= Solution
{c}
Take $u(r)>0$ to be the inward radial speed and select the regular critical solution for a gas reservoir of density $\rho_\infty$ and <adiabatic sound speed> $c_\infty$. Other mathematically subsonic steady solutions need a different inner boundary condition. For the <polytropic equation of state>, let $\gamma=1+1/n$, so
$$
c_s^2=\frac{dp}{d\rho}=\frac{n+1}{n}K\rho^{1/n},
\qquad
h=\int\frac{dp}{\rho}=(n+1)K\rho^{1/n}=nc_s^2.
$$
<Mass conservation> and the <Bernoulli equation> give
$$
\dot M=4\pi r^2\rho u,\qquad
\frac{u^2}{2}+nc_s^2-\frac{GM_{\rm BH}}r=nc_\infty^2.
$$
The gas is at rest at infinity, so its kinetic and gravitational terms vanish there.
Differentiate these two conserved quantities, using $d\log\rho/dr=-2/r-u'/u$. Eliminating the density derivative yields the polytropic <Bondi accretion> equation
$$
\left(u-\frac{c_s^2}{u}\right)\frac{du}{dr}
=\frac{2c_s^2}r-\frac{GM_{\rm BH}}{r^2}.
$$
At a regular <sonic point>, both sides vanish:
$$
u_c=c_c,\qquad r_c=\frac{GM_{\rm BH}}{2c_c^2}.
$$
The <Bernoulli equation> at that point reduces to $nc_\infty^2=(n-3/2)c_c^2$. For $n>3/2$, equivalently $1<\gamma<5/3$,
$$
\boxed{c_c^2=\frac{2n}{2n-3}c_\infty^2
=\frac{2c_\infty^2}{5-3\gamma},\qquad
r_c=\frac{2n-3}{4n}\frac{GM_{\rm BH}}{c_\infty^2}.}
$$
Since $c_s^2\propto\rho^{1/n}$, the critical density is $\rho_c=\rho_\infty[2n/(2n-3)]^n$. Evaluate the conserved <mass accretion rate> at the <sonic point>:
$$
\boxed{\dot M_B=\frac{\pi G^2M_{\rm BH}^2\rho_\infty}{c_\infty^3}
\left(\frac{2n}{2n-3}\right)^{n-3/2}
=4\pi\lambda(\gamma)\frac{G^2M_{\rm BH}^2\rho_\infty}{c_\infty^3},}
$$
where
$$
\lambda(\gamma)=\frac14
\left(\frac{2}{5-3\gamma}\right)^{(5-3\gamma)/[2(\gamma-1)]}.
$$
The isothermal limit $\gamma\to1$ gives $\lambda=e^{3/2}/4$, agreeing with <Isothermal Bondi accretion>.
For $\gamma=5/3$, $n=3/2$ and the Newtonian critical radius tends to zero while $c_c$ diverges. The limiting critical solution approaches <Mach number> one only as $r\to0$; it has no finite-radius Newtonian sonic transition. This is a failure of the Newtonian inner approximation, not an obstruction to accretion onto a <black hole>. A physical <Schwarzschild black hole> has a finite <event horizon>, and the regular relativistic inflow becomes supersonic outside it.
The singular-looking factor in the rate has a finite limit. Writing $\epsilon=n-3/2$, its logarithm is $\epsilon\log[n/\epsilon]\to0$, so the <critical Bondi accretion rate for gamma equals five thirds> is
$$
\boxed{\dot M_{B,5/3}=\frac{\pi G^2M_{\rm BH}^2\rho_\infty}{c_\infty^3}
\quad(\lambda=1/4).}
$$
This is also the leading cold-reservoir limit of <relativistic spherical accretion>; an exact rate for relativistically hot gas requires the relativistic equations, not this Newtonian expression.
For completeness, the finite relativistic <sonic point> can be seen without assigning it an arbitrary inner radius. Let $b=c_s^2/c^2$ denote the relativistic squared sound speed and $g=\gamma-1$. For an isentropic relativistic gas with $p=K\rho_{\rm rest}^{\gamma}$, its dimensionless <specific enthalpy> is $h=g/(g-b)$. Conservation of rest-mass flux and relativistic Bernoulli energy gives the critical relations
$$
r_c=\frac{GM_{\rm BH}}{c^2}\frac{1+3b_c}{2b_c},
\qquad
(g-b_c)\sqrt{1+3b_c}=g-b_\infty.
$$
For $g=2/3$ and $c_\infty\ll c$, expansion gives $b_\infty\simeq(9/4)b_c^2$, hence
$$
\boxed{r_c\simeq\frac{3GM_{\rm BH}}{4cc_\infty}>\frac{2GM_{\rm BH}}{c^2}.}
$$
Here the asymptotic relativistic and Newtonian sound speeds agree to leading order. This finite-radius transition explains why the gas can reach the horizon despite the Newtonian $r_c=0$ result.
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