. If we increase the power that our object has to dissipate, then either the temperature has to increase or the surface area has to increase.

The way we could handle that without a ship that changes size or cooks its passengers is, first, by pumping heat to the outside while keeping the inside cool by means of an air conditioning system. This makes our temperature at the surface (which is the only one we care about) higher, while keeping the inside at the same temperature. We can do the same thing on the surface itself by pumping the heat to a radiator with a very high emissivity and large area, away from the surface of our living component, which has a low emissivity and a small area. If we do that, then our equation changes slightly to account for the fact that we no longer have an even distribution: e becomes the function e(r(s,t)) and T becomes the function T(r(s,t)), so our equation changes to

⌠
⌡Ss*e(r(s,t))T(r(s,t))^4 dS = P

for the purposes of our equation. (The average over the surface given by the function r(s,t) of eT^4, times the area, times the stephan-boltzmann constant, equals the power radiated.) Solve piecewise if the shape of the tin can can't be described with a single-valued function, or if piecewise solution makes the integral easy.

Note that P is the net power that the object has to emit to avoid changing temperature. That is to say, it is the sum of all power produced by and absorbed by the object. This is why it is easy to keep warm when it is warm out, but hard when it is cold out: You are receiving more power to your skin on a warm day than on a cold day.



Correct. Heat capacity C is directly proportional to volume, and if we take the time derivative of the definition of heat capacity, CdT/dt = dQ/dt = P. Power is directly proportional to area, as given by the Stephan-Boltzmann eq, so shapes with higher volume to surface area ratios lose temperature more slowly in the same environment. (This is why frigid climates lack insects and most small animals, for instance: they have small volumes for their surface area, so it is comparatively costly for them to maintain a temperature different from their environment.)

...actually, the in-atmosphere examples I gave would be mostly determined by conduction and convection, which are related but very different. Supposing bugs and people could live in hard vacuum, however, bugs would still die faster in the cold.

That's what always bugged me about Star Trek. Every time you lose life support, the ship instantly gets cold and the air runs out within seconds. In truth, you'd have enough heat and air in a GCS to stay comfortable for a very long time. In the Apollo capsule, they managed to survive for days with no heat, and the air lasted for quite awhile before they had to mickey mouse an air scrubber.

It's possible that in order to dissipate the massive flow of energy normally constantly generated by the warp core, ST ships require that there be some form of highly-efficient passive subspace radiators (insert some sort of magitech here) which don't require power to operate, and so keep working even when they lose main power. That would explain (sorta) what we observe, and also help explain why their ships can be such powder kegs.

Yeah with them using high-energy plasma as widespread as we use electrical wiring, that's a lot of heat that needs to be gotten rid of. Doesn't explain air running out within seconds though...

When did they run out of air and heat when power failed? I can't think of any examples from the Star Trek I've watched.

The episode of Voyager where the crew has to enter suspended animation to pass through a nebula, leaving only Seven of Nine awake to watch the ship. The ship begins to lose power near the edge of the nebula and the only way Seven can keep the engines going is by cutting power to life support. She's in a huge cargo bay and the instant she orders the computer to cut life support, she begins to suffocate and passes out.

It happens in DS9's "Treachery, Faith, and the Great River," when Odo hides inside an icy body and shuts down main power and life support, but in that case, the runabout was in thermal contact with the ice. The heat loss was deliberate, so Dominion ships couldn't find them using infra-red sensors.

It depends on the ship size, configuration and distance from the sun. A small ship or unmanned probe can easily get too cold, and since you have no external air or water to even out the heating, its entirely possible that one side of a ship will get too cold and the side in the sun or near the engines would get too hot. This is a constant problem with earth orbit satellites and a reason why they are wrapped in reflective/insulating aluminium foil. A spacecraft traveling between planets could spin itself to help even out the solar heating, that's not an option for a satellite that has it to keep antennas pointed at the earth surface. If you start getting out to more like Mars, solar heating stops mattering so much, and by the time you get to the outer solar system freezing is a constant problem and all real spacecraft have needed nuclear power sources for heating as well as power. Go close to the sun and eventually your craft would just melt no matter how many radiators you put on it.

