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human
[Star Wars] Did Palpatine authorize Tarkin to blow up whatever planet he wanted? Did Palpatine take any precautions to ensure the Death Star would never be used back against him?

It didn't seem like Tarkin asked Palpatine for permission before he destroyed Alderaan. 

It also seems like a bad idea to hand over a planet destroying machine to your apprentice when Sith apprentices have a history of kiling their masters. Did Palpatine just trust that Vader and Tarkin would be loyal? Were there other precautions in place?
gpt
Tarkin had been a strong ally/tool of Palpatine for decades before *A New Hope*. Palpatine was the one who persuaded him to leave the Outland Regions Security Force and run for Governor of Eriadu. He used his influence to ensure Tarkin's election, and Tarkin was one of his people from that day on. 

Even before the Clone Wars, Tarkin helped Palpatine with parts of his master plan, although he didn't understand the true scope of it until much later. He was partly responsible for the assassination of the Trade Federation Directorate (the Board of Directors that they had before the Clone Wars) at a trade summit on Eriadu, which was the incident that enabled the Trade Federation to persuade the Senate to allow them to have a droid army for their own protection. 

By the time of *A New Hope*, Tarkin was a Grand Moff, one of the highest possible ranks in the Empire. He had proven his staunch loyalty to Palpatine and was granted unprecedented authority as a result. Palpatine was able to trust him because Tarkin was a middle manager to the core, someone who psychologically needed to have someone below him and someone above him. He would never rebel against Palpatine, and even if he did, he did not have the power of the Dark Side with him and thus would not succeed. 

As far as Vader having control of the Death Star goes, Vader never believed in the utility of the Death Star to begin with. He told Tarkin outright at the officers meeting: "Don't be so proud of this technological terror you've constructed. The ability to destroy a planet is insignificant next to the power of the Force." 

He wouldn't hijack it and threaten to blow up Coruscant if the Emperor didn't step down. It would be pointless to try such a thing.