= Solution
Let $v=-u_r>0$, $c=v_s$ and $c_0=v_{s0}$. Steady smooth adiabatic flow has $p=K\rho^\gamma$ with the <entropy> parameter fixed by infinity, and
$$
\dot M=4\pi r^2\rho v,\qquad
\frac{v^2}{2}+\frac{c^2}{\gamma-1}-\frac{GM}{r}=
\frac{c_0^2}{\gamma-1},\qquad c^2=\gamma K\rho^{\gamma-1}.
$$
The second expression follows by integrating the radial momentum equation with $dp/\rho=d(c^2/(\gamma-1))$ and imposing the outer rest condition. Differentiate continuity to give $\rho'/\rho=-2/r-v'/v$, and substitute into momentum. This yields the <Bondi accretion> wind equation
$$
\left(v-\frac{c^2}{v}\right)v'=\frac{2c^2}{r}-\frac{GM}{r^2}.
$$
At a smooth positive-radius <sonic point>, $v=c=c_s$ and both sides' vanishing factors require $GM/r_s=2c_s^2$. Evaluate the Bernoulli relation there:
$$
\frac{5-3\gamma}{2(\gamma-1)}c_s^2=\frac{c_0^2}{\gamma-1},
\qquad
c_s^2=\frac{2c_0^2}{5-3\gamma},\quad
r_s=\frac{5-3\gamma}{4}\frac{GM}{c_0^2}.
$$
Both the squared <sound speed> and sonic radius must be positive and finite. Hence
$$
\boxed{1<\gamma<5/3.}
$$
For $\gamma=5/3$ no finite-radius regular sonic crossing is possible; the critical solution discussed next approaches sonic speed only at the central limit.
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