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Potential spoiler for those who haven't played the game yet, but when you go to Ilum to create your new lightsaber, you learn the Empire comes to Ilum to harvest kyber crystals and are mining the giant strip seen in the top photo! And the crystals, like in the Clone Wars show, amplify sunlight and make ice melt (similar to Starkiller Base using the sun to destroy whole planets). If the Emperor is truly behind all the makings of the First Order, it's very plausible that the FO/Empire continued working on Starkiller Base after the second death star explosion, or the FO just took over free real estate and put a canon in the middle of the planet.