Lonely ‘homeless’ planet found for the first time

Basements don't help. Too close to the surface. You get some insulation from the earth, yes, but it's not much. The gradient is something like 25°C per km of depth/1°F per 70 feet of depth. They live at liquid helium temperatures, so you'll have to get pretty deep to be at a stable temperature.

That's to get to the stable point. It takes a while for the earth to cool down. Assuming 100m depth, ... I tried, but I couldn't figure out the physics. The frost depth in some parts of the world is over 2 meters, so a basement wouldn't be all that useful. Compare the temperature of -50C for winter in Siberia to -270C for the many years of the story. You'll also have liquid air leaking through any cracks and pooling in the bottom - the parts at least which don't drain to cover the frozen sea.

But how deep do you need to go to be ahead of the chill? I found a table in the "permafrost" page on Wikipedia, which linked to http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=Get... . That looks like it has enough details to figure out how long it would take for a surface temperature near absolute zero to penetrate 100m into the ground, causing an initial subsurface temperature of 12C to drop to 0C. Can you figure it out?

In any case, perhaps others might have survived in deep caves, using geothermal heat. It's doesn't mean that the people in this story couldn't have survived using some other means.

The premise to the story is impossible, so it's hard to criticize the author for thinking that the entire atmosphere would freeze.

http://journals.ametsoc.org/doi/pdf/10.1175/1520-0442(1997)0... says that the the Antarctic Circle gets about 120W/sq. meter averaged over the entire year. That's about what I expected since sin(23.5 degrees)snow albedo of 0.63 months/12 months * 1000 W / sq. meter solar power -> 60 W/sq. meter. While not a lot, it's still appreciable.

Assuming everything is radiated out to space, Stefan-Boltzman says that (5.670373e-08)T*4 = 120 W/sq m. => T = 214K, which means Antarctica would have a overall average temperature of about -60C, if only due to solar heating.

Interestingly, "Other scientists have estimated that temperatures on any possible permanently lit spot [on the Moon] would be comparatively balmy, though still a frigid minus 58 Fahrenheit (-50 Celsius), give or take a little." This is in concurrence with my calculation.

Which means that solar heat alone is enough to explain why the temperature does not get to the -180C needed to liquify oxygen.