harry potter - How could Professor Dumbledore break out of Azkaban

There is no direct canon data about how Dumbledore intended to do it but there are some explanations how one can do it.

As a start - one can shield him/herself from the influence of the Dementors without a Patronus. There are several examples in the book about that:

• Sirius Black preserved his sanity just because he knew he was innocent. He learned that Wormtail is in Hogwarts and his lust for revenge + the willingness to protect Harry gave him something that that the Dementors could not take. That way he managed to have enough power to transfigure himself into his animal form and flee.

Here the quote from The Prisoner of Azkaban:

"I don’t know how I did it,” he said slowly. “I think the only reason I never lost my
mind is that I knew I was innocent. That wasn’t a happy thought, so the Dementors
couldn’t suck it out of me . . . but it kept me sane and knowing who I am . . .
helped me keep my powers . . . so when it all became . . . too much . . . I could
transform in my cell . . . become a dog"

• Bellatrix Lestrange was largely unaffected by the Dementors during her stay in Azkaban. This was most probably due to her fanatic devotion to Voldemort which is again not a happy feeling and the Dementors cannot "take" it. As she is sentenced to life in Azkaban she explicitly states that she'll wait for Voldemort to return i.e. she's not afraid for her sanity... that's if we call her "sane" :)

Here are her words to the court (from The Goblet of Fire):

the woman with the heavy-lidded eyes looked up at Crouch and called, “The Dark Lord
will rise again, Crouch! Throw us into Azkaban; we will wait! He will rise again
and will come for us,

These two imply that a strong feeling that is not "happy" per se (like fanaticism, hatred or revenge) can shield one's mind from the Dementors so that he/she can perform some simple spells.
I.e. most probably a disciplined mind can protect him/herself by inducing or imitating such feelings.

We should not forget that Dumbledore is the "most powerful wizard of his time" and also a very intelligent man. He can most probably "direct" his mind so that Dementors are kept at bay and do some simple wandless spells that will allow him to flee.

As a secondary option - he also has the Order of the Phoenix to assist him although I would consider this as a last resort since it will largely discredit them.

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