"They have no idea how much energy it takes to escape the sun's orbit. We use hundred of tons of fuel to orbit one ton communication satellites."
Not sure about this. It's all about constant acceleration. You'll need several years, and this is explained in the movie. We indeed use tons of fuel for satellites, but mainly for getting them to orbit (i.e. to escape Earth's gravity). It's different in the movie, it's all done on the ground, depicted at a larger scale by orders of magnitude.
"It would be far simpler to move underground where it's warmer and use nuclear for energy."
The problem was the sun, which was going to engulf the Earth. The only alternative was to leave the planet (which they did, as plan B).
If they were light out, it's because they were near the engines, which were facing the sun all the time. It's also explained the Earth's rotation was fully stopped. What doesn't make sense to me is that they use this to explain the freezing temperatures. It should be the opposite, if you are facing the sun all the time, temperature will rise (unless you get too far from the sun).
While this movie might seem illogical/stupid, it is no more than, I'd say the vast majority of scifi movies from Hollywood.
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