Sonic density of a polytropic nozzle (${\rho }_{\ast }=[\dot{M}/(A_{\ast }\sqrt{\gamma K}){]}^{2/(\gamma +1)}$)

For steady polytropic flow with $p=K{\rho }^{\gamma }$, $K>0$, $\gamma >1$, a sonic section has $u_{\ast }=c_{\ast }=(\gamma K{\rho }_{\ast }^{\gamma -1}{)}^{1/2}$. Substitution in mass conservation $\dot{M}=A_{\ast }{\rho }_{\ast }{u}_{\ast }$ gives the displayed density. These conditions fix the sonic state for a prescribed flux and throat area, subject to compatible reservoir and downstream boundary conditions.