As $r\to0$, the right-hand side of the Mach-number equation tends to $GM$. Since $f(\mathcal M)\geq4$, a real positive solution at arbitrarily small $r$ requires
$$
4A\dot M^{1/2}\leq GM.
$$
Using the value of $A$ gives
$$
\dot M\leq\frac{(GM)^2}{16A^2}
=\boxed{\frac{\pi\rho_0(GM)^2}{v_{s0}^3}}
=\dot M_{\rm crit}.
$$
This is the <critical Bondi accretion rate for gamma equals five thirds>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 1 / d / Solution

For an adiabatic gas with $\gamma=5/3$, asymptotic density $\rho_0$, and sound speed $v_{s0}$ accreting onto a Newtonian point mass $M$, the largest globally subsonic steady rate is
$$
\dot M_{\rm crit}=\frac{\pi\rho_0(GM)^2}{v_{s0}^3}.
$$


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