In some theoretical models of pulsars, which are rotating neutron stars, the braking torque slowing the pulsar's spin rate is proportional to the \(n^{\text {th }}\) power of the pulsar's angular velocity \(\Omega\); that is, \(\dot{\Omega}=-K \Omega^{n}\), where \(K\) is a constant.

(a) Find a formula for the time rate of change of the pulsar period \(\dot{P}\) in terms of \(P\) itself and the constants \(n\) and \(K\).

(b) For the Vela and Crab pulsars, at least, the product \(P \dot{P}=\) constant. What is their braking index \(n\) ?