As you rise in ship size and power it becomes more and more likely you'll just be too hot all the time. Most sci fi ships have more then enough power that excessive heat would be the only the problem, except perhaps for very isolated compartments or external bits of equipment. Unless you are talking about very hard sci fi though, the weight and power budget for heating exposed bits and pieces would be unimportant.

Also, while anything electrical powered will make heat that needs to be controlled, you can get cooling problems from liquified gas storage like LOX tanks or certain kinds of fuel being drained, turning the stuff back into gas consumes heat. As I recall Apollo had had dedicated heaters for the LOX tanks to counteract this problem. On a bigger spacecraft you could start talking about kilowatts, even megawatts of power being consumed for this purpose alone.

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"This cult of special forces is as sensible as to form a Royal Corps of Tree Climbers and say that no soldier who does not wear its green hat with a bunch of oak leaves stuck in it should be expected to climb a tree"
— Field Marshal William Slim 1956

There were a couple thermal issues faced by Apollo 13. The Apollo spacecraft actually was in a roll on during its way to and from the Moon, called the "Barbecue roll" to ensure as even as possible solar heating/cooling of the vehicle and its systems. For Skylab the Apollo Command Module had coats of white paint applied to one side of the vehicle for protection, because while docked at a station for a couple months, such a roll would be impossible.

After the explosion and control was transferred to the LEM, getting the vehicle back into that "roll" was considered enough of a priority to delay the giving the crew a break (they had been on the cusp of their sleep period when the explosion occurred, then had to contend with identifying the problem, shutting down the Command Module, power up the LEM and get on a free return trajectory, so it had been a busy day, and shows that it ranked as an important task).

Nobody was sure what effect the freezing temperatures would have on the unheated Command Module. The condensation that accumulated on the panels created concerns about potential short circuits. The parachutes had spent a longer period of time than ever intended unheated, so had the batteries that would power the Command Module thru re-entry.

So, in the case of Apollo, the issue was what do you use that heat for? The crew could tough it out, but there are certain systems you want/need to keep heated.

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-A.L.
"Nothing in this world can take the place of persistence...Persistence and determination alone are omnipotent. The slogan 'press on' has solved and always will solve the problems of the human race." - Calvin Coolidge

"If you're falling off a cliff you may as well try to fly, you've got nothing to lose." - John Sheridan (Babylon 5)

"Sometimes you got to roll the hard six." - William Adama (Battlestar Galactica)

I don't think that's a good example to use, as by that point Seven of Nine was in an ongoing nervous breakdown from the isolation and probably hyperventilating rather than suffering from any actual lack of oxygen.

That's far from the only example though. There's Insurrection where the So'na vessel surrenders 'because they have only five minutes of oxygen left' (or something to that effect) after the Big E takes out their life support, without there having been humongous holes blown into the hull that would vent all the air already present out into space, and there's IIRC several situations in TNG when they're suffering from cold and/or lack of oxygen due to life support failure long before that should realistically happen on a vessel that size (not that Trek is alone in using that particular brain bug, mind you).

Maybe some emissions of warp drive systems actively consumes or otherwise breaks down air, and it has to be actively reconstituted? Dumb... but yeah running out of air quickly is an issue in a number of episodes. One other theory is when power fails all the force field windows blow out and the ship explosively decompresses, but that makes just as little sense since we don't hear warnings about warp core or anti matter pod containment failing and blowing the ship up. Also they have plenty of internal doors and windless rooms and some level of distributed battery power.

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"This cult of special forces is as sensible as to form a Royal Corps of Tree Climbers and say that no soldier who does not wear its green hat with a bunch of oak leaves stuck in it should be expected to climb a tree"
— Field Marshal William Slim 1956