Mythological Plotholes: Ganesh's Elephant Head

So, like, Shiva cuts off Ganesh's head because Ganesh wouldn't let Shiva into the house while Parvati was bathing, right? (kind of a "well, that escalated quickly" moment if you ask me)

Then Parvati's all like "You did WHAT?!?!" and is understandably upset that their son's head got cut off.

Trying to get back into his wife's good graces, Shiva gets his son brought back to life.

So far so good, yes?

But here's the part I don't get:

Why is it necessary to cut off an elephant's head to replace Ganesh's decapitated head?

Why not just use his original head?

Religious affiliation of Canadian residents of British Columbia in 2011

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