What publicly happened to Anakin Skywalker?

So, we, the audience, Palpatine, the troops he brought to Mustafar, and maybe a dozen other people know Anakin is Darth Vader.

However, what about the general public of the Star Wars universe? What does the average person in the empire know about what happened to Anakin? He was the hero of the Republic who fought in the Clone Wars against the droid armies of the CIS, and then... what happened to him?

Did he just "die" during Order 66?