3rd order PFD+charge pump PLL pull-in time

I've roughly calculated the pull-in time for PFD+ChargePump PLL based upon Rolland Best's book.

If I did it right, Tp=Δωin /πωn2.

Is it correct? It is only related to ωn?

Pull-in time is a non-linear effect. Good luck trying to find a closed form solution.