 Solution

The intended quantity is $\boxed{N_{\mathrm{th}}^{\ast }}$, the threshold population inversion density of the cavity after switching to low loss. At this value stimulated gain exactly balances the photon-loss rate:Above threshold a nonzero seed grows; below threshold it decays. The unstarred $N_{\mathrm{th}}$ in the subpart is shorthand for the starred threshold quantity in the displayed equations, rather than a different total atomic population. The photon lifetime ${\tau }_c$ fixes the photon-density loss rate $n/{\tau }_c$.