 Solution

At $\alpha =1/2$, setting ${\mathcal{M}}_0=1$ gives $x=1$ in the mass relation, and both sides of the boxed relation in part b equal $\nu$. Moreover,so the stellar surface itself is the sonic point. This conclusion is independent of $\gamma$ throughout $1<\gamma <5/3$.

There is no other value of $\alpha$ giving a smooth solution with ${\mathcal{M}}_0=1$. Indeed, eliminating $\alpha$ in favour of $x$ reduces the Bernoulli relation toThe left side has its unique minimum at $x=1$, where it equals the right side. Thus $x=1$ and $\alpha =1/2$ are forced.