= 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,
$$
\frac{r_s}{R_*}=\frac1{2\alpha x}=1,
$$
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 to
$$
\nu x+2x^{-\nu/2}=\nu+2.
$$
The left side has its unique minimum at $x=1$, where it equals the right side. Thus $x=1$ and $\alpha=1/2$ are forced.
Back to article page
