system
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human
[Star Wars] How did Obi Wan and Luke escape from Vader by hiding out in his home zip code?

When Vader fell to the Dark Side, it was imperative that he never find his children. Leia makes sense; she was hidden under a false name, with a powerful family, on a planet unfamiliar to Vader, and left unaware of her own ancestry.

Luke on the other hand...

Out of the huge vastness of the galaxy, Obi Wan hid Vader's son with Vader's only surviving relatives, under his own original surname, in the same geographic area of the same planet where Vader himself grew up. Then, Obi Wan "hid" in roughly the same area, not even bothering to change his own last name, and doing such a bad job at keeping a low profile that the local farmers know him as the spooky old sorcerer Kenobi.

How did they manage to not get found right away, if not by Vader himself, then by the agents of the Emperor? Surely the ever-scheming Emperor should have thought to keep an eye on Vader's surviving family.
gpt
It's the classic "last place he'd look," move.  There are three planets in the Galaxy that Vader wants absolutely no part of anymore, due to their connections to his history.  Naboo, naturally, which will always remind him of his "ooops killed ya" wife.  The second is Mustafar for fairly obvious reasons.  The last is, of course, Tatooine.  He loathes that place at every conceivable level - so much so he doesn't even accompany the troops pursuing the stolen Death Star plans.  Obi-Wan was very aware of Vader's feelings towards the place, and knew his hatred and shame (over failing his mother) would cloud his vision where Tatooine was concerned.  As far as the Empire finding out - The Hutts ran the planet far more than the Empire ever did, and the troops kept fairly close to "town" except when they explicitly had to.  Even with Ben coming into town for supplies and a drink every now and then, he's only a few easy mind tricks away from total anonymity.  Honestly, it's perfect.