A very large plenum contains a gas at constant stagnation pressure and temperature, $P_t,T_1.$ The flow exhausts to a large tank maintained at vacuum $P_a=0.$ The upstream nozzle area ratio is $\frac{A_2}{A_1}=6,$ and the ratio of exit area to throat area is $\frac{A_e}{A_1}=2.$ The test section has a constant area $A_3=A_2.$ A shock wave is stabilized in the diverging portion of the nozzle. The wall friction coefficient is negligible.

a$)$ Determine $\frac{P_{ic}}{P}{e}_e.$

b$)$ Determine the shock Mach number $M_2.$

Hint $1$: The ambient pressure is zero.

Hint $2$: $A^{**}{P}_0$ is constant across the shock.