The physics of Star Wars' Starkiller Base: could that really happen?

Note: Major Star Wars: The Force Awakens spoilers ahead. In Star Wars: The Force Awakens, the next generation of the Empire -- known as the First Order -- has developed a giant weapon called Starkiller. It's like the Death Star's primary weapon, but on steroids -- this is an actual war with stars. Starkiller is a snowy, forested planet that has been hollowed out; now it's a military base. Its name reflects its function: Starkiller sucks in the materials of nearby stars and stores them in the pla...