God should have installed a fluorescent sun. Those things last FOREVER.
I would imagine a nuclear sub could last until the food runs out. Then the sailors turn to cannibalism. Interesting problem, because you have to feed some crewmembers to other future stewpot-bound crewmembers to keep them alive long enough for you to eat them later. Lots of inefficiency built into the system, kind of like a multistage rocket. I suppose the sub will have some empty freezer space once the food starts to run out, best to kill and freeze as many crewmembers as possible as soon as possible.
If you could figure out a way to harvest tubeworms from the deep ocean vents you might last until you run out of reactor fuel. But I don’t think this is possible, most likely you’ll starve to death. Although…as the oceans freeze over you’ll be forced deeper and deeper to stay out of the ice. It could be the submarine will be forced deep enough for the hull to give out before the food gives out. We’d have to have some wild guesses for how fast/how thick the ice forms, and I suppose the exact depth when a given nuclear sub implodes is classified information.
I had the idea that the sub should surface immediately to stay on top of the ice, but then I recalled that the reactors on the subs take in liquid water for cooling, which means you won’t be able to keep the reactor going. Plus, underwater you’ve got the warm water buffering your temperature, on the surface you’ve just got the thin metal skin exposed to two-digit Kelvin temperatures.
You’d also need to know how well-provisioned a nuclear sub is, which is probably also classified (though likely leaked).
The time for core photons to make their “random walk” to the surface is somewhere between 10,000 and 170,000 years.
Two points: first of all, a star radiating in only the IR bands would be an obviously artificial construct from its spectral lines. Even brown dwarfs radiate as (weak) radio sources (which your presumably metal shell would screen). Second, a “Type II” Dyson sphere (which is not Freeman Dyson’s original concept of a vast number of habitats orbiting a central star in an enclosing orbit, but a single static spherical shell with the star at the center), which I believe is what you are referring to, is dynamically unstable. Because there is no net force due to gravitation inside of a hollow sphere, any relative difference in motion between the star and the sphere (say, asymetrical meteor impacts) will cause one to impact the other. Without an active control system, instability and contact is inevitable. So, Dyson spheres are unlikely.
Also, constructing a Dyson sphere in a system like our solar system would certainly require more materials than are available, even if you could reduce the gas giants to construction materials, so you’d have to drag materials from extrasolar sources, which seems like a pretty pointless waste of energy. More than likely by the time you have anything like that capability, you’ll be less interested in controlling an individual star and more flitting about on your Improbability Drive and discussing what cut of steak to have with the Dish of the Day.
Stranger
Well, I used the term “shell”, because nowadays it seems the term “Dyson Sphere” means a “solid” (of course it couldn’t really be solid) sphere with a biosphere inside. Of course, you’d have to have gravity generators and such, and on and on. A not-neccesarily continuous collection of objects surrounding a star such that only a small fraction of the visible light escapes into interstellar space is what I was thinking of. The ASBs could just set up an incredibly extensive chain of solar power satellites orbiting the Sun along the ecliptic around the orbit of Venus, using the mass of Venus and Mercury. You wouldn’t need much to block 99% of the solar radiation to Earth.
The Sun is about 1.4M km in diameter, and the Earth is 12,800 km. A disk large enough to mask the Sun, orbiting at .85AU (between the orbit of Venus and Earth) would have to be 220x103 km in diameter, or a little more than 17 times the diameter of Earth. This would have a surface area of about 38x109 km. Given concrete with a thickness of ~1m, this would mass 9.2x1019 kg; 1 Ceres or one of Jupiter’s Galilean moons should cover it. The closer you move it to the Sun, the worse it gets. Plus you’ve got a problem: its free orbit is going to be of significantly shorter period than Earth; therefore, you’re either going to have to have it under constant acceleration to keep it in place, or you’re going to have to build a ring structure and spin it at the desired rate of rotation (which, in any case, will have to vary during different stages of the Earth’s orbit). You’re also going to have to figure out how to keep it on station, as we all know that a ring structure spun around a central mass is unstable. And the tension in the structure is going to be beyond the tensile strength of any known material by many orders of magnitude.
If you place your sunshield at the Sol-Earth L1 libration point, the minimum size of a global parasol would be about 27,000km in diameter. This would be the ideal place to put it, rather than in between planetary orbits; it should stay in position with minimal orientation maneuvers and not requiring any thrust or special mechanism to stay on station, plus you could give it a keel pointing toward the Earth that would keep it properly oriented (more or less) by tidal forces. This would only mass about 65x106 tonnes (assuming a flat, 1 meter thick disk, the minimal shape), which could be supplied by a mid-sized Near Earth Object (asteroid). That puts it in the realm of just slightly absurd conjecture rather than highly improbable fantasy.
Either way, it isn’t going to block radiation going out to the universe at large. A true Dyson sphere (be it a Type I collection of individual habitats, or a Type II solid sphere) is going to require massive amounts of material beyond what is found in a system like ours.
Stranger
But why do we insist that the system of shrouds be 1 meter thick? Why not submillimeter thick? You make the screens out of the ASB equivalent of superconducting mylar film. Then you get several layers of solar power satelites freely orbiting around the sun, no rings needed.
I’m not imagining “habitats”, I’m imaging ultrawispy solar collectors. Or mirrors bouncing most of the photons back into the sun. Although they’re gonna radiate back out again sometime. This is why I was imagining that the shell of solar collectors is going to look like a red giant…eventually those solar collecters are going to be glowing red hot.
By Fritz Lieber, by the way.
Only on the Dope would people know this story and get the Cole Porter reference in a science thread.
The Ringworld is unstable, but a complete symmetric sphere is merely of neutral stability, like a ball bearing sitting on a flat surface. That is to say, if something starts it moving, it’ll keep on moving, but it won’t accelerate. So you’d still need to do some dynamical corrections, but you could take your time about them.
Well, I was using conventional materials; assuming just grinding up asteroids and vacuum cementing them into a primative shield. However, you’re going to have a problem with your “ultrawispy solar collectors” or lightweight mirrors in that light pressure will continue to push them outward, and as you note, you’ll still end up radiating at some point. Perhaps you could absorb solar energy and reradiate it as a laser in some some direction, but doing so would still produce waste heat that would have to be radiated away. Thermodynamics is such a bitch.
I was assuming that you’d spin the sphere for centrifugal force along some equator. Otherwise, you’ll have to use some kind of gravity generators, or fill the entire volume with air at 1 atm, both of which pose major technical/logistical problems. Anyway, I’m pretty confident that nobody is really building Type II Dyson spheres, and find it pretty unlikely anybody is concealing stars (at least, via a material barrier) regardless of how advanced they are.
Now, a giant spherical graviational field which created an artificial event horizon (except for a small exit portal, of course)…that would be a terrific way to conceal a star. Of course, it would also have to be of extraordinary size and energy. It’s probably easier just to paint it pink and erect a Somebody Else’s Problem field around it.
Stranger
To what end, though? Assuming we knew what the problem was and that it was not going to be resolved there’d be no point.
The only thing left to do would be to chalk up a decent (or not) run for the human race, call your family and friends to say goodbye, tell them you loved them and then eat the business end of your 9MM. 
I know I ain’t sticking around to slowly freeze to death. 
You should look up because you will see an amazing amount of beautiful stars.
Yeah, if this happened, the extinction of the human race would be invitable, but then, it’s already inevitable. As Terry Pratchett puts it:
Postponing the inevitable is what life’s all about.